3.1. Characteristics of zigzag trajectory

Figure 2. Trajectories of falling spheres with different filament lengths $l/D$ : (a) 0 (plain sphere), (b) 1.0, and (c) 3.0. Here, $\rho _{s}/\rho _{f}= 1.06$ and $I^* = 0.967$ , as for all figures and the table in §§ 3.1–3.3. The upper and lower panels are top and side views, respectively, and the $z$ -axis aligns with the gravitational direction. See supplementary movie 1.

When a filament is attached to the upper side of a sphere falling in a quiescent fluid, the moment of inertia of the model increases, and its centre of mass shifts upwards from the geometric centre of the sphere alone. These altered properties of the sphere induce different hydrodynamic interactions between the sphere and fluid compared with those of a plain sphere with no filament, resulting in distinct falling motions. In this subsection, we analyse the effects of the rear-side filament on the translational dynamics of the falling sphere. The filament length $l$ , which is made dimensionless using the sphere diameter $D$ , is the major parameter in this study, and the sphere exhibits distinct trajectories depending on $l/D$ (see figure 2 and supplementary movie 1). To focus on the effects of $l/D$ , the density ratio of the sphere to the fluid is fixed at $\rho _{s}/\rho _{f}= 1.06$ for all experimental results in §§ 3.1–3.3, regardless of the filament length; the corresponding dimensionless moment of inertia $I^*$ of the sphere is constant at 0.967.

Figure 3. (a) Description of 2-D trajectory plane (blue) and length parameters to represent zigzag trajectory. The upper inset shows the top view of the trajectory. (b) Degree of zigzag motion $w^*$ , with respect to $l/D$ . The error bar represents the standard deviation from the mean for each measurement point.

The falling mode of a plain sphere is determined by the Reynolds number $Re = \bar {v}_{z}D/\nu$ and the density ratio $\rho _s/\rho _f$ , where $\bar {v}_{z}$ is the average vertical velocity of the sphere (Horowitz & Williamson Reference Horowitz and Williamson2010). For our plain sphere, $Re = 2.71 \times 10^3$ and $\rho _s/\rho _f = 1.06$ . According to the previous study (Horowitz & Williamson Reference Horowitz and Williamson2010), these values of $Re$ and $\rho _s/\rho _f$ correspond to the region where the sphere falls vertically. In figure 2(a), the falling trajectory of the plain sphere ( $l/D = 0$ ) is predominantly rectilinear, with a slight horizontal deviation; the scale of the $z$ -axis differs from that of the other axes in the figure. The maximum horizontal distance between any two points projected onto the $xy$ -plane is 0.77 $D$ , which is sufficiently small compared with the vertical travel distance, thus the motion can be classified as vertical falling. The fluid domain depicted in figure 2 is far from the release point controlled by the syringe, so the trajectory does not include the initial transient phase immediately after release; $z/D = 0$ in the figure is located 900 mm below the syringe tip. For our plain sphere, $Re$ generally belongs to the intermediate regime between laminar and turbulent flows, and the minor horizontal displacement in figure 2(a) might be induced by flow instability around the sphere. However, the amplitude of the horizontal displacement is so small that the trajectory of our plain sphere can be considered to be vertical. Although the $Re$ values of the models in this section range from $2.31 \times 10^3$ to $2.71 \times 10^3$ , the falling sphere exhibits a distinct zigzag motion for an intermediate filament length $l/D\approx1.0$ , instead of the almost rectilinear motion observed for $l/D = 0$ (figure 2 b). This zigzag trajectory is not completely within a 2-D plane. In other words, the sphere with the zigzag motion moves periodically in both the major direction of maximum horizontal displacement and the minor direction normal to the major direction (upper panel of figure 2 b).

To quantify the degree of zigzag motion, a 2-D trajectory plane is first defined, on which the horizontal amplitude of the zigzag motion is maximum along the direction parallel to the plane; see the blue plane in figure 3(a). The 2-D trajectory plane is constructed by fitting all positions of the trajectory onto the plane using the least squares method. During the zigzag motion, the sphere periodically reaches peak positions at which the horizontal movement along the major direction reverses. The horizontal displacement between two successive peak positions along the direction parallel to the trajectory plane is denoted as $w$ (figure 3 a). The horizontal displacement in the minor direction has a similar magnitude, regardless of the filament length. The filament length has a pronounced effect on the horizontal displacement in the major direction. Therefore, only the horizontal displacement along the major direction is considered for the degree of zigzag motion. The trajectory height $h$ is the distance between two successive peak positions along the direction perpendicular to the direction defining $w$ , and parallel to the trajectory plane (figure 3 a), indicating how quickly the direction of the trajectory changes. These two parameters are the basic parameters to quantify the zigzag motion. The average horizontal displacement in the major direction, $w_{avg}$ , is made dimensionless using the average trajectory height $h_{avg}$ : $w^* = w_{avg}/h_{avg}$ . Here, $w_{avg}$ and $h_{avg}$ are the averages of $w$ and $h$ , respectively, calculated over multiple periods within a single experimental trial. Rather than using $w_{avg}$ and $h_{avg}$ separately, we present the zigzag motion using a single dimensionless parameter $w^*$ , which effectively characterises the intensity of zigzag motion by quantifying the sphere’s horizontal displacement relative to the distance over which the sphere completes half a cycle of the zigzag motion.

For small $l/D$ (typically $\leqslant 0.4$ ), $w^*$ is close to zero, which indicates that the sphere exhibits almost no zigzag motion (figure 3 b). However, when $l/D$ exceeds 0.4, $w^*$ increases remarkably until $l/D= 0.8$ with the emergence of a distinct zigzag motion. In figure 3(b), the spread of measured values is represented by an error bar, which indicates the standard deviation above and below the mean; throughout this study, all error bars denote the standard deviation from the mean at each measurement point. The large standard deviation of the experimental results for $l/D= 0.6$ signifies a transition regime from non-zigzag to zigzag motion. In this regime, both the zigzag and non-zigzag motions are observed in the experimental trials. As $l/D$ increases from 0.8, the zigzag motion becomes weaker, with a smaller value of $w^*$ . For the sphere with a long filament ( $l/D\approx3.0$ ), the zigzag path almost disappears, as depicted in figure 2(c). Interestingly, $w_{avg}$ remains nearly constant for $l/D= 0.6$ and 1.2, with $w_{avg}/D= 2.43$ and 2.44, respectively. Despite these similar $w_{avg}/D$ values, $w^*$ for $l/D= 0.6$ is much smaller than that for $l/D= 1.2$ due to the greater average trajectory height $h_{avg}$ , which exemplifies that $w_{avg}/D$ alone does not successfully quantify the degree of zigzag motion, and $w^*$ should be employed as a primary parameter.

Figure 4. (a) Falling time $t/t_0$ , (b) trajectory length $L/L_0$ , and (c) average tangential velocity $\bar {v}_t/\bar {v}_{t,0}$ with respect to $l/D$ .

Because the filament displaces the sphere horizontally, it also affects the vertical falling speed of the sphere. Here, the falling time $t$ required for the sphere to move from $z/D=0$ to $z/D= 26.9$ is made dimensionless using the falling time of the plain sphere, $t_0$ ; the $z$ -axis is positive downwards, and hereafter the subscript 0 denotes the result of the plain sphere ( $l/D = 0$ ). All experimental data obtained in this study are measured between $z/D=0$ and $z/D= 26.9$ (top and bottom of the 3-D trajectory domain, respectively). The value of $t/t_0$ is greater than unity for all spheres with a filament, which means that when a filament of any length is attached, the sphere takes a longer time to fall a certain vertical distance than the plain sphere (figure 4 a). As $l/D$ increases from 0 to 0.8, $t/t_0$ increases, reaching a plateau near $t/t_0= 1.2$ within the range $l/D= 0.8{-}1.4$ . Beyond this range, $t/t_0$ decreases with increasing $l/D$ . The trend of $w^*$ versus $l/D$ in figure 3(b) is similar to that of $t/t_0$ . The falling time associated with a strong zigzag motion tends to be greater than that with a weak zigzag motion.

When the sphere follows the zigzag trajectory, the total distance travelled by the sphere (i.e. trajectory length) increases from the vertical falling distance, which contributes to the prolongation of the falling time. In addition to the extension of the trajectory length, the filament changes the tangential velocity of the sphere along the trajectory, which also affects the falling time. The trajectory length $L$ and average tangential velocity $\bar {v}_t$ along the trajectory between $z/D=0$ and $z/D= 26.9$ are computed to examine their effects on the falling time. The trajectory length divided by that of the plain sphere, $L/L_0$ , exhibits a correlation with $t/t_0$ : $L/L_0$ grows from $l/D= 0$ to $l/D= 0.8$ , and thereafter declines from $l/D= 0.8$ to $l/D= 3.0$ (figure 4 b). That is, the extended trajectory length owing to the filament contributes to the prolonged falling time.

In contrast to the trajectory length, the ratio of the average tangential velocity to that of the plain sphere, $\bar {v}_t/\bar {v}_{t,0}$ , exhibits the inverse trend to that of $t/t_0$ (figure 4 c). The upper bar denotes the time-averaged value of the results obtained during the experimental trial. The minimum value of $\bar {v}_t/\bar {v}_{t,0}$ reaches 0.86 near $l/D= 0.8$ and 1.4, and the ratio approaches unity almost monotonically as $l/D$ decreases below $0.8$ or increases beyond $1.4$ . This result indicates that the attachment of a filament induces the falling dynamics that differ from those expected based on previous investigations on the effects of rear-side appendages. For instance, previous studies focused primarily on the role of appendages on drag reduction for fixed bluff bodies in 2-D configurations (Roshko Reference Roshko1955; Apelt et al. Reference Apelt, West and Szewczyk1973; Kwon & Choi Reference Kwon and Choi1996; Anderson & Szewczyk Reference Anderson and Szewczyk1997; Qiu et al. Reference Qiu, Sun, Wu and Tamura2014). If the same mechanism of these appendages was applicable to our experimental model, then one would expect an increase in tangential velocity because of drag reduction. However, this study reveals an unexpected result. The 2-D appendages in the previous studies split the wake of the fixed bluff body, thereby reducing the drag coefficient. By contrast, our sphere is free to move, and the filament does not split the wake in the same manner due to its small cross-sectional area relative to the sphere’s wake. Thus the mechanism by which the filament-shaped appendage affects the freely falling sphere is fundamentally different from how a 2-D appendage influences a fixed bluff body. Indeed, for all filament lengths, the tangential velocity of the sphere with the filament is lower than that of the plain sphere. Furthermore, the reduction in $\bar {v}_t$ is pronounced when the zigzag pattern becomes dominant with greater $w^*$ (figure 3 b).

The attachment of a filament to the sphere results in an extended trajectory and slower movement, leading to prolonged falling time compared to the plain sphere. While changes in both the trajectory length and tangential velocity affect the falling time, the effect of reducing the tangential velocity is more significant than that of elongating the trajectory length. For example, $\bar {v}_t$ decreases by a maximum of 14 % from $\bar {v}_{t,0}$ , while $L$ increases by a maximum of 5% from $L_0$ , according to figures 4(b) and 4(c). To elaborate the trend of the tangential velocity along the zigzag path, the rotational motion of the sphere and its coupling with the translational motion are discussed in § 3.2.

3.2. Mechanism of sphere rotation

The rotational dynamics of the sphere with a filament is now analysed in terms of the angular velocity of the sphere and the orientation of the filament. To represent the angular velocity as a single parameter, instead of expressing it with three Cartesian components, the angular velocity vector of the sphere, $\boldsymbol {\omega }$ , is obtained from the axis–angle notation (Zimmermann et al. Reference Zimmermann, Gasteuil, Bourgoin, Volk, Pumir and Pinton2011; Mathai et al. Reference Mathai, Neut, van der Poel and Sun2016). During the falling process, the rotation of the sphere exhibiting a zigzag translational motion is mostly confined in two opposing directions that share a single axis, which are depicted as red and blue arrows on the top sphere of figure 5(a). This rotation axis is normal to the 2-D trajectory plane described in § 3.1. The unit normal vector of the trajectory plane, $\boldsymbol {n}$ , is defined as the vector perpendicular to this plane with a positive $x$ -component in the global coordinate system. As a representative result, the inner product of the unit angular velocity vector $\boldsymbol {\omega }/|\boldsymbol {\omega }|$ and the unit normal vector $\boldsymbol {n}$ is presented in figure 5(b) for the case of $l/D = 1.0$ , which undergoes a strong zigzag motion. The value of $\boldsymbol {n}\, {{\boldsymbol \cdot}}\, (\boldsymbol {\omega }/|\boldsymbol {\omega }|)$ is close to either 1 or $-1$ , indicating that the rotation axis of the sphere is almost aligned with the normal vector of the trajectory plane during the fall. In contrast to cases with a distinct zigzag motion, the cases with a subtle zigzag motion have weaker alignment, thus the transition of $\boldsymbol {n}\,{\boldsymbol \cdot}\, (\boldsymbol {\omega }/|\boldsymbol {\omega }|)$ between 1 and $-1$ occurs over a longer time, resulting in a smoother plot.

Figure 5. (a) Sign convention of angular velocity $\omega$ and definition of filament angle $\theta$ . (b) Inner product of unit angular velocity vector $\boldsymbol {\omega }/|\boldsymbol {\omega }|$ and unit normal vector of the trajectory plane, $\boldsymbol {n}$ , for $l/D = 1.0$ .

Considering the sign of the rotation direction, the angular velocity of the sphere, $\omega$ , is defined as

(3.1) \begin{equation} \omega = \text{sgn}(\boldsymbol {\omega }\, {\boldsymbol \cdot}\, \boldsymbol {n})\, |\boldsymbol {\omega }|, \end{equation}

where $\text{sgn}$ is the signum function. To characterise the behaviour of the filament with respect to the sphere motion, the filament angle $\theta$ is introduced as the angle between $-\boldsymbol {v}_t$ (the negative value of the tangential velocity vector along the trajectory) and $\boldsymbol {e}_{fil}$ (the vector from the sphere centre to the filament), as shown in figure 5(a):

(3.2) \begin{equation} \theta = \text{sgn}\left [\left \{\boldsymbol {e}_{fil} \times (-\boldsymbol {v}_{t})\right \} {\boldsymbol \cdot}\; \boldsymbol {n} \right ]|\theta |. \end{equation}

In this study, $\omega$ and $\theta$ represent fundamentally different quantities, and $\omega$ should not be interpreted as the time derivative of $\theta$ . This distinction arises because the filament deviates from the trajectory plane, thus $\boldsymbol {e}_{fil}$ is not parallel to this plane.

Figure 6. Time histories of (a) filament angle $\theta$ and (b) sphere angular velocity $\omega /\overline {|\omega _0|}$ for $l/D = 1.0$ . (c) The 3-D trajectory corresponding to (a) and (b). The insets (c) depict the orientation of the filament at each position represented by the red dots.

The time series of the filament angle $\theta$ and the sphere angular velocity ratio $\omega /\overline {|\omega _0|}$ for $l/D = 1.0$ are presented in figures 6(a) and 6(b), respectively. Here, $\overline {|\omega _0|}$ is the time-averaged magnitude of the plain sphere’s angular velocity between $z/D=0$ and $z/D=26.9$ . For a falling plain sphere of the same density and Reynolds number as our model, two vortex rings appear in the wake during one period of wake formation (Horowitz & Williamson Reference Horowitz and Williamson2010). In the present experiment, the plain sphere exhibits slight rotation due to interaction with the unsteady wake, having non-zero $\overline {|\omega _0|}$ . Despite its small magnitude, $\overline {|\omega _0|}$ is used as the denominator in the angular velocity ratio to express the magnitude of the angular velocity of the sphere with a filament relative to that of the plain sphere. For five representative instants during the fall (red dots in the figures), their corresponding positions are marked along the 3-D falling trajectory in figure 6(c). Positions 2 and 4 are the peak positions at which the horizontal displacement of the sphere attains a local maximum. Positions 1, 3 and 5 are determined as the locations that make the vertical distances between neighbouring positions equal.

Although $\omega$ is not the time derivative of $\theta$ by definition (as mentioned above), their time series in figures 6(a) and 6(b) show some correlation. Specifically, $\omega$ approaches zero and changes its sign when $\theta$ reaches its local maximum or minimum value, and $\theta$ approaches zero and changes its sign when $\omega$ reaches its local extreme value. This is because the rotation axis of the sphere almost aligns with $\boldsymbol {n}$ in this case. As explained in § 3.1, the translational motion of the sphere is dominant in the major direction over the minor direction. The alignment between the rotation axis and the normal vector of the trajectory plane constructed based on the translational motion suggests that the translational motion of the sphere is strongly coupled with its rotational motion. Pronounced alignment is observed in cases with a strong zigzag motion (e.g. $l/D = 1.0$ in figure 6). In contrast, for cases with a weak or negligible zigzag motion, no clear correlation is observed between $\theta$ and $\omega$ .

The relationship between the translational and rotational motions can also be identified from the degree of zigzag motion and the angular velocity of the sphere. To reveal the effects of the filament length on the rotational dynamics, the magnitude of the sphere angular velocity, $\overline {|\omega |}$ averaged between $z/D=0$ and $z/D=26.9$ , with respect to $l/D$ is shown in figure 7(a). All spheres with a filament rotate faster than the plain sphere; $\overline {|\omega |}$ increases from $l/D = 0$ to $l/D = 0.8$ , then decreases as $l/D$ increases beyond 0.8. The trend of $\overline {|\omega |}$ is similar to that of $w^*$ in figure 3(b), indicating that the enhanced rotation of the sphere is correlated with the strong zigzag motion. In contrast, the maximum magnitude of the filament angle, $|\theta |_{max}$ , exhibits a similar trend to $\overline {|\omega |}$ for $l/D \geqslant 0.8$ , but a different trend for $l/D \lt 0.8$ (figure 7 b). For $l/D \geqslant 0.8$ , the sphere with a longer filament rotates with a smaller amplitude of the filament angle. However, when $l/D$ is less than 0.8, $|\theta |_{max}$ exhibits significant scatter, despite having large values. A large standard deviation is also observed for $\overline {|\omega |}$ in the range $l/D = 0.2$ –0.6 (figure 7 a). In this range, the filament is not sufficiently long to affect the rotational behaviour of the sphere in a deterministic manner, but flow instability induced by the short filament may be critical.

Figure 7. (a) Average magnitude of sphere angular velocity $\overline {|\omega |}/\overline {|\omega _0|}$ . (b) Maximum magnitude of filament angle $|\theta |_{max}$ for different filament lengths.

Figure 8. (a) Schematic illustrating a wake region (grey) and the components of forces and torques acting on the model. The reference frame is fixed with the sphere. Time histories of sphere angular velocity $\omega$ (solid line) and lift force $F_L$ (dashed line) for (b) $l/D= 1.0$ and (c) $l/D = 2.0$ .

During the fall, a wake region is generated behind the sphere, and the force exerted on the filament is affected by this wake. Lācis et al. (Reference Lācis, Brosse, Ingremeau, Mazzino, Lundell, Kellay and Bagheri2014) assumed that the wake of a bluff body was a simple elliptical region for a fixed but rotatable 2-D cylinder with a rear-side filament in a uniform flow, and showed that the estimated fluid force acting on the filament was in good agreement with experimental results. Lācis et al. (Reference Lācis, Brosse, Ingremeau, Mazzino, Lundell, Kellay and Bagheri2014, Reference Lācis, Olivieri, Mazzino and Bagheri2017) applied this assumption to a falling sphere with an elliptic plate. Similarly, we introduce a simplified wake with an elliptic shape (grey region in figure 8 a) to account for the rotation mechanism of a sphere with a filament in terms of the centre of mass and the relative position of the filament to the wake. In addition to simplifying the wake shape, we also choose the size of the wake region arbitrarily to illustrate the relative position of the filament to the wake as the sphere rotates. The attachment of the filament means that the centre of mass of the entire model ( $\otimes$ symbol in figure 8 a) is above the geometric centre of the sphere; the position of the centre of mass is exaggerated in the figure for clarity. To quantify the shift in the centre of mass by the attachment of the filament, the offset distance $\gamma$ is defined as the distance between the geometric centre of the sphere and the centre of mass of the entire model. Although the value of $\gamma$ is 0.2 % of $D$ for $l/D = 1.0$ (the case exhibiting a strong zigzag motion), it is sufficient to affect the rotational behaviour of the sphere (Will & Krug Reference Will and Krug2021b ).

With the rotation of the sphere, a lift force $F_L$ is generated in the direction normal to the tangential velocity $v_t$ (figure 8 a); in the figure, the reference frame is fixed with the sphere, and $v_t$ is directed towards the sphere. The hydrodynamic force exerted on the filament is negligible compared with that on the sphere because the filament has significantly smaller surface area and volume. Therefore, only the force acting on the sphere is considered in this study. In analysing the force balance of the sphere perpendicular to its tangential velocity vector $\boldsymbol {v}_t$ , we define the lift force vector $\boldsymbol {F}_L$ . This vector encompasses all fluid-dynamic forces exerted on the sphere normal to $\boldsymbol {v}_t$ , including unsteady effects but excluding the hydrostatic (buoyancy) force. It can be acquired from the experimental data as follows:

(3.3) \begin{equation} \boldsymbol {F}_{L} = \left (\frac {1}{6}\rho _{s}\unicode{x03C0} D^3\frac {\mathrm {d}\boldsymbol {v}_{t}}{\mathrm {d}t} - \boldsymbol {F}_{g,net}\right )_{n} = \frac {1}{6}\rho _{s}\unicode{x03C0} D^3\left [\frac {\mathrm {d}\boldsymbol {v}_{t}}{\mathrm {d}t} + \left (\frac {\rho _f}{\rho _s}-1\right )\boldsymbol {g}\right ]_{n}. \end{equation}

The subscript $n$ denotes the component in the direction normal to $\boldsymbol {v}_t$ , and $\boldsymbol {F}_{g,net}$ is the net gravitational force. In this study, $F_L$ is determined by

(3.4) \begin{equation} F_{L} = \text{sgn}\left [\left \{\boldsymbol {F}_{L} \times (-\boldsymbol {v}_{t})\right \} {\boldsymbol \cdot}\; \boldsymbol {n}\right ]\, |\boldsymbol {F}_{L}|. \end{equation}

In (3.3) and (3.4), $\boldsymbol {v}_{t}$ is the tangential velocity of the sphere in the global reference frame.

Figures 8(b) and 8(c) present the angular velocity ratio $\omega /\overline {|\omega _0|}$ and lift force ratio $F_L/\overline {|F_{L,0}|}$ for $l/D= 1.0$ and 2.0, respectively. As shown in figure 2(a), the plain sphere exhibits slight horizontal displacements, and the time-averaged magnitude of the plain sphere’s lift force, $\overline {|F_{L,0}|}$ , is non-zero, albeit small. The angular velocity $\omega$ exhibits a similar temporal profile and period to the lift force $F_L$ with a slight phase delay, implying that the rotation of the sphere is responsible for generating the lift force. This result is in agreement with the finding of Will & Krug (Reference Will and Krug2021b ), who reported that the angular velocity and the lift force are coupled for a plain falling sphere with significant angular velocity because of the shifted centre of mass. As the zigzag motion of the sphere becomes weaker (e.g. $l/D = 2.0$ ), the magnitudes of $\omega$ and $F_L$ decrease. Although not presented here, the correlation between $\omega$ and $F_L$ is almost indistinguishable for $l/D \leqslant 0.4$ and $l/D\approx3.0$ , where the degree of zigzag motion is very small (figure 3 b).

Figure 8(a) describes the fluid-dynamic forces and torques with respect to the centre of mass, allowing us to explain the rotational mechanism of the model in more detail. The drag force $F_D$ acting on the sphere is parallel to the tangential velocity $v_t$ and generates torque $T_D$ in the direction that causes the filament to move away from the centreline, which is defined as the line parallel to the tangential velocity and passing through the sphere centre (vertical dashed line in figure 8 a). By contrast, the flow outside the wake region generates a fluid force on the filament, resulting in torque $T_{ext}$ in the direction towards the centreline. The portion of the filament outside the wake is illustrated as the blue line in figure 8(a). The magnitude of the flow velocity (in the reference frame fixed with the sphere) outside the wake is much greater than that inside the wake, and the area outside the wake is located farther away from the centre of mass than the area inside the wake. Therefore, we focus on the torque generated by the flow outside the wake, which greatly exceeds the torque produced by the flow inside the wake.

As a sphere rotates in a certain direction, a lift force $F_L$ normal to the tangential velocity is generated on the sphere. According to Will & Krug (Reference Will and Krug2021b ), in the case of a falling plain sphere where the centre of mass is positioned below its geometric centre, the torque induced by the lift force opposes the rotating direction of the sphere and suppresses its rotation. Consequently, no zigzag falling trajectory can be identified. By contrast, for a rising plain sphere with the centre of mass below the geometric centre, the external flow in the reference frame fixed with the sphere acts in the opposite direction, so the lift-induced torque is generated in the rotating direction, enhancing the sphere’s rotation, which eventually leads to a zigzag trajectory. That is, the centre of mass and the direction of external flow are important in determining the rotational dynamics and resulting trajectory of the plain sphere. For our model, where the centre of mass is positioned above the geometric centre of the sphere because of the filament, a lift-induced torque $T_L$ is generated in the same direction as the rotation of the sphere, further strengthening its rotation (figure 8 a).

Figure 9. (a) Sequential snapshots of wake visualisation for $l/D = 1.0$ with strong zigzag motion. (b) Corresponding simplified schematics illustrating the rotation process. For (a), see supplementary movie 2.

The sphere with a strong zigzag motion undergoes periodic rotational oscillations, as demonstrated in figure 6 and supplementary movie 1(b). During the fall, the relative position of the filament to the wake region is shifted continuously by the rotation of the sphere. The positions of the filament with respect to the wake region were determined using dye visualisation experiments with fluorescein disodium salt. The visualised wake and filament were filmed by a high-speed camera (FASTCAM MINI-UX50, Photron, Inc.). The experimental trials were repeated to make the direction of maximum horizontal displacement nearly parallel to the field of view of the camera. Figure 9(a) and supplementary movie 2 present visualised wakes in sequence for $l/D = 1.0$ , which produces a strong zigzag motion. In addition, the simplified schematics of the rotation process in the reference frame fixed with the sphere are illustrated in figure 9(b). The schematics are represented in two dimensions because the sphere rotation is dominant with respect to the axis normal to the trajectory plane for $l/D = 1.0$ , and the incoming flow velocity $v_t$ is expressed consistently in the upward direction for ease of comparison between the schematics.

A large portion of the filament is inside the wake when the filament is near the middle of the wake, i.e. when $|\theta |$ is small (figure 9 ai,bi). In this phase, the magnitudes of the angular velocity $\omega$ and the lift force $F_L$ are significant according to figures 8(b) and 8(c). The lift-induced torque $T_L$ acts in the same direction as the drag-induced torque $T_D$ , and the counter-torque produced by the external flow, $T_{ext}$ , is small. As the sphere rotates with increasing $\theta$ , the area of the filament exposed to the flow outside the wake becomes greater, leading to an increase in $T_{ext}$ . Eventually, $\theta$ reaches its maximum magnitude while $\omega$ and $F_L$ almost vanish (figure 9 aii,bii). Thereafter, the sphere begins to rotate in the opposite direction because $T_{ext}$ is still large, generating $F_L$ in the opposite direction from the phase in figure 9(ai), and $T_L$ aligns with the new rotation direction (figure 9 aiii,biii); $T_D$ does not change its direction before the filament returns to the centreline. As the filament moves to the other half of the wake region, reversing the sign of $\theta$ , the directions of the lift force and torques are symmetric to those in the preceding half of the rotation process, and the sphere exhibits symmetric rotational behaviour (figure 9 aiv–avi,biv–bvi).

Figure 10. (a) Snapshots of wake visualisation for a long filament ( $l/D = 2.0$ ). (b) Corresponding simplified schematics of the rotation process.

As the filament extends from $l/D = 1.0$ , the rotational motion of the sphere weakens as the average sphere angular velocity and maximum filament angle decrease (figure 7). Even when the filament is positioned near the middle of the wake, the area of the filament outside the wake becomes significant (figure 10). Consequently, the greater $T_{ext}$ throughout the rotation process yields smaller values of $\overline {|\omega |}$ and $|\theta |_{max}$ . Moreover, when $l/D$ increases to $3.0$ , the moment of inertia of the model increases by 28.8 % from that of $l/D = 1.0$ , which plays a role in reducing the angular velocity. The less-pronounced rotation reduces the lift force acting on the sphere, leading to a weak zigzag motion.

Figure 11. Simplified schematics of the rotation process for a short filament.

Regarding shorter filaments ( $l/D \lt 0.8$ ), with which the degree of zigzag motion is very small (figure 3 b), the filament does not cross to the other half of the wake region, in contrast to cases where the model exhibits symmetric rotational behaviour with a strong zigzag motion (e.g. $l/D = 1.0$ ). Instead, the filament maintains a large $\theta$ within the original half of the wake during the rotation process (figure 11). This asymmetric rotational behaviour results in a smaller $\overline {|\omega |}$ compared with that for $l/D = 1.0$ (figure 7 a). In cases with symmetric rotational behaviour, the angular velocity of the sphere reaches its peak at approximately the time when the filament crosses the centreline ( $\theta=0$ ) by virtue of $T_{ext}$ being exerted in the prior half of the wake (figure 6 a,b). However, for shorter filaments, $T_{ext}$ is minor, and the counter-torque, which is produced by the recirculating wake flow moving towards the rear side of the sphere in the middle of the wake region (grey arrows in figure 11), causes the filament to remain in one half of the wake, having a smaller $\overline {|\omega |}$ . Thus the zigzag motion, which basically requires symmetric rotation crossing the centreline, almost disappears.

Next, we evaluate the drag and lift coefficients of the sphere, and determine their trends in terms of the filament length. In previous studies of a falling or rising plain sphere, the drag coefficient of the sphere is defined based on its vertical velocity: $C_{D,z} = 4D\,|\rho _{s}/\rho _{f}-1|\,g/(3\bar {v}_z^2)$ (Jenny et al. Reference Jenny, Dušek and Bouchet2004; Horowitz & Williamson Reference Horowitz and Williamson2010; Will & Krug Reference Will and Krug2021a , Reference Will and Krugb ). In this study, to include the effects of the tangential velocity directly, the drag and lift coefficients $C_D$ and $C_L$ are calculated using the tangential velocity of the sphere. The drag force vector $\boldsymbol {F}_D$ , which is the total fluid-dynamic force exerted on the sphere in the direction opposite to the tangential velocity vector in the global reference frame, is calculated based on the experimental data. The force balance equation is employed in a manner similar to the lift force $\boldsymbol {F}_L$ in (3.3):

(3.5) \begin{equation} \boldsymbol {F}_{D} = \left (-\frac {1}{6}\rho _{s}\unicode{x03C0} D^3\frac {\mathrm {d}\boldsymbol {v}_t}{\mathrm {d}t} + \boldsymbol {F}_{g,net}\right )_{t} = \frac {1}{6}\rho _{s}\unicode{x03C0} D^3\left [-\frac {\mathrm {d}\boldsymbol {v}_t}{\mathrm {d}t} + \left (1-\frac {\rho _f}{\rho _s}\right )\boldsymbol {g}\right ]_{t}. \end{equation}

The subscript $t$ denotes the component parallel to $\boldsymbol {v}_t$ . Then $C_D$ and $C_L$ are defined in the tangential and normal directions of the trajectory, respectively, and given as

(3.6a) \begin{align} C_D = \frac {2|\boldsymbol {F}_D|}{\rho _{f}v_t^2\unicode{x03C0} (D/2)^2} = \frac {4D\left |\dfrac {\rho _s}{\rho _f}\dfrac {{\rm d}\boldsymbol {v}_t}{{\rm d}t}+\left (1-\dfrac {\rho _s}{\rho _f}\right )\boldsymbol {g}\right |_{t}}{3v_{t}^2}, \end{align}

(3.6b) \begin{align} C_L = \frac {2|\boldsymbol {F}_L|}{\rho _{f}v_t^2\unicode{x03C0} (D/2)^2} = \frac {4D\left |\dfrac {\rho _s}{\rho _f}\dfrac {{\rm d}\boldsymbol {v}_t}{dt}+\left (1-\dfrac {\rho _s}{\rho _f}\right )\boldsymbol {g}\right |_{n}}{3v_{t}^2}. \end{align}

Because $C_D$ and $C_L$ are derived from $\boldsymbol {F}_D$ and $\boldsymbol {F}_L$ , respectively, these coefficients also include the effects of unsteady fluid dynamics, rather than reflecting only quasi-steady forces.

Figure 12. (a) Average drag coefficient $\bar {C}_D$ and (b) average lift coefficient $\bar {C}_L$ with respect to $l/D$ .

When a sphere with a centre fixed in space is placed in a uniform flow, its rotation with respect to an axis transverse to the flow affects the wake structure, resulting in greater drag and lift compared with a fixed sphere without rotation (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009; Kim Reference Kim2009; Poon et al. Reference Poon, Ooi, Giacobello, Iaccarino and Chung2014). The sphere’s rotation deflects the wake in a direction determined by the cross-product of the angular velocity vector and the free-stream vector. As the angular velocity of the sphere increases, the drag and lift coefficients also increase due to changes in the flow structure. Similar trends are observed in our model. In figure 12, both $C_D$ and $C_L$ averaged over the trajectory length between $z/D= 0$ and $z/D=26.9$ show similar trends to the average magnitude of angular velocity $\overline {|\omega |}/\overline {|\omega _0|}$ presented in figure 7(a). Both $\bar {C}_D$ and $\bar {C}_L$ increase in accordance with an increase in the angular velocity. As discussed in § 3.1, the tangential velocity of a falling sphere with a filament is lower than that of a plain sphere, and its trajectory is longer because of the horizontal displacement associated with the zigzag motion. In the presence of the filament, $C_D$ and $C_L$ are enhanced owing to the rotation of the sphere, which eventually leads to the reduced tangential velocity and elongated trajectory, respectively.

3.3. Effects of shifted centre of mass and hydrodynamic interaction of a filament

The attachment of a filament to the sphere enhances rotational dynamics through two mechanisms: shift in the centre of mass for the entire model, and hydrodynamic interaction between the filament and surrounding fluid. In this subsection, we examine quantitatively how each mechanism influences the falling dynamics by employing two additional models (figure 13). In the experimental model discussed in §§ 3.1 and 3.2, the sphere’s centre of mass coincides with its geometric centre when no filament is attached. However, the attachment of a filament shifts the centre of mass of the entire model above the geometric centre of the sphere (figure 13 a). For example, with a filament of length ratio $l/D= 1.0$ , the dimensionless offset distance $\gamma /D$ is 0.2 %.

Figure 13. Schematics of the models with (a) $l/D = 1.0$ and $\gamma /D = 0.2\, \%$ , (b) $l/D = 1.0$ and $\gamma /D = 0$ , and (c) $l/D = 0$ and $\gamma /D = 0.2\, \%$ . The geometric centre of the sphere and the centre of mass of the entire model are denoted by $\bullet$ and $\otimes$ , respectively.

To separate the effects of the shifted centre of mass and the hydrodynamic interaction of the filament, we designed an additional experimental model, in which the centre of mass of the entire model coincides with the geometric centre of the sphere when a filament is attached (figure 13 b). To achieve zero offset distance between the centre of mass and the geometric centre ( $\gamma /D= 0$ ), the centre of mass of a sphere is positioned below the geometric centre by modifying its internal configuration. The offset distance of this sphere was calculated to ensure that the magnitude of this initial downward offset precisely matched the upward shift that would occur upon filament attachment. As a result, when the filament is attached to this sphere, the centre of mass of the entire model coincides with the geometric centre of the sphere.

Experimental results of the models with and without a shifted centre of mass are compared, focusing on a fixed filament length ratio $l/D= 1.0$ , where a strong zigzag motion is observed. For comparison, the results of the model without a shifted centre of mass ( $\gamma /D= 0$ , figure 13 b) are made dimensionless using those of the model with a shifted centre of mass ( $\gamma /D= 0.2\,\%$ , figure 13 a), which is discussed in §§ 3.1 and 3.2 (table 1). In the table, the parameters denoted with subscript 1 refer to those of the model with $l/D= 1.0$ and $\gamma /D= 0.2\,\%$ . For the model with $\gamma /D= 0$ , zigzag motion becomes weaker than in the model with $\gamma /D= 0.2\,\%$ . Specifically, when $\gamma /D$ changes from 0.2 % to 0, the degree of zigzag motion decreases, having $w^*/w^*_1= 0.73$ . In addition to the weakened zigzag motion, the lift force generated by the sphere merely affects the rotation of the sphere due to zero offset distance. Consequently, the sphere rotation is mitigated, as indicated by the average magnitude ratio of angular velocity $\overline {|\omega |}/\overline {|\omega _1|}= 0.76$ and the maximum filament angle ratio $|\theta |_{max}/|\theta |_{max,1}= 0.49$ : slower angular velocity and a narrower range of rotation. The reduced rotational dynamics also influences the translational dynamics, albeit in a minor way. With the change in $\gamma /D$ from 0.2 % to 0, the ratios of the average tangential velocity, trajectory length and falling time are $\bar {v}_t/\bar {v}_{t,1}= 1.04$ , $L/L_1= 0.99$ and $t/t_1= 0.95$ , respectively.

Table 1. Degree of zigzag motion $w^*$ , average magnitude of sphere angular velocity $\overline {|\omega |}$ , maximum magnitude of filament angle $|\theta |_{max}$ , average tangential velocity $\bar {v}_t$ , trajectory length $L$ , and falling time $t$ for two models with different configurations: (i) $l/D = 1.0$ and $\gamma /D = 0$ ; (ii) $l/D = 0$ and $\gamma /D = 0.2\,\%$ . The models of (i) and (ii) correspond to those in figures 13(b) and 13(c), respectively. The subscript 1 denotes the result of the model with $l/D= 1.0$ and $\gamma /D = 0.2\,\%$ in figure 13(a).

To examine the effects of the hydrodynamic interaction of the filament, we manufactured another experimental model (figure 13c), a plain sphere with the same offset distance ( $\gamma /D= 0.2\,\%$ ) as the model with $l/D = 1.0$ in figure 13(a), by adjusting the internal configuration of the sphere. The results for the plain sphere with a shifted centre of mass ( $l/D= 0$ , $\gamma /D= 0.2\,\%$ ) are presented in dimensionless form, using those of the model with $l/D= 1.0$ and $\gamma /D= 0.2\,\%$ (table 1). When the model is influenced solely by the shifted centre of mass without the hydrodynamic interaction of the filament, the zigzag motion becomes weaker, and the rotational dynamics is significantly reduced compared to the model experiencing both effects. Notably, the mitigation of the zigzag motion and the reduction in the rotational dynamics is more pronounced than the model with $l/D= 1.0$ and $\gamma /D= 0$ corresponding to figure 13(b). For $l/D= 0$ and $\gamma /D= 0.2\,\%$ , $w^*/w^*_1= 0.19$ and $\overline {|\omega |}/\overline {|\omega _1|}= 0.29$ , whereas these values are significantly higher at 0.73 and 0.76, respectively, for $l/D= 1.0$ and $\gamma /D= 0$ . Consequently, in the absence of the filament, changes in $\bar {v}_t$ , $L$ and $t$ are greater: $\bar {v}_t/\bar {v}_{t,1}$ , $L/L_1$ and $t/t_1$ for $l/D= 0$ and $\gamma /D= 0.2\,\%$ are 1.19, 0.96 and 0.81, respectively, while they are 1.04, 0.99 and 0.95 for $l/D= 1.0$ and $\gamma /D= 0$ . In summary, these results hint that for the model considered in §§ 3.1 and 3.2, the hydrodynamic interaction of the filament has a more substantial influence on the sphere’s falling dynamics than the shifted centre of mass, under the condition of a strong zigzag motion (e.g. $l/D = 1.0$ ).

3.4. Change in sphere density

The effects of the filament length on the translational and rotational dynamics of the sphere have been discussed in the preceding subsections, with the sphere density remaining constant at $\rho _s/\rho _f = 1.06$ . As the sphere becomes heavier, it is expected that the tangential velocity of the model will increase, affecting its rotational motion. In this subsection, we compare the falling dynamics of the model for higher density ratios. Preliminary experiments with spheres lighter than $\rho _s/\rho _f = 1.06$ yielded significantly scattered results due to their proximity to neutral buoyancy. Therefore, this section focuses on the effects of increased sphere density. We first examine the dynamics of a sphere with a higher density ratio $\rho _s/\rho _f = 1.12$ with respect to filament length $l/D$ , and compare the results with those for a sphere at $\rho _s/\rho _f = 1.06$ . Additionally, to comprehensively understand the effects of sphere density over a broader range, several cases with density ratios exceeding $\rho _s/\rho _f = 1.12$ are investigated while maintaining the filament length constant at $l/D= 1.0$ , where a strong zigzag motion is observed. As stated in § 2, the dimensionless moment of inertia $I^*\ (=I_s/I_{\Gamma })$ remains very similar between the spheres with different $\rho _s/\rho _f$ values ( $I^* \approx 0.96$ ), and the filament properties remain unchanged.

Figure 14. (a) Degree of zigzag motion $w^*$ , and (b) average magnitude of sphere angular velocity $\overline {|\omega |}/\overline {|\omega _0|}$ , for $\rho _s/\rho _f = 1.06$ (black) and $\rho _s/\rho _f = 1.12$ (blue). Here, $I^* = 0.967$ for $\rho _s/\rho _f = 1.06$ , and $I^* = 0.959$ for $\rho _s/\rho _f = 1.12$ .

Regardless of sphere density, the centre of mass of the model is positioned above the geometric centre of the sphere due to the attached filament. The rotational mechanism described in § 3.2 also applies to the model with the greater density. Thus the model with $\rho _s/\rho _f = 1.12$ exhibits a zigzag falling trajectory, similar to the model with $\rho _s/\rho _f = 1.06$ . For both density ratios, the degree of zigzag motion $w^*$ exhibits quite similar trends with respect to $l/D$ (figure 14 a). As the sphere density increases, its velocity also increases. In the reference frame fixed to the sphere, the velocity of the flow surrounding the sphere becomes greater for the heavier sphere. Consequently, the effects of $T_{ext}$ are amplified, resulting in greater magnitudes of the angular velocity and lift force. Because of the enhanced lift force, the magnitudes of the horizontal displacement and the degree of zigzag motion appear to be comparable between the two density ratios in the entire range of $l/D$ , although the heavier sphere falls faster; strictly, $w^*$ for $\rho _s/\rho _f = 1.12$ is slightly smaller than that for $\rho _s/\rho _f = 1.06$ near $l/D = 1.0$ with the strong zigzag motion.

For the heavier sphere, the centre of mass of the model is located closer to the geometric centre of the sphere. This reduced offset distance annihilates the torques induced by the shifted centre of mass. Figure 14(b) compares the average magnitude of the sphere’s angular velocity $\overline {|\omega |}$ divided by that of the plain sphere $\overline {|\omega _0|}$ with the same density for the two cases. In this subsection, results denoted with subscript 0 refer to those of the plain sphere with the same density as the sphere in the numerator of the dimensionless quantities. Although the dimensional $\overline {|\omega |}$ of $\rho _s/\rho _f = 1.12$ significantly exceeds that of $\rho _s/\rho _f = 1.06$ across all $l/D$ values, the dimensionless $\overline {|\omega |} / \overline {|\omega _0|}$ of $\rho _s/\rho _f = 1.12$ is less than that of $\rho _s/\rho _f = 1.06$ , indicating that the filament has less impact on the rotational dynamics for the heavier sphere. For example, at $l/D = 1.0$ , $\overline {|\omega |}/\overline {|\omega _0|} = 3.7$ for $\rho _s/\rho _f = 1.12$ , while $\overline {|\omega |}/\overline {|\omega _0|} = 8.0$ for $\rho _s/\rho _f = 1.06$ .

Figure 15. (a) Trajectory length $L/L_0$ , (b) average tangential velocity $\bar {v}_t/\bar {v}_{t,0}$ , and (c) falling time $t/t_0$ , for $\rho _s/\rho _f = 1.06$ (black) and $\rho _s/\rho _f = 1.12$ (blue).

Figure 16. (a) Degree of zigzag motion $w^*$ , (b) average magnitude of angular velocity $\overline {|\omega |}/\overline {|\omega _0|}$ , (c) trajectory length $L/L_0$ , (d) average tangential velocity $\bar {v}_t/\bar {v}_{t,0}$ , and (e) falling time $t/t_0$ , with respect to $\rho _s/\rho _f$ for the spheres with $l/D= 1.0$ . Here, $I^* \approx 0.96$ for all cases.

In addition to the rotational dynamics, the translational dynamics is less affected by the filament in the case of the heavier sphere. Figures 15(a) and 15(b) present the dimensionless trajectory length $L/L_0$ and average tangential velocity $\bar {v}_t/\bar {v}_{t,0}$ , respectively, for different density ratios. Both $L/L_0$ and $\bar {v}_t/\bar {v}_{t,0}$ exhibit similar trends with respect to $l/D$ , regardless of the sphere density. The dimensional trajectory length $L$ is similar for both density ratios across the entire range of $l/D$ . However, the dimensionless $L/L_0$ is smaller for the sphere with $\rho _s/\rho _f = 1.12$ . In terms of a dimensional quantity, the average tangential velocity $\bar {v}_t$ of the heavier sphere is significantly greater than that of the lighter sphere for all $l/D$ . Moreover, the dimensionless $\bar {v}_t/\bar {v}_{t,0}$ with $\rho _s/\rho _f = 1.12$ exhibits a greater magnitude than that of $\rho _s/\rho _f = 1.06$ . The smaller $L/L_0$ and greater $\bar {v}_t/\bar {v}_{t,0}$ reduce the dimensionless falling time $t/t_0$ for the heavier sphere (figure 15 c).

To further examine the effects of sphere density ratios greater than $\rho _s/\rho _f= 1.12$ , the spheres with four different density ratios, $\rho _s/\rho _f=1.18, 1.24, 1.30, 1.36$ , were manufactured additionally by modifying the internal configurations of the spheres. Similar to the aforementioned results between $\rho _s/\rho _f= 1.06$ and $\rho _s/\rho _f= 1.12$ , for the spheres with $\rho _s/\rho _f \gt 1.12$ , increasing density further reduces the influence of the filament on the falling dynamics because the offset distance between the centre of mass of the entire model and the geometric centre of the sphere becomes smaller. As the sphere density increases, the zigzag motion gradually weakens, and $w^*$ decreases monotonically (figure 16 a). This reduction in the zigzag motion correlates with a decrease in $\overline {|\omega |}/\overline {|\omega _0|}$ (figure 16 b). Consequently, with increasing sphere density, the differences in translational dynamics between a sphere with a filament and a plain sphere of the same density become less pronounced. Specifically, both the dimensionless trajectory length $L/L_0$ and the dimensionless average tangential velocity $\bar {v}_t/\bar {v}_{t,0}$ approach unity as the sphere density increases (figures 16 c,d). This convergence leads to the shorter dimensionless falling time $t/t_0$ for the heavier sphere (figure 16 e). In summary, regarding the dimensionless parameters, the rotational motion of the sphere with the higher density is less distinct than for the lower-density sphere due to the smaller offset distance between the centre of mass and the geometric centre. This change mitigates the effects of the filament on the translational motion.