3 Results

Figure 1 displays high resolution images of two spheres with nearly identical impact conditions ( $U_{0}=2.4~\text{m}~\text{s}^{-1}$ , $D=51$  mm), except that the sphere in (a) has shear modulus $G_{\infty }=5.66\times 10^{5}$  kPa (rigid) while the sphere in (b) has shear modulus $G_{\infty }=12.7$  kPa. The cavity formed by the deformable projectile differs in the oscillatory profile of the cavity walls, in addition to being shallower and wider. Because the cavity physics and projectile dynamics are evidently different for a deformable sphere, characterizing the initial deformation and resulting material oscillation in the sphere is critical to understanding water entry physics for deformable objects.

Figure 3(a,b) lends additional insight into why a cavity produced by a deformable sphere deviates from that formed by a rigid sphere. At 12 ms after impact, the sphere has deformed significantly into an oblate spheroid, creating a wider cavity than a rigid sphere. Elastic forces cause the sphere to rebound from this initial deformation into a prolate spheroid with its major axis aligned with the vertical ( $t=29$  ms). The continually oscillating sphere now proceeds to a second radial expansion that penetrates through the cavity wall, forming a smaller cavity within the first ( $t=43$  ms), resulting in a so-called matryoshka cavity (Hurd et al. Reference Hurd, Fanning, Pan, Mabey, Bodily, Hacking, Speirs and Truscott2015; Belden et al. Reference Belden, Hurd, Jandron, Bower and Truscott2016). For this case pinch off occurs within the second cavity ( $t=96$  ms).

For each experimental test, the position of the bottom of the sphere $y_{b}$ was tracked through a series of high-speed images. In figure 3(c), $y_{b}/D$ is plotted as a function of dimensionless time $t/t_{p}$ ( $t_{p}$ : time to pinch off), in which the sphere oscillation is evident. Figure 3(d) shows the measured value of $\unicode[STIX]{x1D706}$ , which reaches an initial large peak due to the impact event ( $t=12$  ms), and then decays throughout the water entry. This decay in $\unicode[STIX]{x1D706}$ is typical for all deformable sphere water entry events studied.

Figure 3.

(a) A sphere deforms significantly as it impacts and enters the water ( $G_{\infty }=6.70$  kPa,  $D=51$  mm and $U_{0}=5.3~\text{m}~\text{s}^{-1}$ ); after the initial deformation the sphere oscillates between oblate and prolate shapes, creating a second cavity within the first. Pinch off occurs within this second, smaller cavity ( $t=96$  ms, supplemental movie 1, available at https://doi.org/10.1017/jfm.2017.365). (b) The water entry event captured from a top view highlights the changing diameter and splash curtain dome over event. (c) The measured position of $y_{b}$ is plotted against dimensionless time where vertical lines correspond to the images above. (d) Plotting the parameter $\unicode[STIX]{x1D706}$ as a function of time portrays a decaying sinusoid. Image sequences (a,b) are both shown in supplemental movie 1.

Figure 4(a) presents a simplified description of the sphere oscillation in which the sphere deforms into an oblate spheroid with symmetry about the $y$ -axis. Here, $\unicode[STIX]{x1D706}$ represents the principal stretch in the $x$ and $z$ directions, and by conservation of volume the principal stretch in the $y$ direction is $\unicode[STIX]{x1D706}_{y}=1/\unicode[STIX]{x1D706}^{2}$ . Defining $\unicode[STIX]{x1D706}$ in this way is based on the observation that the primary mode of deformation in the sphere during water entry is equi-biaxial tension, and $\unicode[STIX]{x1D706}$ is a measure of the principal stretch in the sphere. The parameter $\unicode[STIX]{x1D706}_{pN}$ represents the maximum stretch of the sphere in the $x$ – $z$ plane for the $N$ th deformation period.

Figure 4.

(a) When a deformable sphere impacts the water surface it flattens into an oblate spheroid with an increased cross-sectional diameter at $t=T/4$ ( $\unicode[STIX]{x1D706}_{p1}D$ ) before rebounding back into a sphere at $T/2$ . The sphere then forms a prolate spheroid at $3T/4$ , then returns to a spherical shape in a single period $T$ (not shown). The principal stretch $\unicode[STIX]{x1D706}$ defines the deviation of the sphere from its un-stretched diameter $D$ . The subscript $pN$ refers to the maximum value of $\unicode[STIX]{x1D706}$ within the $N$ th period. (b) The oscillation period of $\unicode[STIX]{x1D706}$ scales with $D/\sqrt{G_{\infty }/\unicode[STIX]{x1D70C}_{s}}$ (the slope of the linear fit is 1.4). The analytical model predicts a slightly smaller slope (1.2). Large data points represent large spheres ( $D=100$  mm) and small data points represent small spheres ( $D=51$  mm). (c) The peak value of $\unicode[STIX]{x1D706}$ for a given period and given sphere radius appears to depend only on $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}$ . Shading denotes period number ( $N$ ) as indicated in the legend. Symbols represent experimental data and lines represent model prediction. Thin lines represent small spheres ( $D=51$  mm) and thick lines represent large spheres ( $D=100$  mm). (d) The measured values of $\unicode[STIX]{x1D706}$ in time, for the experimental case featured in figure 3, are represented by grey triangles. The behaviour predicated by the analytical model is represented by a dashed line. The solid line portrays a time-adjusted model with frequency shifted to correspond with the scaling in (b).

Based on the decaying behaviour of $\unicode[STIX]{x1D706}$ during water entry, we aim to address if the source of damping is in the sphere material, water or both. First, we isolate the response of the sphere by performing a series of tests in which the spheres are dropped onto horizontal rigid surfaces (appendix A, figure 12, supplemental movie 2). Impact with the rigid surface results in an initially large sphere deformation that decays in time. Based on these observations, we apply a viscoelastic model to the sphere material (Bergström & Boyce Reference Bergström and Boyce1998) as summarized in appendix A. The model includes parameters to account for viscous damping, but the equilibrium stress is still governed by the hyperelastic neo-Hookean model (parameterized by shear modulus  $G_{\infty }$ ). The rigid surface impact data are used to calibrate the dynamic parameters of the viscoelastic model. Second, to see if the damping in the material model can explain the decay observed in the water entry events, we construct a simplified model of the sphere oscillation (derived in appendix A). The sphere is prescribed an initial stretch $\unicode[STIX]{x1D706}_{0}=\unicode[STIX]{x1D706}_{p1}$ at $t=t_{0}$ and allowed to oscillate freely for $t>t_{0}$ . The analysis is performed for all experimental cases and the results are summarized in figure 4.

Figure 4(b) shows that the oscillation period predicted by the model is slightly less than that observed in the experiments. Because the viscoelastic model parameters were calibrated to an experimental test isolated from the water, we suggest the observed lower frequency (longer period) of the sphere in water is attributable to the added mass experienced by the sphere. Some mass of water has to be accelerated during portions of the sphere oscillation period (e.g. between $t=3T/4$ and $t=5T/4$ in figure 4 a). However, if we scale the response in time by a ratio of experimental to modelled periods, $T_{exp}/T_{model}$ , then the predicted oscillations show good agreement with the experiments. Despite the difference in period, the magnitude of the predicted peaks in $\unicode[STIX]{x1D706}$ are consistent with experiments (figure 4 c,d), suggesting that the dominant source of damping is in fact in the sphere material. We note that the model agrees more accurately at the peaks than the valleys because an oscillating sphere complies with the idealization of the model (ellipsoidal assumption) more closely in its oblate shape than its prolate shape as is observed in figure 5(d).

Figure 5.

(a–d) The water entry of four spheres with identical diameter ( $D=51$  mm), density ( $\unicode[STIX]{x1D70C}_{s}=1070~\text{kg}~\text{m}^{-3}$ ) and impact velocity ( $U_{0}=6.5~\text{m}~\text{s}^{-1}$ ) but varying shear moduli: (a)  $G_{\infty }=5.66\times 10^{5}$  kPa, (b)  $G_{\infty }=70.2$  kPa, (c)  $G_{\infty }=6.70$  kPa, (d)  $G_{\infty }=1.12$  kPa. (e) For the largest and most compliant spheres tested ( $D=100$  mm,  $U_{0}=6.5~\text{m}~\text{s}^{-1}$ and $G_{\infty }=1.12$  kPa) spheres decelerate more rapidly, occupying the space where pinch off would occur. The attached cavity recedes upward along the sphere, pinching at the top of. Image sequences (a–e) correspond to supplemental movies 3–7 respectively.

Figure 5(a–d) shows the cavity growth and pinch off resulting from the impact of four spheres ( $D=51$  mm, $U_{0}=6.5~\text{m}~\text{s}^{-1}$ ) with $G_{\infty }$ decreasing from $5.66\times 10^{5}$  kPa (a) to 1.12 kPa (d). The cavity in image sequence (b) is created by a sphere with a shear modulus of $G_{\infty }=70.2$  kPa. The resulting cavity and pinch off strongly resembles those created by the rigid sphere in (a), except for the presence of small-scale undulations on the cavity walls due to sphere vibration. The sphere in (c) has a shear modulus an order of magnitude smaller than that in (b); it deforms significantly upon impact creating a much wider cavity and shallower pinch-off event. The smaller $G_{\infty }$ results in higher magnitude but lower frequency oscillations, creating a second impact-like event within the first cavity. Deformations are even more pronounced in sequence (d). Pinch off occurs within the second cavity formed for image sequences (c) and (d). Spheres with lower values of $G_{\infty }$ are often observed to decelerate so rapidly that they occupy the space where a deep seal would normally occur as seen in figure 5(e). In this instance the contact line of the second cavity recedes up the surface of the sphere and pinches off at the top.

Water entry events are often classified by cavity characteristics, with a common parameter being time to pinch off ( $t_{p}$ ). The dimensionless time $t_{p}U_{0}/D$ is plotted against Froude number ( $Fr=U_{0}/\sqrt{gD}$ ) in figure 6(a) for all tested cases, which is the same non-dimensionalization employed by Aristoff et al. (Reference Aristoff, Truscott, Techet and Bush2010) for decelerating rigid spheres (dashed line). However, the scaling does not provide an effective data collapse for deformable elastomeric spheres. Instead we normalize using a new term $D_{eff}=\unicode[STIX]{x1D706}_{pN}D$ , which represents the maximum deformed diameter that the sphere assumes within the cavity in which pinch off occurs. For example, in the case seen in figure 5(d), pinch off occurred within the second cavity resulting in $D_{eff}=\unicode[STIX]{x1D706}_{p2}D$ . This adjustment provides a more convincing data collapse as can be seen in figure 6(b), where the solid line is a fit to the data (slope  $=$  1.3).

Figure 6.

(a) Dimensionless pinch off plotted against $Fr$ does not produce a convincing collapse as was the case with rigid spheres in the study by Aristoff et al. (Reference Aristoff, Truscott, Techet and Bush2010). The dashed line represents the theoretical scaling proposed in the same study. (b) Rather a more accurate scaling is achieved by a dimensionless pinch-off time scaled by $D_{eff}=\unicode[STIX]{x1D706}_{pN}$ , which represents the maximum deformed diameter that the sphere assumes within the cavity in which pinch off occurs (slope of linear fit  $=$  1.3).

We have described the effects of elastomeric sphere deformation on the global features of water entry, and now turn attention to sphere dynamics. Based on the description of the sphere as an ellipsoid (figure 4 a), the position of the centre of mass is defined as

where, as already discussed, $y_{b}$ is tracked from images (figure 3 c). Any noise in measurements of $\unicode[STIX]{x1D706}$ would be amplified in (3.1); therefore, we use the time-scaled results of the model simulations to define $\unicode[STIX]{x1D706}$ . The velocity and acceleration of the centre of mass, ${\dot{y}}_{c}$ and $\ddot{y} _{c}$ , are computed from derivatives of smoothing splines fit to $y_{c}$ , as was done in Truscott, Epps & Techet (Reference Truscott, Epps and Techet2012). Figure 7(a–c) displays ${\dot{y}}_{c}$ as a function of dimensionless time for all values of shear modulus $G_{\infty }$ , where $D=51$  mm and $U_{0}$ ranges from 3.0 to $6.5~\text{m}~\text{s}^{-1}$ . The values of ${\dot{y}}_{c}$ for rigid spheres ( $G_{\infty }=5.66\times 10^{5}$ ) are plotted as blue curves. The vertical lines indicate the end of the first oscillation period for the elastomeric spheres (corresponding grey shades match the legend in (a)). Elastomeric spheres experience a greater deceleration than the rigid spheres as $\unicode[STIX]{x1D706}D$ increases. However, the deformable spheres quickly transition to a deceleration rate similar to that of a rigid sphere as the magnitude of $\unicode[STIX]{x1D706}$ decreases (compare slopes of grey curves to blue curve after the first oscillation period marked by the vertical lines). Finally, after pinch off ( $t/t_{p}>1$ ), a steady state is reached and spheres fall at nearly constant velocity ( $\unicode[STIX]{x1D706}\rightarrow 1$ ). Notice that in (c) the softest spheres (lightest grey) lose nearly all of their velocity during the first deformation cycle, whereas more rigid spheres lose a significantly smaller portion.

Figure 7.

Dimensionless velocity ( $|{\dot{y}}_{c}|/U_{0}$ ) is plotted against dimensionless time ( $t/t_{p}$ ) for spheres impacting with three different velocities: (a)  $U_{0}=3.0~\text{m}~\text{s}^{-1}$ , (b)  $U_{0}=5.3~\text{m}~\text{s}^{-1}$ and (c)  $U_{0}=6.5~\text{m}~\text{s}^{-1}$ . Compared to rigid spheres (blue curves), deformable spheres experience a larger deceleration rate after impact over the first cycle of sphere deformation. After the first oscillation period, deformable spheres follow a deceleration similar to rigid spheres, and then transition to a nearly constant velocity after pinch off. (d) The large initial change in velocity is investigated by plotting $\unicode[STIX]{x0394}U/U_{0}=(U_{1}-U_{0})/U_{0}$ , where $U_{1}$ is the velocity of the sphere centre of mass after one oscillation period, against $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}$ . For $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}\gtrsim 0.2$ , $\unicode[STIX]{x0394}U/U_{0}$ scales with $(G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2})^{-1/2}$ , as predicted by a scaling analysis from the equation of motion for the sphere. For $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}\lesssim 0.2$ , $\unicode[STIX]{x0394}U/U_{0}$ asymptotes to the limit of 1, with nearly all of $U_{0}$ being lost over the first period of oscillation.

We perform a scaling analysis of the water entry event to gain insight into the sphere deceleration over the first deformation period. For simplicity, added mass is neglected and thus the dominant forces include drag, gravity and buoyancy. Because the spheres are nearly neutrally buoyant, gravitational and buoyant terms cancel and a simple equation of motion for the impacting sphere can be expressed as

where $\forall$ represents the volume of the sphere, $A$ the cross-sectional area, $U$ the velocity of the centre of mass and $C_{D}$ the coefficient of drag. We simplify this expression by defining a characteristic acceleration $(U_{1}-U_{0})/T=\unicode[STIX]{x0394}U/T$ , where $U_{1}$ denotes the velocity of the sphere centre of mass after the first deformation period. By noting that $\forall \sim D^{3}$ , $A\sim D^{2}$ and $C_{D}\sim 1$ , we can approximate (3.2) as

We previously showed that $T\sim D/\sqrt{G_{\infty }/\unicode[STIX]{x1D70C}_{s}}$ (figure 4 b), and for tested spheres $\unicode[STIX]{x1D70C}_{w}/\unicode[STIX]{x1D70C}_{s}\sim 1$ . This allows us to rearrange (3.3) to

The velocity $\unicode[STIX]{x0394}U/U_{0}$ is plotted against $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}$ in figure 7(d). This dimensionless number, which is a ratio of material shear modulus to impact hydrodynamic pressure, collapses the data. For $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}\gtrsim 0.2$ , the data follow the scaling predicted by (3.4). However, in the limit of small $G_{\infty }$ and large $U_{0}$ spheres deform significantly, and the argument $A\sim D^{2}$ no longer holds as there is a more complicated dependence of $\unicode[STIX]{x1D706}$ on the material properties and impact conditions. Furthermore, it is likely that added mass plays a more significant role as $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}\rightarrow 0$ (see appendix B). When $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{0}^{2}<0.2$ , we find $\unicode[STIX]{x0394}U\rightarrow U_{0}$ within the first oscillation period. Nonetheless, the experimental data follow the proposed scaling well, and this allows us to predict how the impact dynamics of deformable spheres will differ from their rigid counterparts based on material properties and impact conditions.

At this point, it is worth commenting on the expected role of added mass during the water entry event. Prior research on rigid sphere water entry has shown that forces arising from added mass are significant in the early moments of impact, primarily at times before the entire sphere has passed the free surface (Faltinsen & Zhao Reference Faltinsen and Zhao1998; Truscott et al. Reference Truscott, Epps and Techet2012, Reference Truscott, Epps and Belden2014). As discussed earlier, it is likely that the added fluid mass is responsible for the longer oscillation period of the spheres in water. This added mass would be expected to resist sphere acceleration in the direction of travel as the sphere oscillates. While added mass undoubtedly affects the physics of deformable sphere water impact, we argue that it is unlikely to significantly affect the trends in $\unicode[STIX]{x0394}U/U_{0}$ for $G_{\infty }/\unicode[STIX]{x1D70C}_{w}U_{o}^{2}\gtrsim 0.2$ (see appendix B for more details). This is supported by the good agreement between the experimental data and the predicted trend in figure 7(d).

Figure 8.

Force coefficients for deformable spheres ( $\overline{C}_{F}$ ) calculated by averaging over each deformation period ( $y$ direction) are plotted as a function of dimensionless time (black symbols). The data encompass all sphere diameters, shear moduli and impact velocities tested. Force coefficients ( $C_{F}$ ) for rigid spheres entering the free surface with the same specific gravity as the deformable spheres are plotted as a function of dimensionless time (blue curves). The period-averaged values for deformable spheres $\overline{C}_{F}$ follow the instantaneous values for rigid spheres $C_{F}$ , except during the first sphere deformation period in which deformable spheres experience larger drag from increased $\unicode[STIX]{x1D706}D$ .

To further investigate how the water entry of a deformable elastomeric sphere differs from that of a rigid sphere, we calculate the total force coefficient acting on the sphere in the $y$ direction as a function of time. The oscillating behaviour of the sphere results in a varying instantaneous force coefficient $C_{F}$ . Therefore, we calculate a period-averaged force coefficient,

where $\overline{\ddot{y} }_{c}$ and $\overline{{\dot{y}}}_{b}$ are the acceleration and velocity of the centre of mass and sphere bottom averaged over a single oscillation period, respectively. Using (3.5), values for $\overline{C}_{F}$ are plotted in figure 8 as a function of dimensionless time $t/t_{p}$ for all experimental cases. The period-averaged values $\overline{C}_{F}$ follow the instantaneous experimental values $C_{F}$ for three cases of rigid sphere water entry (blue curves). This trend holds except for the first sphere deformation period ( $t/t_{p}\approx 0.2$ to 0.4 depending on $G_{\infty }$ ), for which the deformable spheres experience larger drag from increased $\unicode[STIX]{x1D706}D$ . Over this period, the spheres deform into ellipsoids with a large aspect ratio and thus we expect the force coefficient to be larger. For example, for an ellipsoid with $\unicode[STIX]{x1D706}=1.3$ , we expect the force coefficient to be between 3–7 times larger than that of a sphere, depending on Reynolds number (Daugherty & Franzini Reference Daugherty and Franzini1977).