3.1 Overall description of the free-surface deformation

Figures 2 and 3 show the overall flow structures above and below the free surface, induced by the water entry of a projectile (smooth surface) with different $AR$ , impacting at $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively (see also supplementary movie S1 available at https://doi.org/10.1017/jfm.2018.1026). Since the smooth acrylic surface had a static contact angle smaller than $90^{\circ }$ and the capillary number was less than 0.1, the typical low-speed cavity whose boundary is attached to the nose of the body is not observable (Truscott et al. Reference Truscott, Epps and Belden2014). Rather than a splash accompanied by a cavity, jet flow is evident above the free surface. For the cases of $AR$ 1–4 (at both $U_{\circ }=2.5$ and $4.2~\text{m}~\text{s}^{-1}$ ), a vertical jet appears immediately after the projectile is fully submerged (e.g. at $t^{\ast }=2.0$ in figures 2 b and 3 b), which breaks up into small droplets later. Here, the dimensionless time is defined as $t^{\ast }=tU_{\circ }/D$ and the instant of impact is set to $t=0$ . As the jet falls, the free surface slightly sags, which is clearer for the higher $U_{\circ }$ of $4.2~\text{m}~\text{s}^{-1}$ (at $t^{\ast }=25.0$ in figure 3 b), in which case the contribution of inertia is greater. As $AR$ increases, the maximum jet height tends to decrease, except when $AR=8.0$ . As shown in figure 2(d), the water film that rises along the surface of the projectile ( $AR$  8) does not converge to form a jet, but rather descends together with the body. As the body passes the free surface, it is dragged further downward due to surface tension to form a shallow cavity in the wake behind the projectile ( $t^{\ast }=8.0$ in figure 2 d). This behaviour is consistent with the description provided by Aristoff & Bush (Reference Aristoff and Bush2009), who explained that the free surface is attached to the rear edge of a cylinder as it enters water and is pulled down to form a cavity. The radial size of this cavity is similar to the cylinder diameter, and cavity pinch-off occurs when the cylinder base reaches a depth equal to the capillary length. This type of cavity is called a quasi-static cavity and has quite low air entrainment. Quasi-static cavities are not created until the tail of the body is fully submerged and are different from those formed by the entry of a body with a roughened front part, as we will discuss in § 4. After the shallow cavity detaches from the body tail, a long jet is induced due to the high pressure in the collapsing cavity ( $t^{\ast }=10.0$ in figure 2 d). Thus, this jet has a different origin from those appearing at smaller $AR$ . When $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ , the same flow structures (jet and quasi-static cavity) are induced but the cavity shape for $AR$  8 is more distorted due to the increased inertia (after the collapse, most of the cavity bubble descends with the body being attached to it) and thus the jet induced by the collapsing cavity is not well established and becomes shorter (figure 3 d).

Figure 4.

Jet formation by the water entry of a body with a smooth surface at $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ (a–c) and $4.2~\text{m}~\text{s}^{-1}$ (d–f): (a,d) $AR=1.0$ , (b,e) 2.0, (c,f) 4.0. The numbers in each image denote $t^{\ast }$ .

Figure 5.

Formation of thin and thick jets due to the water entry of a body with a smooth surface.

Figure 4 shows the sequence of jet formation for the cases of $AR$ 1–4 and demonstrates that jet formation can be understood as a two-step process, i.e. the formation of thin and thick jets. This behaviour is clearly visible for the cases with $AR$ 1–2 at $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ (figure 4 a,b), and the $AR$  4 case shows a more complex jet (breakdown into crown-like droplets and another jet due to the collapse of a shallow cavity) (figure 4 c). The formation of thin and thick jets in the water entry of a sphere was previously mentioned by Kuwabara et al. (Reference Kuwabara, Tanba and Kono1987) and is similar to the behaviour occurring in granular impacts (Marston et al. Reference Marston, Seville, Cheun, Ingram, Decent and Simmons2008; Royer et al. Reference Royer, Corwin, Conyers, Flior, Rivers, Eng and Jaeger2008) and laser-induced jets (Chen et al. Reference Chen, Yu, Su, Chen and Chen2013), but their mechanisms have not been addressed in detail. Figure 5 illustrates the sequential process of thin and thick jet formation. As we will explain in § 3.2, the water film first rises along the body surface after impact and converges at the rear pole to form a thin jet. As the body sinks, the surrounding water is pushed and dragged around the body tail to induce a stagnation point below the free surface. From this stagnation point, vertical flow is generated and the upward component forms a thick jet over the free surface (§ 3.3).

3.2 Water film rise and thin jet formation

Figure 6 shows the sequential movement of a thin water film along the smooth surface of a projectile with different $AR$ at $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ . For $AR~1$ , the ascending film gathers at the rear pole and forms a thin, long jet over the free surface (figure 6 a), which will be referred to as a thin jet. Similar apex jet formation upon the impact of a viscous glycerin droplet onto a lower viscosity liquid was reported previously by Marston & Thoroddsen (Reference Marston and Thoroddsen2008). They explained that the impact speed directly determines the rise speed of a liquid film and it should be sufficiently fast to give momentum to the liquid film to ascend to the pole, but not so much as to cause separation from the surface. When $AR=2.0$ , the water film still receives enough momentum to meet at the rear pole and generate a thin jet before the projectile fully submerges (figure 6 b). The time taken for jet formation increases as $AR$ increases, because it takes longer for the projectile to submerge. When $AR$ increases to 4.0, the water film rises along the side wall of the projectile, stays at a constant height for some time and then falls together with the body (figure 6 c). In this case, the convergence of the water film is delayed. Since the water film cannot meet at the apex of the projectile, the thin jet formation is not as organized as in the cases of $AR$  1–2, but the water film breaks into a finger-like shape ( $t^{\ast }=3.6$ in figure 6 c). Below the free surface, a shallow (quasi-static) cavity is formed, and its collapse induces another jet emerging through the water fingers (figure 4 c). For the highest $AR$ of 8.0, similar to the case of $AR~4$ , the water film stays at a constant height but falls into the free surface before the projectile submerges completely (figure 6 d). Thus, it is understood that at higher $AR\;({>}4.0)$ , the water film does not fully converge at the pole, leaving a sagging space (i.e. cavity) under the free surface.

Figure 6.

Movement of a thin water film along the body surface (smooth) after impact ( $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ): (a) $AR~1$ ; (b) $AR~2$ ; (c) $AR~4$ ; (d) $AR~8$ . The numbers in each image denote $t^{\ast }$ .

Figure 7.

Temporal variation of the position of the end of a water film: (a) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ; (b) $4.2~\text{m}~\text{s}^{-1}$ . ●, $AR~1$ ; ○, $AR~2$ ; ▫, $AR~4$ ; ♢, $AR~8$ . The origin is defined as the location on the undisturbed free surface that the nose of the body first contacts, and the data are plotted with a time interval $\unicode[STIX]{x0394}t^{\ast }$ of $0.05$ .

Figure 8.

Temporal variation of the liquid film height ( $h^{\ast }=h/D$ ) from the free surface: (a) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ; (b) $4.2~\text{m}~\text{s}^{-1}$ . ●, $AR~1$ ; ○, $AR~2$ ; ▫, $AR~4$ ; ♢, $AR~8$ . The solid lines denote the present estimates (for $AR~8$ ) (see figure 9 and (3.5)).

Figure 7 tracks the end of a water film (i.e. three-phase contact point) while rising along the body surface. Here, $r$ and $z$ denote the radial and vertical directions, respectively, and the origin is defined as the point on the undisturbed free surface that the nose of the body first contacts. The raw images were binarized (Otsu Reference Otsu1979) to distinguish the interfaces between the ambient air, water film and solid surface before measuring the position of the water film. For both $U_{\circ }=2.5$ and $4.2~\text{m}~\text{s}^{-1}$ , the water film follows the surface of the falling projectile and converges near the rear pole ( $z/D\simeq 0.5$ ) for the cases of $AR~1{-}2$ . For $AR~4$ , the last location of the film end is around $z/D=0.2{-}0.3$ , which corresponds to the root of a finger-like jet ( $t^{\ast }=3.6$ in figure 6 c). Thus, the complete convergence of the water film as a jet is not achieved before the body submerges. The water film rises and then simply falls down vertically, for the case of $AR~8$ . The initial speed of the water film is not affected much by variations of $AR$ and $U_{\circ }$ , but its descending speed later changes considerably (figure 7). In addition, the maximum height of the water film is independent of $AR$ when it is sufficiently high that the water film cannot converge completely before the projectile fully submerges. To investigate this feature further, the liquid film height ( $h/D$ ), i.e. the $z$ -position in figure 7, is shown as a function of time in figure 8. For both $U_{\circ }$ values, the general trend of the variation of $h/D$ with $AR$ is similar. That is, the initial speed (at $t^{\ast }<1.0$ ) of a rising film is almost identical ( $\unicode[STIX]{x2202}h^{\ast }/\unicode[STIX]{x2202}t^{\ast }\simeq 0.7$ and 0.6 for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively) for all $AR$ values, but the slope during the falling stage is different. After the maximum height, the falling velocity decreases with increasing $AR$ ; $\unicode[STIX]{x2202}h^{\ast }/\unicode[STIX]{x2202}t^{\ast }=-0.51$ , $-0.44$ , $-0.21$ and $-0.17$ for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , and $-0.61$ , $-0.52$ , $-0.20$ and $-0.10$ for $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ when $AR=1.0$ , 2.0, 4.0 and 8.0, respectively. The maximum height of the water film remains constant at $h/D\simeq 0.8$ . For $AR~1{-}2$ , the film height decreases sharply after the water film closes at the rear pole of the projectile; thus, the maximum height is less than $0.8D$ (figure 8). As $AR$ increases to 4.0, $h/D$ increases to the maximum of 0.8 and then decreases gradually. At $t^{\ast }\simeq 3.5$ , similar to the cases with lower $AR$ values, the water film experiences a sharp fall after it gathers around the rear pole. For $AR~8$ , the water film falls down to the free surface at a constant rate after reaching the maximum height of $0.8D$ . Interestingly, the critical non-dimensional time for classifying these different rising water film behaviours depending on the $AR$ is similar even if the impact velocity increases to $4.2~\text{m}~\text{s}^{-1}$ (figure 8 b).

Figure 9.

Schematic diagram of the ascending water film on the surface of an impacting body: $h$ is measured from the undisturbed free surface. The thickness of the water film is defined as  $e$ .

If a thin jet is not formed through water film convergence at the rear pole, the maximum height of a water film is fixed at $h/D\simeq 0.8$ . To study this behaviour, we simplified the problem of a water film rise along the surface of an impacting body (figure 9). As shown, it is assumed that the water film rises along the body surface with a constant thickness ( $e$ ), which is very thin compared to the body size ( $D=2R$ ), while the body moves downward at $U_{\circ }$ . The distance to the end of water film from the undisturbed free surface is defined as $h$ . In this situation, the inertia transferred from the impacting body drives the motion and the capillary ( $F_{c}$ ), viscous ( $F_{v}$ ) and gravitational ( $F_{g}$ ) forces resist it (3.1):

(3.1) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\left(m_{f}\frac{\text{d}h}{\text{d}t}\right)=F_{c}-F_{v}-F_{g}.\end{eqnarray}$$

Here, $m_{f}$ is the mass of the water film, and the capillary and gravitational forces are given by $F_{c}=2\unicode[STIX]{x03C0}R\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{D}$ and $F_{g}=m_{f}g=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}Rehg$ , respectively ( $\unicode[STIX]{x1D70E}$ : surface tension, $\unicode[STIX]{x1D703}_{D}$ : dynamic contact angle, $g$ : gravitational acceleration). To calculate the viscous force, it is necessary to know the velocity distribution inside the water film. Since the water film is quite thin, it is assumed that the vertical ( $z$ ) component of the velocity ( $v=v(x)$ ) is dominant. We also consider that the film flow on a moving surface is fully developed, with boundary conditions of $v(0)=-U_{\circ }$ and $\text{d}v/\text{d}x(e)=0$ . Thus, the water film velocity at the interface to the ambient air is considered to be the same as the velocity at which the water film rises (similar to a Couette flow). The velocity distribution inside the water film and the corresponding viscous force at the body surface are given by (3.2) and (3.3), respectively:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle v(x)=\left(\frac{\text{d}h}{\text{d}t}+U_{\circ }\right)\left(2-\frac{x}{e}\right)\frac{x}{e}-U_{\circ }, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle F_{v}=\int \unicode[STIX]{x1D70F}_{w}=a_{1}\frac{4\unicode[STIX]{x03C0}R\unicode[STIX]{x1D707}}{e}h\frac{\text{d}h}{\text{d}t}+a_{2}\frac{4\unicode[STIX]{x03C0}R\unicode[STIX]{x1D707}U_{\circ }}{e}h. & \displaystyle\end{eqnarray}$$

Here, the pre-factors of $a_{1}$ and $a_{2}$ are introduced to compensate for the possible deviations from the assumptions of fully developed flow and constant water film thickness. Das et al. (Reference Das, Chanda, Eijkel, Tas, Chakraborty and Mitra2014) used similar pre-factors to compensate for the deviations of the velocity profiles from the assumed one in the capillary rise of an electrolytic solution. These pre-factors play the role of fitting parameters in actual application, and they were determined by the least square error method. Collecting these relations, equation (3.1) can be written as (3.4):

(3.4) $$\begin{eqnarray}\frac{\text{d}^{2}h}{\text{d}t^{2}}=\left(\frac{\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{D}}{\unicode[STIX]{x1D70C}e}\right)\frac{1}{h}-a_{1}\frac{2\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70C}e^{2}}\left(\frac{\text{d}h}{\text{d}t}\right)-\left(a_{2}\frac{2\unicode[STIX]{x1D707}U_{\circ }}{\unicode[STIX]{x1D70C}e^{2}}+g\right)-\frac{1}{h}\left(\frac{\text{d}h}{\text{d}t}\right)^{2}.\end{eqnarray}$$

This equation can be further normalized for the dimensionless height ( $h^{\ast }=h/D$ ) as

(3.5) $$\begin{eqnarray}\frac{\text{d}^{2}h^{\ast }}{{\text{d}t^{\ast }}^{2}}=\left(\frac{\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{D}}{\unicode[STIX]{x1D70C}e}\right)\frac{1}{U_{\circ }^{2}}\frac{1}{h^{\ast }}-a_{1}\frac{2\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70C}e^{2}}\frac{D}{U_{\circ }}\left(\frac{\text{d}h^{\ast }}{\text{d}t^{\ast }}\right)-\left(a_{2}\frac{2\unicode[STIX]{x1D707}U_{\circ }}{\unicode[STIX]{x1D70C}e^{2}}+g\right)\frac{D}{U_{\circ }^{2}}-\frac{1}{h^{\ast }}\left(\frac{\text{d}h^{\ast }}{\text{d}t^{\ast }}\right)^{2}.\end{eqnarray}$$

For $e$ and $\unicode[STIX]{x1D703}_{D}$ , we obtained the values from the measured data. First, we measured the film thickness over time, which was almost constant ( $e/D\simeq 0.02$ ) during the rise and got thicker ( $e/D\simeq 0.03$ ) as the film fell (see figure S1 in the supplementary material). Based on this, we assumed that $e$ is constant and used a time-averaged thickness of 0.8 mm and 0.11 mm for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively. On the other hand, $\unicode[STIX]{x1D703}_{D}$ would change over time as well, but we were not able to measure it accurately (measured values were scattered in the range of $110^{\circ }{-}150^{\circ }$ ) due to the limited resolution near the contact line. Thus, we measured the static contact angle of a water droplet ( $\unicode[STIX]{x1D703}_{c}=74^{\circ }$ ) on the side wall of the $AR~8$ projectile, and used a dynamic contact angle model given by $\unicode[STIX]{x1D703}_{D}^{3}=\unicode[STIX]{x1D703}_{c}^{3}+144Ca$ (Cox Reference Cox1986). As a result, $\unicode[STIX]{x1D703}_{D}$ was estimated as $110^{\circ }$ and $125^{\circ }$ for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively. These values were used as a representative (constant) $\unicode[STIX]{x1D703}_{D}$ .

Designating $t^{\ast }=0$ as the instant at which the rise of the water film can first be distinguished visually, the initial conditions of $h^{\ast }(0)$ and $\text{d}h^{\ast }/\text{d}t^{\ast }(0)$ were calculated from the measured data. These initial conditions were found to have positive values, indicating that the initial driving force of the film rise is the strong inertia transferred from the impacting body. After putting these empirical values into (3.5), it was numerically solved using the fourth-order Runge–Kutta method. The pre-factors were found to be $(a1,a2)=(15,2.8)$ and $(77,5.6)$ for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively. The results of the estimated water film height for the case of $AR~8$ are plotted together in figure 8, in which the present estimates show reasonable agreement with the measured water film heights. This also indicates that our assumptions are reasonable. As the impact velocity increases, the deviation between the measurement and estimate slightly increases. Since we assumed a constant film thickness, the effect of a curved edge near the free surface would be non-negligible with increasing inertia of the projectile. The first, second, third and fourth terms on the right-hand side of (3.5), on the other hand, represent the contributions of surface tension, viscosity, gravity and added water film mass from the pool, respectively. After comparing the magnitudes of these components, it was found that the inertia drives the initial motion at $t^{\ast }<1.0$ , which is counteracted by a capillary force and large initial mass of rising water film. As time progresses, the inertia of the rising water film is reduced, and it is balanced by the surface tension, viscous force and gravity at $t^{\ast }\simeq 1.2{-}1.5$ . After that, the viscous force (together with gravity) drives the downward fall of the water film (figure 8). Since the inertia dominates the flow for a very short time only during the initial stage, as shown above, the general trend of a moving water film does not vary significantly with $U_{\circ }$ (figures 7 and 8).

Figure 10.

Generation of a thick jet (a–d) and the flow structures below the free surface (e–h) at: (a,e) $t^{\ast }=0.8$ ; (b,f) 1.2; (c,g) 1.6; (d,h) 2.0. The considered case is $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , $AR=1$ with a smooth surface.

3.3 Thick jet formation

As it is shown most clearly in figure 2(b), a thick jet is induced after thin jet formation. While its shape differs slightly depending on the $AR$ , it can be said that the tip of a thick jet tends to collapse into a disk-like shape below the thin jet for $AR=1.0{-}4.0$ . When $AR=8.0$ , the jet flow originates from a different mechanism, i.e. the collapse of a quasi-static cavity. To understand the process of thick jet formation, we measured the liquid (water) phase velocity below the free surface at the instants at which thick jets were formed above the free surface, using particle image velocimetry. The measured velocity and vorticity fields are shown together with the corresponding jet flow in figures 10–12. As the sphere ( $AR~1$ ) moves through the water surface, it pushes the surrounding water away (figure 10 e), which turns around toward the rear side as it descends (figure 10 f,g,h). Thus, the water flow induced by the sinking sphere resembles that of a doublet, as mentioned by Kubota & Mochizuki (Reference Kubota and Mochizuki2011). Due to the interaction between this flow structure and a free surface, a stagnation point appears below the free surface (indicated with an arrow in figure 10 h), and substantial upward flow is induced above this point, resulting in a thick (secondary) jet. As shown in the figures, the generation of a thick jet is well synchronized with the appearance of a stagnation point below the free surface. Previously, Kuwabara et al. (Reference Kuwabara, Tanba and Kono1987) observed a similar phenomenon and mentioned that the secondary jet is related to high-speed water flow toward the sphere centre. As this thick jet falls down into the free surface, it forces the free surface to sag (figure 3 a,b). For the cases of $AR~2{-}4$ , the same thick jet formation mechanism is applied as the stagnation point shows up between the free surface and body tail (figures 11 h, 12 h). When $AR=4.0$ , the flow displaced by the front side of the body is not directly connected to the flow toward the rear side (figure 12 e) but the low pressure region in the wake behind the sinking body induces a stagnation point in the flow, resulting in a thick jet (figure 12 b,f,c,g,d,h). Since the motion of a displaced fluid around the front part of the body contributes strongly to the generation of upward flow, it becomes weaker as $AR$ increases. Thus, the height of a thick jet decreases with increasing (decreasing) $AR$  ( $U_{\circ }$ ).

Figure 11.

Generation of a thick jet (a–d) and the flow structures below the free surface (e–h) at: (a,e) $t^{\ast }=1.8$ ; (b,f) 2.2; (c,h) 2.6; (d,h) 3.0. The considered case is $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , $AR=2$ with a smooth surface.

Figure 12.

Generation of a thick jet (a–d) and the flow structures below the free surface (e–h) at: (a,e) $t^{\ast }=3.6$ ; (b,f) 4.2; (c,g) 4.8; (d,h) 5.4. The considered case is $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , $AR=4$ with a smooth surface.

3.4 Jet-tip breakdown

So far, we have discussed the processes of thin and thick jet formation. Interestingly, the jet tip breaks down in some cases (figures 2 a,b, and 3 a). When $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , breakdown occurs for $AR~1$ and $AR~2$ , and it is clearly visible for $AR~1$ at $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ . In figure 13, the temporal variation of a continuous (without a breakdown) jet height ( $H_{jet}$ ) is plotted. Since the jet flow for the case of $AR~8$ has a different origin than that discussed here, it is not included in this figure. Roughly speaking, $H_{jet}$ decreases with increasing $AR$ , because the momentum of the impacting body is transferred to the jet faster as $AR$ decreases. For $AR~1$ , thin jet breakdown occurs at $t^{\ast }\simeq 4.0$ and 23.0 for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively (sharp decreases of $H_{jet}$ are denoted with star symbols in the figure). For $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , $H_{jet}$ was measured for thin and thick jets before and after breakdown, respectively. When $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ , the initial steep slope indicates the rise of a thin jet and the second gentle increases corresponds to the rise of a thick jet (figure 13 b). In the case of $AR~2$ , distinguishable jet breakdown occurs only when $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ (at $t^{\ast }\simeq 9.5$ ); at $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ , the thin jet breaks into small droplets as soon as it is formed.

Figure 13.

Temporal variation of the continuous jet height ( $H_{jet}$ ) measured from the free surface: (a) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ; (b) $4.2~\text{m}~\text{s}^{-1}$ . ●, $AR~1$ ; ○, $AR~2$ ; ▫, $AR~4$ . The star symbols denote the breakdown of the corresponding jet.

The necking of liquid jets is well known in relation to the role of surface tension, such that the perturbation on a liquid filament surface grows at a critical condition (Rayleigh–Plateau instability) (Plateau Reference Plateau1873; Rayleigh Reference Rayleigh1878). To assess the breakdown of the present jet flows, the relevant parameters are summarized in table 2. The equivalent radius ( $R_{t}$ ) of the jet tip was measured at the moment that the jet reached the maximum height ( $H_{jet,max}$ ). We measured $R_{t}$ by assuming that the jet-tip shape is spherical. The corresponding value of the Ohnesorge number ( $Oh=\unicode[STIX]{x1D707}/(\unicode[STIX]{x1D70C}R_{t}\unicode[STIX]{x1D70E})^{0.5}$ ) was obtained in each case, which indicates the ratio of the viscous time scale to the capillary time scale and has been employed as an important variable in the pinch-off of viscous liquid filaments (Notz & Basaran Reference Notz and Basaran2004; Eggers & Villermaux Reference Eggers and Villermaux2008; Driessen et al. Reference Driessen, Jeurissen, Wijshoff, Toschi and Lohse2013). As $AR$ increases, $Oh$ tends to decrease for both impact velocities (table 2).

Finally, the aspect ratio of a jet is defined as $L_{\circ }=H_{jet,max}/(2R_{t})$ . Notz & Basaran (Reference Notz and Basaran2004) investigated the critical condition of liquid filament breakup and found that the critical $L_{\circ }$ for jet breakdown increases with increasing $Oh$ , below which the liquid film does not break. They reported that the critical $L_{\circ }$ is $5.5\pm 0.5$ for $Oh\simeq 0.001$ . In the present results, jet pinch-off occurred in the cases of $AR~1{-}2$ when $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $AR~1$ when $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ , in which cases $L_{\circ }$ is larger than about $5.0$ and $Oh$ is slightly larger than 0.001 (table 2). As shown in figure 4, the present breakdown occurs mostly in the form of end pinching (jet-tip breakdown) (Stone, Bentley & Leal Reference Stone, Bentley and Leal1986), which has been suggested as a main cause at $Oh\lesssim 0.1$ (Driessen et al. Reference Driessen, Jeurissen, Wijshoff, Toschi and Lohse2013). Thus, it is understood that the viscosity influenced the jet breakdown in the present cases. At $Oh\gtrsim 0.1$ , breakdown will occur due to the Rayleigh–Plateau instability. The existence of a critical aspect ratio also indicates that there is a relation between the inertia and capillary forces. That is, $R_{t}$ is associated with the capillary force and $H_{jet,max}$ is related to the lifetime (inertia) of a jet. As $AR$ increases, $R_{t}$ increases and the jet rise velocity decreases. As $R_{t}$ decreases, the jet becomes more susceptible to breakdown. Jet pinch-off was not observed for $AR\geqslant 4.0$ and $AR\geqslant 2.0$ when $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively, which supports this interpretation.

Table 2.

Measurement of $R_{t}$ at $H_{jet,max}$ . The corresponding values of $Oh$ and $L_{\circ }=H_{jet,max}/(2R_{t})$ were calculated.

4.1 Overall description of the free-surface deformation

In this section, we describe the water entry of a projectile with a rough front. Except the roughness (figure 1 b), the other conditions were same as those considered in § 3. The representative temporal evolutions of a splash (jet) and cavity are shown for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ (figure 14) and $4.2~\text{m}~\text{s}^{-1}$ (figure 15). Compared to the case with a smooth front part (figures 2 and 3), the differences in the free-surface deformation are clear in terms of an underwater cavity and a splash above the free surface. For the cases considered, a splash is generally formed after impact ( $t^{\ast }<1.0$ and $t^{\ast }<2.0$ for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and $4.2~\text{m}~\text{s}^{-1}$ , respectively) (figures 14 and 15), which developed from horizontal ejecta, as explained by Truscott et al. (Reference Truscott, Epps and Belden2014). As time progresses, the splash evolves further and converges into a dome, followed by the necking (or pinch-off) of an underwater cavity. For the case of $AR~8$ , the projectile itself interferes with the processes of dome closure and cavity pinch-off (figures 14 d, 15 d). The cavity pinch-off induces two vertical jets: one is forced upward through the dome and the other is ejected downward toward the falling projectile. Later, the splash and upward jet break down in a complex manner. Compared to the case of $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ (figure 15), the shapes of the splash and cavity, for $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , are less organized (i.e. axisymmetry is disturbed and the dome closure is not completed); thus, their volumes are reduced at a lower $U_{\circ }$ (figure 14).

Figure 14.

Splash and cavity formation by the water entry of a body with a rough surface at $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ : (a) $AR=1.0$ ; (b) 2.0; (c) 4.0 and (d) 8.0. The numbers in each image denote $t^{\ast }$ .

Figure 15.

Splash and cavity formation by the water entry of a body with a rough surface at $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ : (a) $AR=1.0$ ; (b) 2.0; (c) 4.0 and (d) 8.0. The numbers in each image denote $t^{\ast }$ .

In figure 16, the process of splash formation for the case of $AR~1$ is shown for both impact velocities. With a smooth surface, the water film rises along the body surface and converges at the rear pole (figure 6); however, it cannot move along the rough surface and instead separates early to create a splash shortly after the projectile impacts the free surface. It should be noted that the separation of a water film also occurs from a smooth body surface if the inertia of the impacting body is sufficiently large (Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara and Takano2004). Latka et al. (Reference Latka, Strandburg-Peshkin, Driscoll, Stevens and Nagel2012) performed an experiment in which a liquid droplet impacted a rough surface, producing a splash immediately after the collision. As the roughness increases, droplets are strongly prevented from transforming into thin liquid sheets and early splashing at the advancing contact line is encouraged. This phenomenon has been attributed to a large force required for the liquid to penetrate into the voids between roughness features while climbing the body surface (Kubiak et al. Reference Kubiak, Wilson, Mathia and Carval2011). Similarly, in the present case, the roughness of the sphere surface prevents water from forming a stable liquid film on it but encourages an early separation to create a splash. On the other hand, the speed of a water film is lower at a lower impact velocity, enabling it to move further along the body surface without being affected by the surface roughness (figure 16 a). Zhao et al. (Reference Zhao, Chen and Wang2014) reported that a splash is not formed in the water entry of a sphere even when the surface is rough if the impact velocity is sufficiently low that the water film can fully fill the voids between roughness features. Since we applied randomly distributed roughness features, the separation position of a water film differs locally around the periphery (figure 16 a). This non-uniformity of the roughness causes asymmetry of the splash (and underwater cavity), which decreases with increasing $U_{\circ }$ (figure 16 b).

Figure 16.

Splash formation above a free surface upon the water entry of a body (with a rough surface) at (a) $U_{o}=2.5~\text{m}~\text{s}^{-1}$ and (b) $4.2~\text{m}~\text{s}^{-1}$ . The numbers in each image denotes $t^{\ast }$ .

Figure 17.

Temporal variation of $H_{s}$ : (a) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ; (b) $4.2~\text{m}~\text{s}^{-1}$ . ●, $AR~1$ ; ○, $AR~2$ ; ▫, $AR~4$ ; ♢, $AR~8$ .

4.2 Splash and underwater cavity formation

To quantity the effects of $AR$ and $U_{\circ }$ on splash and cavity formation, the temporal variations of the height ( $H_{s}$ ) and width ( $W_{s}$ ) of a splash are plotted in figures 17 and 18, respectively. Here, we used the portion of a continuously connected water film in a splash, detected by applying a binarization and a Wiener filter to the raw images. $H_{s}$ is defined as the vertical distance from the free surface to the highest position in the detected water film and the horizontal distance between them was designated as $W_{s}$ (see the insets in figures 17 and 18). After dome closure, the positions at which the Worthington jet crosses the dome are used to determine $H_{s}$ , and $W_{s}$ is measured until the dome is closed. The splash size tends to increase with increasing $AR$ for lower $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ (figures 17 a, 18 a), but the tendency is reversed for $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ (figures 17 b, 18 b). As mentioned above, the water film separates later by filling some of the voids between roughness features when $U_{\circ }$ is low. Since the sinking velocity of a projectile increases with increasing $AR$ (see figure 23), there is not enough time for the liquid film to fill the voids on a rough surface. The downward velocity of a nose immediately after impact ( $t^{\ast }<5.0$ ) exhibits the same trend as that shown in figure 23. Consequently, the separation occurs earlier and the splash volume increases. Thus, the case of $AR~1$ has the shortest splash height and width when $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ . At this $U_{\circ }$ , the splash height increases to a peak and then decreases (figure 17 a). The maximum height increases with increasing $AR$ and becomes almost the same for $AR~4$ and $AR~8$ . $W_{s}$ follows a trend similar to that of $H_{s}$ and remains constant at later times (figure 18 a). The maximum width also increases with increasing $AR$ . When $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ , on the other hand, the water film separates earlier and thus the splash size increases sharply at $t^{\ast }<3.0$ . The variation among $AR$ values is negligible, indicating that the separation position of the water film is not affected, and the splash grows to almost the same maximum height and width (figures 17 b, 18 b). After the splash converges into a dome, its height decreases ( $5.0\lesssim t^{\ast }\lesssim 7.0$ ). Subsequently, the upward Worthington jet is ejected to the free surface due to the pinch-off of an underwater cavity, by which the air contained in the cavity pops up, resulting in a slight increase of $H_{s}$ at $t^{\ast }=7.0{-}15.0$ . In figure 15 (at $t^{\ast }=9.0$ and 14.0), the air inside the upper cavity after the pinch-off moves together with the vertical jet and the dome expands. The Worthington jet also exists at $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ , but no secondary peak of $H_{s}$ exists (figure 17 a) due to the incomplete dome closure (figure 14). Finally, $H_{s}$ decreases as the dome collapses at $t^{\ast }>15.0$ . In the case of $AR~8$ , the secondary peak is not clearly observable and its magnitude is smaller than it is in the other cases. As shown in figure 15(d), the long projectile shape disturbs the completion of dome closure (and cavity pinch-off). However, the increase in jet height at $t^{\ast }=19.0$ (figure 15 d) is attributed to the annular jet caused by the entry of a long cylindrical body. On the other hand, the slopes of decreasing $H_{s}$ are almost the same for all $AR$ values (figure 17 b). As the splash forms, $W_{s}$ increases to the maximum (again the ascending slopes are not affected by $AR$ ) and then decreases to form a dome (figure 18 b). As $AR$ increases, it takes less time to form a dome (i.e. $W_{s}/D=0$ ). Since the splash impacts the side wall of the projectile while it transforms into a dome, the splash width saturates to $W_{s}/D=1.0$ for the case of $AR~8$ .

Figure 18.

Temporal variation of $W_{s}$ corresponding to $H_{s}$ : (a) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ; (b) $4.2~\text{m}~\text{s}^{-1}$ . ●, $AR~1$ ; ○, $AR~2$ ; ▫, $AR~4$ ; ♢, $AR~8$ .

Figure 19.

Process of a cavity pinch-off behind a body with a rough front surface at (a–d) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ and (e–h) $4.2~\text{m}~\text{s}^{-1}$ : (a,e) $AR~1$ , (b,f) $AR~2$ , (c,g) $AR~4$ , (d,h) $AR~8$ . The numbers in each image denote $t^{\ast }$ .

Figure 20.

Temporal variation of $V_{c}$ : (a) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ; (b) $4.2~\text{m}~\text{s}^{-1}$ . ●, $AR~1$ ; ○, $AR~2$ ; ▫, $AR~4$ ; ♢, $AR~8$ .

Under the free surface, cavity pinch-off occurs, as shown in figure 19 (see supplementary movie S2 as well). A typical deep seal is observable for $AR~1{-}2$ , but an additional pinch-off occurs first at the side wall of the projectile for $AR\geqslant 4.0$ . As $U_{\circ }$ increases, the cavity formation and subsequent pinch-off process occur symmetrically. Figure 20 depicts the cavity volume ( $V_{c}$ ) calculated by assuming its axisymmetry; that is, the outer profiles of the cavity and projectile were revolved and their volumes were subtracted to obtain these values (see the inset in figure 20). The projectiles of $AR~4{-}8$ at $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ have very small (and almost constant) cavity sizes initially, but the size suddenly increases (figure 20 a). In these cases, the low impact velocity provides a momentum that is insufficient to create a well-defined cavity; thus, the initial cavity is small. As the cavity shrinks after the pinch-off, a portion of air bubble sticks to the tail, which is dragged by the falling body (at $t^{\ast }=4.0{-}4.6$ in figure 19 c). Consequently, the cavity size appears to increase suddenly. On the other hand, the projectiles of $AR~1{-}2$ have much larger cavities, which increase in size from the beginning. In these cases, the body occupies smaller portions of the cavity than in the cases of $AR~4{-}8$ . For $AR~1{-}2$ , the cavity expands to the peak, after which the air pressure becomes lower than the hydrostatic pressure of the surrounding water, and then shrinks gradually and necking occurs (figure 20 a). When $AR=1.0$ , the cavity is larger and necking occurs earlier than $AR~2$ .

When $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ (figure 20 b), the initial ( $t^{\ast }\lesssim 3.5$ ) cavity size is almost the same. At $t^{\ast }>4.0$ , the $V_{c}$ increases as $AR$ increases (except $AR~8$ ), because the additional air is pushed down to the cavity while the dome is formed above a free surface. As shown in figure 17(b), $H_{s}$ decreases at $t^{\ast }=4.0{-}7.0$ , while the splash crown collapses into a dome. As the splash converges, air is pushed around inside the cavity. $H_{s}$ decreases with increasing $AR$ , inducing cavity expansion. For the largest $AR$ of 8.0, both dome closure and cavity pinch-off occur on the side wall of the body, indicating that they occur earlier than they do with the other $AR$ values. Consequently, the cavity with $AR~8$ is less developed and small. The cavity is the smallest when $AR=1.0$ , because dome closure occurs the latest among the test cases, and the cavity contraction due to hydrostatic pressure is stronger than the expansion due to the dragged air.

Meanwhile, the instant at which the cavity size reaches the maximum occurs earlier as $AR$ decreases, except for $AR~8$ , agreeing with the observation that cavity contraction and pinch-off occur earlier as $AR$ decreases. After the cavity pinch-off, the Worthington jet pops up and the splash height increases ( $t^{\ast }=7.0{-}13.0$ in figure 17 b). In particular, the domes are considerably larger for $AR~2{-}4$ , because the cavity below the free surface is larger (figure 20 b). This finding agrees with the fact that $H_{s}$ increases to almost the same position at $t^{\ast }=12.5$ in figure 17(b) when $AR=1.0{-}4.0$ . To support this observation, we calculated the sum of the cavity and dome volumes, $V_{c}/D^{3}$ and $V_{d}/D^{3}$ , respectively, at $t^{\ast }=5.0$ , which has similar values of 11.0–12.5 for $AR~1{-}4$ .

Figure 21.

Airflow above a free surface during splash formation by a sphere when $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ . The splash has been masked, and the numbers in each image denote $t^{\ast }$ .

Table 3.

Measured values of $t_{d}$ , $t_{p}$ , $H_{p}$ , $H$ and $H_{p}/H$ when $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ .

Since the measurement repeatability was better for $U_{\circ }=4.2~\text{m}~\text{s}^{-1}$ , it was analysed further. We measured the instants of the dome closure ( $t_{d}$ ) and cavity pinch-off ( $t_{p}$ ). The positions of cavity pinch-off ( $H_{p}$ ) and the centre of the hemispherical head ( $H$ ), both measured from the free surface, and their ratio ( $H_{p}/H$ ) were also calculated (table 3). For $AR~1{-}4$ , dome closure occurs earlier but cavity pinch-off is delayed as $AR$ increases. Since the underwater speed of a sinking projectile increases with increasing $AR$ (see figure 23), $H$ increases, resulting in a decrease of $H_{p}/H$ . This behaviour is the same as that resulting from increasing the density of a sphere, as reported by Aristoff et al. (Reference Aristoff, Truscott, Techet and Bush2010). They showed that cavity pinch-off occurs faster and closer to the free surface as the sphere density decreases. Thus, it is understood that the effective density of a projectile increases with increasing $AR$ . For $AR~1{-}4$ , only the nose of the projectile touches the cavity, but the effective mass increases with increasing $AR$ . For $AR~8$ , $t_{d}$ and $t_{p}$ are strongly disturbed by the projectiles and are therefore the shortest. Since the dome and cavity dynamics are affected by the surrounding pressure, we measured the air velocity induced by the entry of a sphere ( $AR~1$ ) above the free surface (figure 21). It can be seen that a substantial amount of the surrounding air is dragged into the splash, due to the cavity expansion. Likewise, the velocity of the entrained air ( $v_{a}$ ) was measured (immediately above the splash) for other $AR$ values when the body tail passed the end of a splash. When averaged for $-0.35\leqslant r/D\leqslant 0.35$ , $v_{a}/U_{\circ }$ was found to be 0.14, 0.20 and 0.27 for $AR~1$ , $AR~2$ and $AR~4$ , respectively. According to the Bernoulli equation, the increased $v_{a}$ induces a lower pressure, i.e. a greater pressure difference across the splash crown; thus, the dome closure is accelerated, as discussed by Marston et al. (Reference Marston, Truscott, Speirs, Mansoor and Thoroddsen2016).

Figure 22.

Temporal variation of the vertical ( $z$ ) position of the centre of mass of the body: (a) $U_{\circ }=2.5~\text{m}~\text{s}^{-1}$ ; (b) $4.2~\text{m}~\text{s}^{-1}$ . The open and closed symbols correspond to smooth and rough surfaces, respectively. $z=0$ denotes the position of the undisturbed free surface.