3.1. Reduction in impact force

A body falling on a free surface experiences an impact force that peaks for a short duration. General force profiles for a flat disk (without a surface pattern) and a surface-patterned disk are shown in figure 2. Here, a 5 mm-thick flat disk of diameter $D = 50\ \textrm {mm}$ was used. For the surface-patterned disk, a 2 mm-thick perforated disk ($r = 1.8\ \textrm {mm}$ and $d = 4.2\ \textrm {mm}$) with the same $D$ was attached to the flat disk, giving the total thickness of 7 mm. They were dropped from a height of $H = 15\ \textrm {cm}$, resulting in an impact speed of $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$. Compared with the weight of the entire falling unit, the difference in weight between the flat disk and the surface-patterned disk is negligible, and thus it is assumed that the falling units have the same momentum. Here, $t$ = 0 is designated as the instant of free-surface impact, and the time at which the maximum impact force $F_{max}$ occurs is denoted by $t_{peak}$. The flat disk experiences $F_{max} = 193.8\ \textrm {N}$ at $t_{peak} = 0.575\ \textrm {ms}$, while the surface-patterned disk yields $F_{max} = 92.1\ \textrm {N}$ at $t_{peak} = 0.750\ \textrm {ms}$ (figure 2). The $F_{max}$ value of the surface-patterned disk is less than half that of the flat disk, and the peak appears at a later time. That is, the surface pattern notably reduces the maximum impact force. Although these two disks have different thicknesses, the comparison of the maximum impact forces is not affected by the disk thickness. We confirmed that the maximum impact force for the flat disk did not change when its thickness was increased to 7 mm. Furthermore, the acrylic disk of thickness 5 mm was rigid enough to avoid elastic deformation affecting the maximum impact force.

Figure 2. General impact force profiles for (a) a flat disk ($D = 50\ \textrm {mm}$) and (b) a surface-patterned disk ($D = 50\ \textrm {mm}$, $r = 1.8\ \textrm {mm}$, $d = 4.2\ \textrm {mm}$, $h = 2\ \textrm {mm}$) impacting a free surface at $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$. The impact duration $t_{imp}$ is defined as the time interval during which the impact force exceeds $0.2F_{max}$ and is discussed in § 3.4. High-speed images for the water entry of the two disks are provided in the supplementary movie 1 available at https://doi.org/10.1017/jfm.2021.123.

After the peak, the force profile shows fluctuations for both the flat and surface-patterned disks (figure 2). Owing to the limitations of our experimental set-up, the force sensor picks up oscillations in the rod arising from the impulsive load. That is, the fluctuations do not originate from flow phenomena such as hydrodynamic loading as a result of reflected shock waves. Residual oscillations after the peak impact force were also reported by Vincent et al. (Reference Vincent, Xiao, Yohann, Jung and Kanso2018), who measured the impact forces on wedges and noted that oscillations were due to the drop mechanism of the model. Because the initial peak of the impact force is our primary interest, subsequent fluctuations after the initial peak are not considered in the analysis of the magnitude and duration of the impact force.

The impact force was measured five times for each surface-patterned disk, and the average of the maximum impact force is presented in figure 3. In figure 3(a), the value of $F_{max}$ decreases with $d$ for the same $r$ (the same colour in figure 3a), with $D = 50\ \textrm {mm}$, $h = 2\ \textrm {mm}$ and $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$ kept constant. The total area of the holes in the surface pattern, $A_{hole}$, becomes larger as the side length $d$ of the holes increases, and the increase in $A_{hole}$ reduces the maximum impact force $F_{max}$. Furthermore, the $F_{max}$ values for different surface-patterned disks are similar as long as they have similar $A_{hole}$, despite different pattern details. In other words, the maximum impact force seems to be determined by the total area of the holes, regardless of the shape of the pattern. Here, the flat disk is considered as a disk with no holes, $A_{hole} = 0$ (see the black solid circle in figure 3). Compared with $F_{max}$ for the flat disk, which is 188.3 N, the surface-patterned disks have $F_{max}$ values that could be as low as 79.3 N (a 58 % reduction) (figure 3a). The attachment of a surface pattern to a flat disk consistently reduces the maximum impact force, and the magnitude of this reduction becomes greater with increasing $A_{hole}$.

Figure 3. Maximum impact force $F_{max}$ for different surface-patterned disks with $D = 50\ \textrm {mm}$ at $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$ as a function of $A_{hole}$. Here $F_{max}$ for a 5 mm-thick flat disk is located at $A_{hole}=0$ in black solid circles ($\bullet$). (a) Cases with the same $r$ are shown in the same colour; $h = 2\ \textrm {mm}$ in all cases. (b) Cases with the same $h$ have the same shape: circles, $h = 2\ \textrm {mm}$; five-pointed stars, 3 mm; six-pointed stars, 4 mm. For detailed information on the dimensions, see table 1.

Additional experiments were conducted by changing the hole depth $h$ for a given $A_{hole}$ to clarify whether the reduction in $F_{max}$ is simply due to a decrease in the area that is in contact with the free surface, $A_{pattern}$ (grey area in figure 1b, $({\rm \pi} D^2)/4 = A_{pattern}+A_{hole}$), or is caused by some complex flow phenomena that are induced by the holes. Although the surface-patterned disks have the same $d$ and $r$ (and thus the same $A_{hole}$), $F_{max}$ decreases with the increase in $h$ (figure 3b). A reduction in the maximum impact force of up to 42 % is achieved as $h$ increases from 2 to 4 mm for the same surface pattern (and the same $A_{hole}$). In other words, figure 3 demonstrates that both $A_{hole}$ and $h$ are important geometric parameters of the surface-patterned disks that determine $F_{max}$.

To address the combined effect of $A_{hole}$ and $h$, the maximum impact forces are plotted versus $V_{hole}/D^3$ for various surface-patterned disks, where $V_{hole} (= A_{hole} h)$ is the total volume of the holes created by the surface pattern (figure 4). Here, the coefficient of maximum impact force is introduced as $F_{max}/{\rho _w}cU A$, where $\rho _w = 998\ \textrm {kg}\,\textrm {m}^{-3}$ is the density of water, $c = 343\ \textrm {m}\,\textrm {s}^{-1}$ is the speed of sound in air, $U$ is the impact speed and $A = ({\rm \pi} D^2)/4$ is the frontal area of the disk. This non-dimensionalization comes from the theoretical approach for the free-surface impact of a flat-bottomed body. To predict the slamming pressure of a flat plate, von Kármán (Reference von Kármán1929) used momentum conservation to suggest that $p_{max}={\rho _w}c_wU$, in which the pressure rise propagates at the speed of sound in water, $c_w$. In a later study, Verhagen (Reference Verhagen1967) considered the effect of a thin layer of air trapped between the flat plate and the free surface at the moment of impact, and modified the relation to $p_{max}\sim {\rho _w}cU$, where $c$ is the speed of sound in air. In figure 4, the maximum impact force coefficients $F_{max}/{\rho _w}cU A$ collapse onto a single monotonically decreasing curve with respect to the dimensionless total volume of the holes, $V_{hole}/D^3$.

Figure 4. Coefficient of maximum impact force, $F_{max}/{\rho _w}cU A$, versus dimensionless total volume of holes, $V_{hole}/D^3$. The extreme case with the maximum hole volume is denoted by the black cross symbol ($\times$) for each of $h = 2$, 3 and 4 mm; the disk image in the figure is for $h = 3\ \textrm {mm}$. The cases of a 5 mm-thick flat disk, shown in black, are located at $V_{hole}/D^3 = 0$. The dashed line is a theoretical prediction from (3.14) derived in § 3.3 for the surface-patterned disks.

Let us consider the case of an extremely large hole. For special cases where $d$ is very close to $D$, only a 2 mm rim remains with a single large hole inside, giving the maximum hole volume for given disk diameter $D$ and hole depth $h$; e.g. the disk in the inset of figure 4. For each of $h = 2$, 3 and 4 mm ($D = 50\ \textrm {mm}$, $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$), the case with the maximum hole volume corresponds to each of the three black cross symbols located in $0.02 < V_{hole}/D^3 < 0.06$ (figure 4). The values of $F_{max}/{\rho _w}cU A$ in these extreme cases follow the same trend as those for the mesh-like surface-patterned disks. Note that the disk with the maximum value of $V_{hole}/D^3$ for given values of $D$ and $h$ has a flat surface inside the rim, and the only difference from the flat disk with $V_{hole}/D^3 = 0$ is the existence of the rim. Interestingly, despite this similarity in shape, the values of $F_{max}/{\rho _w}cU A$ are much smaller than those of the flat disks; compare two groups of data coloured black between $V_{hole}/D^3 =0$ (flat disk) and $0.02 < V_{hole}/D^3 < 0.06$ (disk with the maximum hole volume) in figure 4.

Figure 4 reveals that, regardless of the shape of the pattern, the total volume of the holes created by the surface pattern, $V_{hole}/D^3$, characterizes the reduction in maximum impact force, $F_{max}/{\rho _w}cU A$, in the case of a surface-patterned disk. To support this argument, we tested other regular and irregular surface patterns and obtained consistent experimental results; see the Appendix. In addition to the experimental measurements, the trend in figure 4 will be established by theoretical analysis in § 3.3.

3.2. Air entrapment

In this section, based on flow visualization, we present a qualitative explanation of why the maximum impact force on a surface-patterned disk decreases as the total volume of holes in the surface pattern increases. The air bubbles trapped inside the holes at the beginning of the impact process are mainly responsible for this result. From the visualization of the bottom and side of a disk impacting a free surface, we can observe the trapping of air inside the holes and identify the subsequent complex behaviour of the air bubbles (figure 5 and supplementary movies 2 and 3). After the moment of impact, the trapped air becomes visible as it escapes from the holes (figures 5aiii,aiv and 5bii,biii). As the disk descends farther inside the water, vented air bubbles remain beneath the disk, moving radially towards the edge of the disk. From the side view, the radial motion of the bubbles is clear; a bubble inside the red circle is tracked and observed to move to the edge of the disk in figure 5(biv–bvi). When the bubble reaches the edge of the disk, it joins the air cavity that begins at the disk edge and forms behind the disk; the bubble inside the blue circle in figure 5(biv,bv) disappears in figure 5(bvi).

Figure 5. Visualization of air bubbles beneath a surface-patterned disk upon water entry in (a) bottom view and (b) side view: $D = 50\ \textrm {mm}$, $r = 3.8\ \textrm {mm}$, $d = 4.2\ \textrm {mm}$, $h = 2\ \textrm {mm}$, and $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$. In the bottom view, the focal plane of the images is set near the free surface. Thus, in panels (av, avi), the images become blurred after the disk has passed the focal plane. See supplementary movies 2 and 3. The times are (ai) $t = 0.00\ \textrm {ms}$, (aii) $t = 0.45\ \textrm {ms}$, (aiii) $t = 1.15\ \textrm {ms}$, (aiv) $t = 1.80\ \textrm {ms}$, (av) $t = 11.90\ \textrm {ms}$, (avi) $t = 19.65\ \textrm {ms}$, (bi) $t = 0.35\ \textrm {ms}$, (bii) $t = 1.25\ \textrm {ms}$, (biii) $t = 4.00\ \textrm {ms}$, (biv) $t = 9.25\ \textrm {ms}$, (bv) $t = 15.00\ \textrm {ms}$, (bvi) $t = 20.60\ \textrm {ms}$.

For the correlation between the generation of impact force and the motion of trapped air bubbles during the initial impact phase, the force profile is analysed in accordance with the corresponding images of a surface-patterned disk (figure 6a). At the instant of impact (figure 6ai) and during the steep rise in the impact force (figure 6aii,aiii), air is compressed inside the holes. As the impact force reaches its maximum and begins to diminish, air bubbles start to escape from the edges of each hole (figure 6aiv). Thereafter, the bubbles continue to be vented from the holes (figure 6av,avi).

Figure 6. (a) Profile of impact force for a surface-patterned disk ($D = 50\ \textrm {mm}$, $r = 1.8\ \textrm {mm}$, $d = 10.2\ \textrm {mm}$, $h = 2\ \textrm {mm}$ and $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$) and corresponding images of the bottom surface focusing on four holes at the disk centre (bottom view). (b) Schematic diagrams of the dynamics of air bubbles inside the holes. The grey block and the white area denote a surface pattern and a single hole, respectively. Subpanels (i–vi) in panel (a) correspond to subpanels (i–vi) in panel (b). See supplementary movie 4 for panel (a).

To elucidate the dynamics of the air bubbles inside the holes, an advanced optical set-up is required for the clear visualization of the bubbles. Alternatively, in this study, we conjecture the characteristic motion of the trapped air bubbles (figure 6b) based on our experimental observations (bottom view) and the results of previous studies. As the disk continues to descend through the water immediately after the impact (figure 6bi), the patterned part (grey colour in figure 6b) acts locally as a flat plate of width $r$. It is well known that the pinning of a contact line is universally observed during cavity formation behind an impacting body (Truscott et al. Reference Truscott, Epps and Belden2014). Similar to the initial stage of air-cavity and splash formation in the impact of a flat plate (Mayer & Krechetnikov Reference Mayer and Krechetnikov2018), the pinning of the contact line at the corner of the pattern occurs upon impact, as shown in figure 6(bii,biii). Simultaneously, the water displaced by the patterned part moves into the hole, and therefore the air inside the hole becomes compressed. When the air pressure inside the hole reaches a certain value, the pressurized air penetrates the contact line at the corner of the pattern (figure 6biv). Afterwards, the air is depressurized and flows out of the gaps at the edge of the holes (figure 6bv), as observed in the bottom view (figure 6a). Once a significant volume of the trapped air has escaped from the hole, the hole is filled with water (figure 6bvi).

When the air bubble penetrates the contact line and vents out of the hole, it mostly escapes through the side that is farther from the centre of the disk, as demonstrated in figure 6(aiv–avi) (magnified views of the four holes at the disk centre). For a flat disk, the pressure at the centre of the surface is greater than that at the edge (Okada & Sumi Reference Okada and Sumi2000; Ma et al. Reference Ma, Causon, Qian, Mingham, Mai, Greaves and Raby2016). Similarly, for our model, we may assume that the pressure on the surface is not uniform, but decreases radially from its centre. Thus, as the pressurized air overcomes the water pressure inside the hole, the air bubble starts to vent through the side farther from the centre, where the pressure is lower.

The characteristic behaviour of trapped air bubbles in a surface-patterned disk (figures 5 and 6) differs notably from that of a flat-bottomed structure. It is well known that, when a flat-bottomed structure impacts a free surface, a thin layer of air becomes trapped and is broadly distributed. Verhagen (Reference Verhagen1967) confirmed this trapping of a thin layer of air below a flat plate using high-speed images, and considered the compressibility of air to explain the peak pressure exerted on the plate. The behaviour of the trapped layer of air beneath the flat plate was clearly visualized, with initial touchdown at the edge and collapse of air from the edge to the centreline (see Mayer & Krechetnikov (Reference Mayer and Krechetnikov2018), figure 24). Similar dynamics were also observed for a flat disk in which the trapped air layer had a circular shape and collapsed toward the centre of the disk (see Ermanyuk & Gavrilov (Reference Ermanyuk and Gavrilov2011), figure 4).

The impact force exerted on a surface-patterned disk is the spatial integral of pressure over the entire frontal area of the disk. This frontal area of the disk can be decomposed into $A_{pattern}$ and $A_{hole}$ (figure 1b). The pressure over $A_{hole}$ compresses the trapped air, imposing a pressure on the disk until the air eventually vents out at some point during the impact process. The compression of air contributes to a reduction in peak pressure, and therefore disks with a greater $A_{hole}$ (and $V_{hole}$) generally yield a reduced peak force for the same model conditions (figures 3 and 4). The role of confined air in reducing the peak pressure was also described by Mathai et al. (Reference Mathai, Govardhan and Arakeri2015), who found that a concave-nosed body exhibited a smaller peak pressure than a flat or convex-nosed body because of the presence of trapped air. In summary, our systematic experimentation has revealed that the compression and depressurization of trapped air bubbles during the initial impact phase leads to the rise and fall of the impact force and accounts for a reduction in the peak impact force.

3.3. Theoretical prediction of impact force

Although the previous sections described the effects of a surface pattern and trapped air inside the holes based on force measurements and flow visualization, theoretical analysis is also necessary to provide quantitative predictions of how the impact force is affected by the volume of holes on a surface-patterned disk. Previous studies on the water impact of a flat-bottomed body addressed the preimpact deformation of the free surface by the approaching body (Verhagen Reference Verhagen1967; Ermanyuk & Ohkusu Reference Ermanyuk and Ohkusu2005; Mayer & Krechetnikov Reference Mayer and Krechetnikov2018). The deformation of the free surface enables the entrapment of a thin, but widespread, air layer at the instant of impact, producing the air-cushioning effect. However, in our analysis of a surface-patterned disk, the preimpact deformation of the free surface is assumed to be negligible because of its minor magnitude; the volume of the entrapped air layer between the disk bottom surface and the free surface is so small compared with the total volume of the holes. That is, the free surface is assumed to be flat before the impact.

When the surface-patterned disk impacts the free surface, air is trapped inside the holes (figure 6). The impact force experienced by the surface-patterned disk is the spatial integral of the pressure of the water that the disk contacts as well as that of the trapped air inside the holes. Let $p_{hole}$ be the pressure on the holes, $p_{pattern}$ be the pressure on the pattern surface and $p_0$ be the atmospheric pressure. Then, the upward force $F$ exerted on the disk is

(3.1)\begin{align} F=\int p_{hole} \,\mathrm{d} A_{hole}+\int p_{pattern} \,\mathrm{d} A_{pattern}-p_{0}A=\bar{p}_{hole}A_{hole}+\bar{p}_{pattern} A_{pattern}-p_{0}A, \end{align}

where $\bar {p}_{hole}$ and $\bar {p}_{pattern}$ denote the mean pressure over the surface area for the holes and the patterned part, respectively. Here, we assume that the air pressure is uniform inside each hole and the water pressure on the patterned surface changes linearly between two nearby holes. Accordingly, the relation between $\bar {p}_{hole}$ and $\bar {p}_{pattern}$ is given as $\bar {p}_{pattern} = \bar {p}_{hole} = \bar {p}$, and the upward force $F$ reduces to

(3.2)\begin{equation} F=(\bar{p}-p_{0})A. \end{equation}

After the instant of impact, water enters the hole and compresses the trapped air. Although the trapped bubbles exhibit complex behaviour, we simplify the decrease in volume of the trapped air at each hole as an increase in the mean water penetration $x(t)$ with respect to the bottom surface of the pattern (figure 7a) and assume that $x(t)$ is uniform in all holes of the disk. Without a surface pattern, water on the disk–water interface accelerates to a downward speed of $U_v$, the same as the speed of the disk, soon after the impact. However, for a surface-patterned disk, some portion of the water moves upward into the holes through $A_{hole}$ at a speed of $\dot {x}(t)$ relative to the disk. That is, the interface in contact with the surface pattern moves downward with a speed of $U_v$, and the interface in contact with air in the holes moves downward with a speed of $U_v - \dot {x}$. The spatially averaged effective speed $U_{eff}$ of the water on the interface is then

(3.3)\begin{equation} U_{eff} = \frac{[(A-A_{hole}) U_v + A_{hole}(U_v-\dot{x})]}{A} = U_v-\frac{A_{hole}}{A}\dot{x}. \end{equation}

In other words, in comparison with the speed of the disk $U_v$, the effective (net) downward speed of water on the free surface is $U_v-A_{hole}\dot {x}/A$ due to the existence of the holes (figure 7b).

Figure 7. Schematic diagram after impact. (a) After the impact, air is trapped and compressed inside the hole. The mean depth of water penetration inside the hole is denoted as $x(t)$. (b) Added mass of the disk is pushed downward with an effective speed of $U_{eff}(=U_v-A_{hole}\dot {x}/A)$.

The peak impact force is generally explained in terms of the added mass of water that is accelerated by a solid object. The added mass of a flat disk upon water entry is expressed as $(\rho _w D^3)/6$ (Glasheen & McMahon Reference Glasheen and McMahon1996b). In this study, however, the presence of the surface pattern may alter the fluid behaviour beneath the disk, as discussed in § 3.2. Therefore, we express the added mass of the surface-patterned disk as $k \rho _w D^3$, where the unknown constant $k$ will be assigned later. With (3.2), the equation of motion for the added mass becomes

(3.4)\begin{equation} (\bar{p}-p_0)A= k\rho_w D^3 \frac{\mathrm{d}}{\mathrm{d}t}\left(U_v-\frac{A_{hole}}{A}\dot{x}\right)={-}k \rho_w D^3 \frac{A_{hole}}{A}\ddot{x}. \end{equation}

Because the falling unit is heavy and the phase for the first peak of the impact force is very short, typically $O$(1 ms) (figure 2b), the impact speed $U_v$ of the disk is assumed to be constant as $U$ during the impact phase: $\mathrm {d}U_v/\mathrm {d}t = \mathrm {d}U/\mathrm {d}t =0$. The initial conditions are $x(0)=0$ and $\dot {x}(0)=UA/A_{hole}$ because the water initially has zero speed: $U-A_{hole}\dot {x}(0)/A = 0$.

According to Verhagen (Reference Verhagen1967), air entrapped between a falling plate and a deformed free surface becomes compressed even before impact. Although we neglected the free surface deformation before the impact, air inside the holes may be compressed before the impact. Let a new variable $\bar {p}_c$ denote the pressure of air inside the holes right at the moment of impact. Because the air inside the holes is compressed very quickly as water penetrates, from $\bar {p}_c$ to $\bar {p}$ in (3.4), the adiabatic relation $\bar {p}_ch^\gamma = \bar {p}(h-x)^\gamma$ holds. Then, $\bar {p}=\bar {p}_c(1-x/h)^{-\gamma }\approx \bar {p}_c(1+\gamma x/h)$, and (3.4) becomes

(3.5)\begin{equation} (\bar{p}_c-p_0)+\frac{\gamma \bar{p}_c}{h}x+\frac{k\rho_w D^3}{A}\frac{A_{hole}}{A}\ddot{x}=0. \end{equation}

Solving (3.5) with the initial conditions, we obtain

(3.6a)\begin{gather} x(t)={-}\frac{\bar{p}_c-p_0}{\gamma \bar{p}_c/h}+\frac{\bar{p}_c-p_0}{\gamma \bar{p}_c/h} \cos(\omega t)+\frac{A}{A_{hole}}\frac{U}{\omega} \sin(\omega t), \end{gather}

(3.6b)\begin{gather}\dot{x}(t)={-}\frac{\bar{p}_c-p_0}{\gamma \bar{p}_c/h} \omega \sin(\omega t) + \frac{A}{A_{hole}}U\cos(\omega t), \end{gather}

where

(3.7)\begin{equation} \omega=\left(\frac{\gamma \bar{p}_c/h}{(k \rho_w D^3/A)(A_{hole}/A)}\right)^{{1}/{2}}. \end{equation}

Substituting (3.6a) into (3.4), the impact force $F$ is

(3.8)\begin{equation} F(t)=k \rho_w D^3 \frac{A_{hole}}{A} \left(\frac{\bar{p}_c-p_0}{\gamma \bar{p}_c/h}\omega^2\cos{\omega t}+\frac{A}{A_{hole}} U \omega\sin{\omega t} \right). \end{equation}

To determine the added-mass coefficient $k$, the impulse $I$ imposed on the surface-patterned disk is computed by integrating $F(t)$ over impact duration, $t = 0\text {--}T$:

(3.9)\begin{equation} I=\int_{0}^{T} F \,\mathrm{d}t = k \rho_w D^3 \frac{A_{hole}}{A} \left(\frac{\bar{p}_c-p_0}{\gamma \bar{p}_c/h}\omega\sin{\omega T}+\frac{A}{A_{hole}} U (1-\cos{\omega T}) \right). \end{equation}

As will be discussed in § 3.4, this impulse should be equal to that imposed on a flat disk at the instant of impact ($I_{flat}$ = $(\rho _w D^3 U)/6$). Because the impulse should be independent of $T$ and the time history of the impact force during impulse duration looks like the first half-period of a harmonic function according to our experiment (figure 2b), $T={\rm \pi} /\omega$ is a likely theoretical prediction of the impulse duration. The impulse $I$ is then given by

(3.10)\begin{equation} I=\int_{0}^{{\rm \pi}/\omega} F \,\mathrm{d}t = 2 k \rho_{w} D^3 U, \end{equation}

which results in $k = {1}/{12}$; $I$ should be equal to $I_{flat}$ (= $(\rho _w D^3 U)/6$).

When seen by an observer fixed on the ground, the free surface, which is initially in a stationary state, is accelerated by the disk to an effective downward speed of $2U(=U-A_{hole}\dot {x}({\rm \pi} /\omega )/A$) at $t = T(={\rm \pi} /\omega )$. When seen by an observer riding on the disk, the free surface approaches the observer with a speed of $U$ at $t=0$ and moves away from the observer with a speed of $U$ at $t = T(={\rm \pi} /\omega )$. In this sense, the role of the air bubbles inside the holes may be described as a spring. Before impact, the spring (air bubble inside a hole) absorbs a loading by an object (water) with an incoming speed $U$ relative to the spring, and at the end of the impact phase the object (water) moves in the opposite direction with a speed $U$ relative to the spring. Because the speed of the disk is $U$ in the ground reference, the effective downward speed $U_{eff}$ of the water eventually should become $2U$ in the ground reference. Thus, the equivalence of impulse between the flat disk and the surface-patterned disk implies that the added mass $(\rho _wD^3)/12$ of the surface-patterned disk is half that of the flat disk with no surface pattern, $(\rho _wD^3)/6$.

According to (3.8) and the relation between $F$ and $T$ such that $F(T) = F({\rm \pi} /\omega ) = 0$, $\bar {p}_c$ should be equal to $p_0$. This relation can also be obtained using our experimental data for maximum impact force. The maximum impact force corresponds to the maximum amplitude of the solution in (3.8), and the maximum impact force coefficient $F_{max}/{\rho _w}cU A$ is expressed as

(3.11)\begin{equation} \frac{F_{max}}{\rho_w cU A}=\left[\left(\frac{\bar{p}_c-p_0}{\rho_w c U}\right)^2 + \frac{\gamma \bar{p}_c}{12\rho_w c^2}\frac{D^3}{V_{hole}}\right]^{{1}/{2}}. \end{equation}

To determine $\bar {p}_c$ from our experimental data, the mean absolute percentage error is evaluated for a given $\bar {p}_c$ (figure 8a), which is defined as $\varSigma _{i=1}^{n} |(F_{{max, exp}, i}-F_{{max,theo},i})/F_{{max,exp},i}|/n$, where $n$ is the total number of experimental cases and $F_{{max, exp}, i}$ and $F_{{max, theo}, i}$ are the experimental and theoretical values, respectively, of the maximum impact force for each case $i$. In figure 8(a), the mean absolute percentage error is smallest at $\bar {p}_c/p_0=1$, which indicates that (3.11) is best fitted with our experimental data when $\bar {p}_c = p_0$ (figure 8b).

Figure 8. (a) Mean absolute percentage error between experimental and theoretical values of the maximum impact force $F_{max}$ as a function of $\bar {p}_c/p_0$. (b) Comparison between experimental and theoretical values of the maximum impact force coefficient $F_{max}/{\rho _w}cU A$ for $\bar {p}_c/p_0=1$.

Interestingly, the relation $\bar {p}_c = p_0$ shows that the air inside the gap between the free surface and the surface-patterned disk is not compressed, and maintains the atmospheric pressure before impact, unlike for the flat disk with no pattern. For the flat-disk case, the air in the gap begins to be compressed when the radial velocity at the rim of the disk is equal to the speed of sound before impact. Thus, the air pressure at the instant of impact increases significantly beyond $p_0$ (Verhagen Reference Verhagen1967). However, for the surface-patterned disk, the holes inside the surface pattern buffer the pressure increase of air between the disk and the free surface. The pressure inside the gap changes little before impact and can be approximated as $p_0$. Note that $\bar {p}_c=p_0$ means the air remains uncompressed before impact and does not contradict our previous argument that the air becomes compressed like a spring after impact.

With $\bar {p}_c = p_0$, (3.6a), (3.8) and (3.11) are simplified as follows:

(3.12)\begin{gather} x(t) =\frac{A}{A_{hole}}\frac{U}{\omega} \sin(\omega t), \end{gather}

(3.13)\begin{gather}F(t)=\frac{1}{12}\rho_w D^3 U \omega\sin{\omega t}, \end{gather}

(3.14)\begin{gather}\frac{F_{max}}{\rho_w cU A} = \left(\frac{\gamma p_0}{12\rho_w c^2}\right)^{{1}/{2}}\left(\frac{V_{hole}}{D^3}\right)^{-{1}/{2}}. \end{gather}

Although (3.12) and (3.13) are expressed in the form of an oscillation, the equations are only physically meaningful until the penetration height $x$ returns to zero at $t = T (={\rm \pi} /\omega )$. After $t = T$, $x$ is negative in (3.12), which means that the air has escaped from the holes, and our analysis becomes invalid.

The time history of the impact force acting on the disk is compared between the theoretical prediction (3.13) and the experimental measurement for two different cases (figure 9). For comparison, the peak of the theoretical curve is positioned at the same $x$ value (time) as that of the experimental curve. The theoretical prediction matches well with the experimental data if arbitrary phase shift is applied to the theoretical curve of (3.13). The phase shift is necessary because the rise in the experimental impact force is more gradual and lasts longer. The theoretical profile is symmetric with respect to its peak, whereas the profile obtained experimentally is asymmetric. Our theoretical approach might not capture important interface dynamics that determine the impact force at the early rise phase of the impact force. Future research should examine exactly how trapped air bubbles behave in the holes of a surface pattern and improve the current theoretical model to account for the asymmetric time history of the impact force. Furthermore, the maximum impact force coefficient (3.14) provides the dashed line in figure 4. Although we made several assumptions to simplify the problem, (3.14) is in good agreement with most of the experimental data for surface-patterned disks.

Figure 9. Time history of the impact force for a surface-patterned disk: (a) $V_{hole}/D^3 = 0.014$ ($D = 50\ \textrm {mm}$, $r = 1.8\ \textrm {mm}$, $d = 4.2\ \textrm {mm}$, $h = 2\ \textrm {mm}$ and $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$); (b) $V_{hole}/D^3 = 0.025$ ($D = 50\ \textrm {mm}$, $r = 1.8\ \textrm {mm}$, $d = 6.2\ \textrm {mm}$, $h = 3\ \textrm {mm}$ and $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$). The red dashed line is given by (3.13) with a phase shift that matches the phase of the peak.

3.4. Impact duration and impulse

We now consider the impact duration and impulse to examine the water impact of a surface-patterned disk from the perspective of momentum transfer. In the experimental results, the impact duration $t_{imp}$ is defined as the time during which the impact force exceeds $0.2F_{max}$ before and after its peak (figure 2b), and the impulse $I$ is calculated by integrating the impact force over the impact duration (see the shaded area in figure 2b). Because it was difficult to determine, with the necessary accuracy, the time at which the impact force becomes non-zero from the experimental results, a threshold was adopted to obtain the impact duration, and this was varied to find an appropriate value. The computed impulse was similar when the threshold value was between $0.1F_{max}$ and $0.3F_{max}$, but decreased sharply beyond 0.4$F_{max}$. The main arguments in this section are not affected when the threshold is between $0.1F_{max}$ and $0.3F_{max}$, and so $0.2F_{max}$ was chosen to define $t_{imp}$. Because of the difficulty in specifying an exact contact time in water impact problems, previous studies on the water entry of solid objects have tended to compute the impulse exerted on the objects using different time ranges (Huera-Huarte, Jeon & Gharib Reference Huera-Huarte, Jeon and Gharib2011; Ma et al. Reference Ma, Causon, Qian, Mingham, Mai, Greaves and Raby2016; Speirs et al. Reference Speirs, Belden, Pan, Holekamp, Badlissi, Jones and Truscott2019). In that the impulse refers to momentum loss by water impact, the dimensionless impulse is obtained by dividing the impulse by the momentum of the falling unit at impact, $MU$. Here $M$ is the total mass of the falling unit (dashed box in figure 1a), and this is set to be $M = 1.3\ \textrm {kg}$, regardless of which surface-patterned disk is used, because each disk accounts for less than 2 % of the mass of the falling unit.

In contrast to $F_{max}/{\rho _w}cU A$ (figure 4), the dimensionless impulse $I/MU$ is not strongly affected by $V_{hole}/D^3$ (figure 10a). That is, while the size of the disk is critical for the impulse, the shape of the surface pattern itself has no significant effect. For surface-patterned disks with $D = 50\ \textrm {mm}$, the values of $I/MU$ are mostly within a narrow range between 0.015 and 0.021, regardless of the shape of the surface pattern and the depth of the holes. Moreover, surface-patterned disks with $D = 30\ \textrm {mm}$ undergo much smaller momentum loss than those with $D = 50\ \textrm {mm}$, and exhibit an almost constant $I/MU \approx 0.005$. Note that $I/MU$ for the flat disk ($V_{hole}/D^3 = 0$) is also within the range of the surface-patterned disks in figure 10(a), which validates our approach using $I$ = $I_{flat}$ to determine the added-mass coefficient $k$ in § 3.3. The value of $I/MU$ depends on the disk diameter $D$ rather than $V_{hole}/D^3$. For a disk with larger $D$, a greater volume of water per unit disk area is displaced and accelerated by the impact, leading to an increase in the momentum loss. A similar result was reported by Glasheen & McMahon (Reference Glasheen and McMahon1996b). Note that the impulses in the present study (0.0098 N s $\leq I \leq 0.045\ \textrm {N}\,\textrm {s}$) lie in the same range as the results of Glasheen & McMahon (Reference Glasheen and McMahon1996b), who used flat disks with diameters between 25.4 and 61.4 mm and impact velocities from 0.562 to $3.11\ \textrm {m}\,\textrm {s}^{-1}$, which cover the ranges of $D$ and $U$ in our experimental models (table 1).

Figure 10. (a) Dimensionless impulse $I/MU$ versus $V_{hole}/D^3$ for two different disk diameters, $D = 50\ \textrm {mm}$ and 30 mm. (b) Maximum impact force coefficient $F_{max}/{\rho _w}cU A$ versus dimensionless impact duration $t_{imp}/(D/c)$. In panels (a,b), the dashed line shows the theoretical prediction, $I/MU =\rho _w D^3/6M$ and (3.15). The notation of the symbols is the same as in figure 4.

The impact duration $t_{imp}$ defined above for the experimental results corresponds to $T = {\rm \pi}/\omega$ theoretically, according to our analysis in § 3.3: $F(0)=F({\rm \pi} /\omega )=0$ from (3.13). The impulse during the period $t=0\text {--}{\rm \pi} /\omega$ is $I=(\rho _w D^3U)/6$ in (3.10). The corresponding dimensionless impulse $I/MU (=\rho _w D^3/6M)$ is approximately 0.016 and 0.003 for $D = 50\ \textrm {mm}$ and 30 mm, respectively, which slightly underestimate the experimental measurements (figure 10a). Nevertheless, (3.10) can capture the aforementioned trend based on our experimental results: $I/MU(=\rho _w D^3/6M)$ is independent of $A_{hole}$ and $V_{hole}$, but is affected by $D$ for a given mass of the falling unit.

From (3.13), the maximum impact force coefficient is expressed in terms of the impact duration $t_{imp}(\approx T ={\rm \pi} /\omega )$ as

(3.15)\begin{equation} \frac{F_{max}}{\rho_w c U A} = \frac{1}{3} \left(\frac{t_{imp}}{D/c}\right)^{{-}1}. \end{equation}

For the same impulse (same $D$ and $U$), the maximum impact force is inversely proportional to the impact duration. As predicted by (3.15), for our experimental data, $F_{max}/{\rho _w}cU A$ depends solely on the dimensionless impact duration $t_{imp}/(D/c)$, and seemingly collapses onto a single curve of (3.15) for all surface patterns ($d$, $r$ and $h$), disk diameters $D$ and impact velocities $U$ considered in this study (figure 10b). That is, the application of a surface pattern to the disk can reduce $F_{max}/{\rho _w}cU A$ by increasing $t_{imp}/(D/c)$.

In figures 4 and 10(b), we find that $F_{max}/{\rho _w}cU A$ decreases inversely with $V_{hole}/D^3$ and $t_{imp}/(D/c)$, respectively. From (3.14) and (3.15),

(3.16)\begin{equation} \frac{t_{imp}}{D/c} = \left(\frac{4\rho_w c^2}{3 \gamma p_0}\right)^{{1}/{2}}\left(\frac{V_{hole}}{D^3}\right)^{{1}/{2}}, \end{equation}

and the correlation between $V_{hole}/D^3$ and $t_{imp}/(D/c)$ is clearly shown in figure 11(a). A larger volume of trapped air inside the holes monotonically extends the impact duration. While $t_{imp}/(D/c)$ is approximately 3 for the flat disks ($V_{hole}/D^3 = 0$), it can increase up to 7 for the surface-patterned disks. Here, the theoretical curve for the surface-patterned disks underestimates the impact duration obtained from the experiment. This underestimation occurs because the impact force rises more moderately than predicted by the theoretical model as demonstrated in figure 9, and thus the theoretical impact duration serves as a lower bound for the experimental results. Regarding the time duration of the rise phase of the impact force ($t_{rise}$ in figure 9), the values of the experimental measurements are greater than the theoretically predicted values, whereas they exhibit much less discrepancy for the time duration of the fall phase ($t_{fall}$ in figure 9).

Figure 11. (a) Dimensionless impact duration $t_{imp}/(D/c)$ as a function of $V_{hole}/D^3$. The notation of the symbols is the same as in figure 4. The dashed line is given by (3.16). (b–d) Comparison of time at which the volume of the air bubbles that have escaped from the centre holes reaches a maximum for three surface-patterned disks with $V_{hole}/D^3$ = 0.008, 0.018 and 0.027, respectively ($D = 50\ \textrm {mm}$ and $U = 1.71\ \textrm {m}\,\textrm {s}^{-1}$).

To further examine the relation between the impact duration and the volume of trapped air inside the holes, we selected three experimental models with $D = 50\ \textrm {mm}$ and different values of $V_{hole}/D^3$: figure 11(b) for $V_{hole}/D^3$ = 0.008 ($r = 2.8\ \textrm {mm}$, $d = 3.2\ \textrm {mm}$, $h = 2\ \textrm {mm}$); figure 11(c) for $V_{hole}/D^3 = 0.018$ (1.8 mm, 8.2 mm, 2 mm); figure 11(d) for $V_{hole}/D^3 = 0.027$ (1.8 mm, 8.2 mm, 3 mm). We measured the time at which the volume of the air bubbles that had escaped from the holes near the centre of the disk after the depressurization process reached a maximum from the visualization images (figure 11b–d); $t/(D/c) = 0$ at the instant of impact. For the case of $V_{hole}/D^3 = 0.008$, it takes $t/(D/c) = 6.52$ for the trapped air to be vented. For the other values of $V_{hole}/D^3 = 0.018$ and 0.027, the venting of the trapped air is delayed: $t/(D/c)$ = 7.89 and 8.92, respectively. Although the time required for venting differs for each hole because of the non-uniform spatial distribution of pressure over the disk and the resultant spatial variation in the dynamics of air bubbles through the holes, a hole with a greater volume generally requires a longer time for the trapped air to be maximally vented. During the prolonged venting period, the trapped air inside the holes functions to cushion the impact, thereby extending the impact duration.

Using the intermediate parameter $t_{imp}/(D/c)$, we can now interpret the result in figure 4, where the increase in $V_{hole}/D^3$ leads to the monotonic decrease in $F_{max}/{\rho _w}cU A$. Neither the existence of the surface pattern nor its shape changes the magnitude of the impulse (figure 10a). Therefore, an extended impact duration would lead to a decrease in the maximum impact force because the impulse remains unchanged (figure 10b). Moreover, the amount of trapped air determines the impact duration (figure 11a). Increasing the volume of holes in the surface pattern allows more air to be trapped upon water impact, which extends the impact duration and subsequently reduces the maximum impact force (figure 4).