# The water entry surface seal behavior of spheres at intermediate speed
regimes

Akihito Kiyama1,2  Rafsan Rabbi1  Nathan Speirs3  Jesse Belden3  Yoshiyuki
Tagawa2,4  Tadd T. Truscott1,5<EMAIL_ADDRESS>1 Department of
Mechanical and Aerospace Engineering, Utah State University, Logan, UT 84322,
USA 2 Institute of Global Innovation Research, Tokyo University of Agriculture
and Technology, Fuchu, Tokyo 183-8538, Japan 3 Naval Undersea Warfare Center
Division Newport, 1176 Howell Street, Newport, Rhode Island 02841, USA. 4
Department of Mechanical Systems Engineering, Tokyo University of Agriculture
and Technology, Koganei, Tokyo 184-8588, Japan 5 Mechanical Engineering
Program, Physical Science and Engineering Division, King Abdullah University
of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia

###### Abstract

This research focuses on the water entry of spheres in the surface seal
regime. Herein, surface seal occurs in the wake of a sphere impact with the
water surface and is characterized by splash dome over and cavity pull-away
between the _low-speed_ ($U<30$ m/s) and _high-speed_ ($U>165$ m/s) regimes
for various sphere diameters $D$. The established empirical scaling laws for
pinch-off time $t_{p}$ hold well for both low and high speeds (i.e.,
respectively $t_{p}UD^{-1}~{}C_{d}^{-1/2}=const$ and
$t_{p}UD^{-1}~{}C_{d}^{-1/2}\propto Ca^{-1}$) but the transition deserves
refinement. The intermediate regime in particular scales as
$t_{p}UD^{-1}~{}C_{d}^{-1/2}\propto Ca^{-1/2}$, which means that the pressure
reduction inside the cavity determines the surface seal dynamics as speeds
increase. We also report on the transition from deep to surface seal, where
the seal time is scaled more like a deep seal. The partial pull-away, ripples
on the cavity surface, and the secondary pinch-off are also reported.

##  1 Introduction

Water entry of a sphere involves complex fluid motion (e.g., Truscott et al.
(2014)). In general, the sphere forms a slender cavity in its wake if the
sphere impacts the water surface fast enough (Duez et al., 2007). The cavities
are typically axisymmetric, though they can vary in shape and length. Aristoff
& Bush (2009) revealed that these cavities can be categorized into four types:
quasi-static, shallow, deep, and surface seals. Speirs et al. (2019) enlarged
the study revealing that the sphere-water wetting angle and surface roughness
play a large role in the determination of these cavity types (based on the
work of Duez et al. (2007); Zhao et al. (2014)).

Herein we focus on surface seal which occurs between _low_ and _high_ impact
speeds where the cavity seals off at the surface from the inward collapse of
the splash curtain. The cavity then follows the sphere deeper into the pool.
In particular, we look at the seal time, which is an important parameter for
understanding the physics behind the evolution of this cavity type. Surface
seal cavities are often analyzed by their time to seal, which is defined in
three ways. May (1952) was perhaps the first to film three noticeable moments
of surface seal, namely the time of “apparent” surface seal $t_{a}$ (defined
as the moment when any part of the splash starts to fall downwards inside the
cavity), the completion of surface seal $t_{c}$ (defined as the moment when
the splash fully collapses on itself, the cavity closes and the overall cavity
starts to move downward), and the cavity pull-away $t_{p}$ (defined as the
moment when the cavity separates from the free surface, see section 3).
Hereafter, we use $t$ as the general expression of the surface seal time,
while we use $t_{a},t_{c}$, and $t_{p}$ for denoting a specific time as
described.

Lee et al. (1997) classified surface seal based on impact speed $U$ into three
regimes: _low-speed_ ($U<30$ m/s), _high-speed_ ($U>165$ m/s), and _very high-
speed_ ($U>400$ m/s) with most of the past studies focused on the _low-speed_
regime. Gilbarg & Anderson (1948) performed a _low-speed_ surface seal
experiment up to $U\approx 30$ m/s and found an empirical relationship for the
time of surface seal

$t\propto\frac{D}{\rho_{g}U},$ (1)

where $\rho_{g}$ is the density of the surrounding gas. This scaling is
estimated using a Bernoulli assumption that the pressure drop inside the
cavity is due to the air inflow speed into the cavity that is assumed to be
equal to the sphere speed. Recently, Eshraghi et al. (2020) have estimated the
airflow speed by considering the rate of cavity expansion and found that
sphere speed (they tested speeds from 2.0 to 6.0 m/s) is not always a good
measure of the airflow speed. However, the empirical relationship proposed in
equation 1 has been used to predict the surface seal time for a wide range of
parameters. May (1952) and Marston et al. (2016) experimentally verified that
the empirical relationship for the apparent ($t_{a}$) and pull-away ($t_{p}$)
time of surface seal $tUD^{-1}\propto\rho_{g}^{-1}$ holds for various sphere
diameters and gas densities. Marston et al. (2012) found the surface seal time
to be $tUD^{-1}\approx 5.75$ under standard atmospheric conditions for spheres
heated up to the Leidenfrost temperature and velocities up to 7 m/s. One
exception in the _low-speed_ regime is surface seal that occurs near the
transition between deep and surface seal entry speeds. Eshraghi et al. (2020)
reported the established scaling law significantly underestimates the surface
seal time, which we also discuss in this paper.

More importantly, there are only a few studies on the surface seal time for
faster speeds ($U>30$ m/s). Lee (2000) proposed a scaling law that predicts a
break-down of equation 1 at faster sphere speeds. Lee (2000) considered the
characteristic cavity radius $R_{max}$ and the speed of the radial cavity
expansion $V_{r}$ based on the previous consideration (Lee et al., 1997) as

$t\sim 2\frac{R_{max}}{V_{r}}\propto\frac{\rho_{l}DU}{\Delta
P}~{}C_{d}^{1/2},$ (2)

where $C_{d}$ is the drag coefficient, see also Supplemental Information. In
brief, the model assumes that the pressure difference $\Delta
P~{}=P_{0}-P_{c}$ is the driving force of the surface seal, where $P_{0}$ and
$P_{c}$ are respectively the atmospheric pressure and the pressure inside the
cavity. For lower speeds, assuming $\Delta P$ increases as $\Delta P\propto
U^{2}$ based on the Bernoulli equation regarding the airflow, the model
predicts $V_{r}\propto U$ while $R_{max}\propto
D\sqrt{(\rho_{l}/\rho_{g})C_{d}}$ is insensitive to $U$ (see supplemental
information), and thus the surface seal time $t$ decreases as $U$ increases.
This is consistent with the established empirical scaling for _low-speed_
surface seal (equation 1). For faster speeds, the model assumes $\Delta P$
stops changing once the pressure inside the cavity reaches the liquid vapour
pressure $P_{v}$, and $V_{r}$ is not a function of $U$ anymore while $R_{max}$
increases as $U$ increases. The model implies that the surface seal time
switches from decreasing to an increasing trend at $U=51$ m/s (Lee, 2000) but
Abelson (1970) showed that this was not the case through pressure measurements
where they tried impact speeds of up to 76 m/s and observed a decreasing trend
of cavity pressure $P_{c}$. Further, Guo et al. (2020) also reported that
their data at $U$ = 80 to 470 m/s did not conform to the theory of Lee (2000);
Lee et al. (1997), but did suggest that there was a relationship between the
surface seal time and the Cavitation number
($Ca=(P_{0}-P_{v})/(\frac{1}{2}\rho_{l}U^{2})$). We clarify this issue through
a visual representation in figure 1, where equation 2 is plotted over a data
set of pull away surface seal time $t_{p}$ and impact velocity $U$. The
fitting is insensitive to $D$ but there is an inflection point where the
velocity is assumed to reduce the cavity pressure below the vapor pressure.
The data from this study and Guo et al. (2020) are overlaid to show that the
theory does a decent job at low velocities but does not account for the
transition between low and high speeds. The surface seal time of this
transition region (i.e., before the cavity is at vapor pressure) remains
worthy of study.

Figure 1: A comparison between the cavity pull-away time $t_{p}$ and sphere
impact speed $U$ for various sphere diameters $D$. Circles represent the water
entry of sphere, where the experimental conditions are described in section 2.
Squares represent the water entry of a cylinder data from Guo et al. (2020).
The solid line shows equation 2, where the input pressure $\Delta P$ is
switched at $U=34$ m/s and $C_{d}=0.384$ (Lee et al., 1997). A dashed line
shows an extended relationship $t_{p}\propto U^{-1}$. Vertical dotted lines
indicate _low-speed_ and _high-speed_ regimes.

Therefore, this work examines the surface seal at $U>30$ m/s. Our setup allows
us to employ the sphere impact speeds up to 128 m/s, which corresponds to the
transition regime between _low-speed_ and _high-speed_ per Lee’s definition
(Lee et al., 1997), where experimental data are lacking in the literature.
Surface seal time is experimentally evaluated through high-speed imaging
observations. As suggested in equation 2, the pressure difference $\Delta P$
plays an important role in determining the surface seal time. Using $Ca$ to
scale the non-dimensional surface seal time makes physical sense since
increasing the speed of entry towards total vaporization of the cavity should
be considered in the scaling approach. We discuss the influence of $\Delta P$
on the surface seal time, and show that the scaling laws (equations 1 & 2) are
connected to each other through different interpretation of $\Delta P$ in
different speed regimes which yields different $Ca$ based scaling.

This work fills the gap of experimental understanding up to the _high-speed_
surface seal regime and lays the groundwork for future studies in the _very
high-speed_ regime, where deep seal is predicted to occur rather than surface
seal (Lee et al., 1997).

##  2 Experimental Methods

Figure 2: The experimental set-ups used; (a) the free-fall system, and (b) the
air rifle set-up. The impact speed of the sphere was adjusted by changing drop
heights for (a), or the number of air pumps for (b). The drops heights for (a)
were set at up to five different levels for each of nine different sphere
diameters and the number of pumps for (b) were set at several different levels
for one sphere diameter. Illustrated definitions of time associated with (c) a
slower and (d) a faster surface seals, $t_{a}$, $t_{c}$, and $t_{p}$
(corresponding to the 4-6th frames in figure 3(a-c)). Black arrows indicate
the features associated with each definition (section 3).

Two different methods were employed for the varied impact speeds: a free-fall
system for the relatively lower impact speeds and an air rifle for the higher
speeds as shown in figure 2. The range of parameters are summarized in the
Supplemental Information.

The free-fall system is based on that used in Speirs et al. (2019) (figure
2(a)). Stainless steel spheres ($D$ = 1–18 mm, density
$\rho_{s}\approx~{}$7.8$\times 10^{3}$ kg/m3) were dropped from an
electromagnet at a given resting height, resulting in impact speeds from
$U=4.43$ to 10.39 m/s. Spheres are made hydrophobic by applying Glaco Mirror
Coat Zero, with an advancing static contact angle of $141^{\circ}$. Two high-
speed cameras (Photron SA-3) are used to film both splash and cavity dynamics
at 2,500 frames per second (fps).

The air rifle setup (figure 2(b)) is based on that used in Kiyama et al.
(2019). We first use copper-coated spheres (diameter $D\approx$ 4.5 mm,
density $\rho_{s}\approx~{}$6.8$\times 10^{3}$ kg/m3) to shoot from an air
rifle (Crossman 760) to the water surface, where the surface seal dynamics are
filmed by a high-speed colour camera (Phantom v2510) at 100,000 fps and a
resolution of $\approx$ 0.1 mm/pix. The sphere impact speed ranges from
$U=8.13$ to 128 m/s controlled by the number of air pumps. The spray coating
was not applied for faster speed impacts ($U\geq 30$ m/s), since it was
reported that the surface property does not play a significant role in the
cavity dynamics in the high-speed conditions (May, 1951; Duez et al., 2007;
Zhao et al., 2014). We also capture the overall cavity dynamics with a typical
frame rate of 50,000 fps and resolution $\approx$ 0.3 mm/pix in a separate
experiment. In addition to the copper-coated spheres ($D~{}\approx 4.5$ mm),
we employ two different smaller stainless steel spheres ($D=~{}2.5$ & $3.5$
mm, $\rho_{s}~{}\approx~{}$7.8$\times 10^{3}$ kg/m3).

The image processing including adjustment of the image brightness/contrast was
performed through ImageJ. The surface seal times were determined manually
based on the definitions described in section 3.1. The sphere impact speed $U$
for the free-fall set-up was estimated by considering the drop height of the
sphere (Speirs et al. (2019)), while that for the air-rifle set-up is
calculated by adopting a custom Matlab code that determines velocity from
sphere position before impact (Epps et al., 2010). For the estimated
uncertainty in the image-based measurement, see Supplemental Information.

##  3 Results and discussion

###  3.1 Surface seal observations

Figure 3: Surface seal water entry of three spheres ($D=4.5$ mm) at
velocities: (a) $U=9.58$ m/s, (b) $U=82.2$ m/s, and (c) $U=128$ m/s. $t=0$ is
set to the moment when the camera first detects the sphere penetrating the
free surface. Arrows point to the residual of the mist-like splash as
discussed in section 3. Supplemental movies 1-3 correspond to (a-c).

We begin by investigating high-speed imagery of surface-seal cavity creation
at different speeds. Typical surface seals for three speeds $U=$ 9.58, 82.2,
and 128 m/s of a $D=4.5$ mm sphere are shown in figure 3 (see also
Supplementary videos). For the slower speed (figure 3(a)) a part of the splash
sheet starts to fall down when the splash dome has almost closed (1.96 ms).
The splash crown closes on itself at 2.69 ms and then the cavity detaches from
the free surface at 4.11 ms, where the downward vertical jet forms inside the
cavity. In contrast, the higher velocities alter the time of surface closure
and introduce different features (e.g., Shi et al. (2000)). The higher
velocities introduce a mist-like splash and the misty jet inside the cavity
(figure 3(b)). This is more pronounced in the fastest case (figure 3(c)) where
the mist also occurs right above the free surface (indicated by the black
arrows). The conical bubbly regions between the main cavity and free surface
(see the last frame of figure 3(b) & (c)) suggests that the air-fluid mixture
is still rushing into the cavity, even if the cavity has pulled away from the
free surface.

Classification of surface seal cavity types begins by looking at the times
associated with the closure features as sketched in figure 2(c & d). We define
the “apparent” surface seal $t_{a}$ as the moment when any part of the splash
sheet start to flow downward corresponding to the airflow direction marked by
a single arrow. The “complete” seal $t_{c}$ occurs when the splash crown domes
over meeting itself at the central axis. The dome often starts to move
downwards at this moment (double arrows). The cavity “pull-away” $t_{p}$
occurs when the air cavity descends below the original free surface level. The
4th, 5th and 6th frames of figure 3 respectively correspond to the times
$t_{a},t_{c}$ and $t_{p}$ outlined in figure 2(c & d).

We note that in the literature, “complete” seal (i.e., splash dome-over) is
often considered as the indicator of surface seal and “pull-away” is not
(Marston et al., 2012, 2016; Eshraghi et al., 2020). Regarding our images,
figure 3 indicates that the cavity “pull-away” $t_{p}$ is the easiest to
measure; whereas the other two ($t_{a}$ & $t_{c}$) are relatively challenging
to measure from these side-view images. Therefore, we focus on the cavity
“pull-away” time $t_{p}$ in this manuscript. We provide values for other two
times ($t_{a}$ & $t_{c}$) in supplemental information.

###  3.2 Time of surface seal for varying sphere speeds and diameters

Figure 4: The dimensionless time associated with the surface seal
$t_{p}UD^{-1}C_{d}^{-1/2}$ as a function of (a) $Ca$, while $t_{p}UD^{-1}$ is
plotted against Weber number $We$ in (b). The size of the markers corresponds
to sphere size. Circles represent the data taken by using the set-up shown in
figure 2(a) and diamonds the set-up shown in figure 2(b). Sphere diameters
smaller than the capillary length ($l_{c}\approx 2.7$ mm) are represented by
filled markers. Squares represent Guo et al. (2020) data, where we assume
$P_{0}=101.325$ kPa. In (a), the vertical solid and dot-dashed lines
correspond to $Ca=1$ and $Ca=6.1\times 10^{-3}$ for a sphere. The horizontal
dot-dashed and dotted lines represent the conventional scaling
($t_{p}UD^{-1}C_{d}^{-1/2}=17.6\pm$3.2) for spheres ($C_{d}=0.32$). The error
bar reflects variation of $C_{d}$ values for diamonds (see SI). The inclined
dashed and inclined solid lines show the slopes of equations 3.1 & 3.2
respectively. The solid vertical line in (b) indicates the transition
threshold from deep seal to surface seal ($We=640$, Aristoff & Bush (2009))
and the dashed horizontal line indicates the conventional scaling
$t_{p}UD^{-1}=9.97$. The inset shows a parameter space on Weber $We$ \- Bond
$Bo$ numbers.

Our experimental data show that there is a transition regime where the $t_{p}$
scaling proposed by Gilbarg & Anderson (1948) (for low speed, equation 1) and
Lee (2000) (for high speed, equation 2) does not explain intermediate impact
speeds (figure 1). In figure 4, the experimental results of $t_{p}UD^{-1}$ are
shown as a function of dimensionless parameters $Ca$ and $We$ (figures 4(a &
b)). We also present high-speed experimental data taken from Guo et al.
(2020), who investigated the water entry of a cylinder ($U=80-470$ m/s, $D=6$
mm diameter). Figure 4(a) indicates that the dimensionless surface seal time
$t_{p}UD^{-1}C_{d}^{-1/2}$ for spheres is a constant for $Ca>1$. Note that few
data points for $Ca>1$ do not fit the trend because they have low Bond numbers
as discussed later. A clear separation of data at $Ca\sim O(1)$ suggests that
the vaporization of water alters the pull-away mechanism. Here we consider the
scaling law proposed by Lee et al. (1997); Lee (2000), where they considered
the cavity pressure $P_{c}$ as a constant once it reaches the liquid vapour
pressure $P_{c}=P_{v}$. Substituting $\Delta P=P_{0}-P_{v}$ into equation 2
leads to the relationship,

$\displaystyle\frac{t_{p}U}{D}\propto\frac{\rho_{l}U^{2}}{P_{0}-P_{v}}~{}C_{d}^{1/2},$
(3) $\displaystyle\frac{t_{p}U}{D}C_{d}^{-1/2}\propto Ca^{-1},$

which scales similar to the data from Guo et al. (2020) (inclined dash-dot
line in figure 4(a), where the best fit for $Ca<0.01$ was
$t_{p}UD^{-1}C_{d}^{-1/2}\propto Ca^{-0.89}$, see SI for $C_{d}$ values used).

Though pressure measurement in our system was not feasible, we found that
Abelson (1970) measured the pressure inside the cavity during the water entry
of a conical-head projectile (76.2 mm diameter) up to 76.2 m/s. They reported
that the pressure inside the cavity $P_{c}$ starts to drop when the projectile
speed exceeds 10.4 m/s and $P_{c}$ decreases linearly up to 76.2 m/s.
Substituting $\Delta P\propto U$ (Abelson, 1970) into equation 2 leads to
another scaling,

$\frac{t_{p}U}{D}~{}C_{d}^{-1/2}\propto Ca^{-1/2},$ (4)

which scales well with our data (inclined solid line of figure 4(a), where the
best fit for $0.01<Ca<0.1$ was $t_{p}UD^{-1}C_{d}^{-1/2}\propto Ca^{-0.45}$).
Extending the linear relationship until $P_{c}$ reaches zero pressure, Abelson
(1970) predicted the approximate critical velocity of 163 m/s or $Ca\sim
6.1\times 10^{-3}$ (vertical dot-dashed line) where a transition to _high-
speed_ occurs (i.e., between the solid line of the _intermediate speed_
spheres and the dashed line of _high-speed_ cylinders).

Figure 4(b) presents the same data as a function of $We$, indicating that the
time of pull-away $t_{p}UD^{-1}$ is divided into three regimes. First, the
scaling law ($t_{p}UD^{-1}$=const.) holds well for the range $2\times
10^{3}<We<3\times 10^{4}$, where we found the averaged value
$t_{p}UD^{-1}=9.97\pm 1.80$ (marked by the horizontal line). All the data in
this regime had sphere speed $U<27$ m/s. This is consistent with the condition
in which no significant cavity pressure drop was recorded by Abelson (1970)
(up to 10.4 m/s). However, the scaling law ($t_{p}UD^{-1}$=const.) does not
work well outside of this range.

At $We$ between 640 and 2000, $t_{p}UD^{-1}$ decreases as $We$ increases. This
range corresponds to the few outlier cases with low Bond numbers in the $Ca>1$
regime. At $We=640$ surface seal occurs at nearly the same time as deep seal
pinch-off resulting in larger $t_{p}UD^{-1}$. As $We$ increases the inertial
effects become more important than surface tension and the $t_{p}UD^{-1}$
values eventually merge to the constant empirical scaling as expected ($We$
between $2\times 10^{3}$ and $3\times 10^{4}$). The overall trend is
consistent with the data reported in Eshraghi et al. (2020), who studied the
surface seal dynamics at slower speeds (up to 6 m/s) and discussed the
contribution of the airflow to the surface seal time. They reported the dome
over occurred at around $We=1.7\times 10^{3}$. This slight increase of surface
seal time might be qualitatively explained in terms of $\Delta P$ as well. A
cavity at $We=640$ likely experiences both the surface and deep seals at the
same time. The airflow could be blocked as the deep seal progresses, resulting
in relatively higher pressure inside the cavity and thus smaller $\Delta P$
than that predicted by the Bernoulli approach. The surface seal takes longer
to complete as predicted by equation 2.

At faster speeds ($We>3\times 10^{4}$), the pull-away time $t_{p}UD^{-1}$
increases as $We$ increases. This regime largely corresponds to $Ca<1$ in
figure 4(a), suggesting that the presence of some water vapor alters the
surface seal dynamics.

###  3.3 Partial pull-away of the cavity and the ripples on the cavity surface

The partial pull-away is observed when the splash becomes a mist of small
droplets as mentioned in the last frames of figure 3(b) & (c). This phenomenon
is triggered because of the emergence of the misty splash sheet upon the fast
ejection of a thin splash sheet. A well-known pull-away is accompanied by the
contiguous splash sheet (e.g., figure 3(a)), with clear separation between the
cavity and the free-surface apparent after pull-away happens. In contrast, the
misty splash is not contiguous, with air filaments connecting the main cavity
to the surface upon sheet folding (figure 5). The airflow can enter into the
cavity through the filaments until they finally collapse on themselves. We
qualitatively observed that the faster sphere speeds can increase the size or
the amount of air bubbles entrapped in the conical filaments connecting the
main cavity and the surface (e.g., figure 3). The size of the conical portion
might also depend on the amount of air bubbles brought from the surface by the
low cavity pressure as marked by the arrows in figure 5(a).

The texture seen on the cavity walls (figure 5(b)) comes from the impact of
small droplets that have broken off of the collapsed splash crown (also seen
by Mansoor et al. (2014)). Each impact forms small, approximately
hemispherical cavities that propagate out radially from the impact sites.
These impacts begin near the top of the cavity, where droplets reach first,
then move down toward the sphere. At faster speeds, the crown breaks up into
numerous finer droplets that create a rough texture on the cavity wall.

Figure 5: (a) Partial pull-away process for $D=4.5$ mm and $U=124$ m/s. The
time interval is 0.1 ms. Arrows indicate the air bubbles elongating from the
free surface towards the main cavity. (b) Development of the ripples on the
cavity wall for $D=18$ mm and $U=7.67$ m/s. The ripples emanate from droplets
from the collapsed splash crown. The time interval is 0.4 ms.

###  3.4 Secondary pinch-off

Figure 6: Secondary pinch-off of surface seal cavities for (a) $D=4$ mm,
$U=$4.43 m/s, $Fr=5.0\times 10^{2}$, $Ca=8.5$, (b) $D=4$ mm, $U=$6.26 m/s,
$Fr=1.0\times 10^{3}$, $Ca=4.3$, (c) $D=4.5$ mm, $U=$80.7 m/s, $Fr=1.5\times
10^{5}$, $Ca=2.6\times 10^{-2}$, (d) $D=4.5$ mm, $U=$126 m/s, $Fr=3.6\times
10^{5}$, $Ca=1.1\times 10^{-2}$. (e) The dimensionless pinch-off time of the
secondary pinch-off $t_{d}~{}\tau^{-1}$ (corresponds to the 6th frames in
(a-d)) as a function of Froude number $Fr$. Supplemental movies 4-7 correspond
to (a-d).

Lastly, we report the dynamics of the secondary pinch-off (i.e., deep seal
after surface seal), labeled as $t_{d}$. A slower impact leads to surface seal
followed by deep seal about half way between the sphere and top of the cavity
(see figure 6(a), $t=26.4$ ms). At higher speeds, the impact introduces
different pinch-off dynamics. The cavity does not collapse at half of its
length but near the very top (between frames $t=11.2$ & $13.6$ ms in figure
6(b)). The cavity tail shrinks radially and then collides with the vertical
jet inside the cavity, perhaps due to the reduced cavity pressure. The trend
holds for much faster speeds as shown in figures 6(c & d). Although the
overall appearance at their pinch-off moments ($t=3.46$ ms in (c) and 3.82 ms
in (d)) are similar in shape to the slower case (figure 6(a)), the pinch-off
dynamics are different. The upper portion of the cavity collapses further and
forms residual microbubbles. Since the cavitation number for figures 6(c & d)
is much smaller than unity, the residual microbubbles may contain not only the
streaming air but also water vapour.

Figure 6(e) shows $t_{d}$ measured from the time of sphere impact normalized
by $\tau=2.06\sqrt{D/(2g)}$ (Duclaux et al., 2007). The time
$t_{d}~{}\tau^{-1}$ is close to unity for smaller $Fr$ values as expected but
decreases as $Fr$ increases (also seen by Mansoor et al. (2014)) until $Fr\sim
1.0\times 10^{4}$.These findings reveal that the reduced cavity pressure makes
the secondary pinch-off events depart from typical deep seal behavior as $Fr$
increases.

##  4 Conclusion

Surface seal dynamics between the _low-speed_ and _high-speed_ regimes
($U=4.43$ to 128 m/s) reveal a strong connection to the decreasing pressure
inside the cavity as the water vapour becomes a larger factor in the seal
behavior with increasing speed. We define the surface seal time by the “pull-
away” phenomenon $t_{p}$ and noticed that the dimensionless behavior is more
easily scaled by the cavitation number $Ca$. We found that the established
semi-empirical scalings hold well for low speeds,
$t_{p}UD^{-1}C_{d}^{-1/2}=const$ for $Ca>1$ ($t_{p}UD^{-1}=9.97\pm 1.80$ for
$2.0\times 10^{3}<We<3.0\times 10^{4}$), and high speeds,
$t_{p}UD^{-1}C_{d}^{-1/2}\propto Ca^{-1}$ where $Ca<6.1\times 10^{-3}$ for
cylinders (Guo et al., 2020). However, we found the scaling breaks down in the
intermediate range and suggest that $t_{p}UD^{-1}C_{d}^{-1/2}\propto
Ca^{-1/2}$ at intermediate speeds by assuming $\Delta P\propto U$ as measured
by Abelson (1970). The conventional Bernoulli approach, which describes the
overall cavity pressure, is not a good measure to scale the surface seal
phenomenon. Instead, as speeds increase beyond $Ca<1$ the cavitation number
must be considered, indeed even the recent data of Guo et al. (2020) indicates
that this is so. The cavity dynamics including the partial pull-away (figure
5(a)), ripples on the cavity surface (figure 5(b)), and the secondary pinch-
off (figure 6) were also discussed. Increasing sphere speed $U$ also alters
the mechanism of the secondary pinch-off from typical deep seal behavior.

##  5 Acknowledgments

A.K. came up with this research while at TUAT. A.K. is currently at USU as
JSPS Overseas Research Fellow. Y.T. acknowledges the financial support from
JSPS KAKENHI Grant No. 20H00223. T.T.T. & R.R. acknowledge a small portion of
the funding from the Office of Naval Research, Navy Undersea Research Program
(Grant No. N000141812334). Declaration of Interests: the authors report no
conflict of interest.

## References

  * Abelson (1970) Abelson, H. I. 1970 Pressure measurements in the water-entry cavity. Journal of Fluid Mechanics 44 (1), 129–144.
  * Aristoff & Bush (2009) Aristoff, Jeffrey M & Bush, John W M 2009 Water entry of small hydrophobic spheres. Journal of Fluid Mechanics 619, 45–34.
  * Duclaux et al. (2007) Duclaux, V, Caillé, F, Duez, C, Ybert, C, Bocquet, L & Clanet, C 2007 Dynamics of transient cavities. Journal of Fluid Mechanics 591, 177–19.
  * Duez et al. (2007) Duez, Cyril, Ybert, Christophe, Clanet, Christophe & Bocquet, Lydéric 2007 Making a splash with water repellency. Nature Physics 3 (3), 180–183.
  * Epps et al. (2010) Epps, Brenden P, Truscott, Tadd T & Techet, Alexandra H 2010 Evaluating derivatives of experimental data using smoothing splines. In Proceedings of Mathematical Methods in Engineering International Symposium. MMEI, Lisbon Portugal, pp. 29–38.
  * Eshraghi et al. (2020) Eshraghi, Javad, Jung, Sunghwan & Vlachos, Pavlos P 2020 To seal or not to seal: The closure dynamics of a splash curtain. Physical Review Fluids 5 (10), 104001.
  * Gilbarg & Anderson (1948) Gilbarg, David & Anderson, Robert A 1948 Influence of Atmospheric Pressure on the Phenomena Accompanying the Entry of Spheres into Water. Journal of Applied Physics 19 (2), 127–139.
  * Guo et al. (2020) Guo, Zitao, Chen, Tuo, Mu, Zhong Cheng & Zhang, Wei 2020 An investigation into container constraint effects on the cavity characteristics due to high-speed projectile water entry. Ocean Engineering (210), 107449.
  * Kiyama et al. (2019) Kiyama, Akihito, Mansoor, Mohammad M, Speirs, Nathan B, Tagawa, Yoshiyuki & Truscott, Tadd T 2019 Gelatine cavity dynamics of high-speed sphere impact. Journal of Fluid Mechanics 880, 707–722.
  * Lee (2000) Lee, Minhyung 2000 Water-entry induced cavity pressure. KSME International Journal 14 (5), 562–568.
  * Lee et al. (1997) Lee, M, Longoria, R G & Wilson, D E 1997 Cavity dynamics in high-speed water entry. Physics of Fluids 9 (3), 540–550.
  * Mansoor et al. (2014) Mansoor, M M, Marston, J O, Vakarelski, I U & Thoroddsen, S T 2014 Water entry without surface seal: extended cavity formation. Journal of Fluid Mechanics 743, 295–326.
  * Marston et al. (2016) Marston, J O, Truscott, T T, Speirs, N B, Mansoor, M M & Thoroddsen, S T 2016 Crown sealing and buckling instability during water entry of spheres. Journal of Fluid Mechanics 794, 506–529.
  * Marston et al. (2012) Marston, J O, Vakarelski, I U & Thoroddsen, S T 2012 Cavity formation by the impact of Leidenfrost spheres. Journal of Fluid Mechanics 699, 465–488.
  * May (1951) May, Albert 1951 Effect of Surface Condition of a Sphere on Its Water-Entry Cavity. Journal of Applied Physics 22 (10), 1219–1222.
  * May (1952) May, Albert 1952 Vertical Entry of Missiles into Water. Journal of Applied Physics 23 (12), 1362–1372.
  * Shi et al. (2000) Shi, H H, Itoh, M & Takami, T 2000 Optical observation of the supercavitation induced by high-speed water entry. Journal of Fluids Engineering-Transactions of the Asme 122 (4), 806–810.
  * Speirs et al. (2019) Speirs, Nathan B, Mansoor, Mohammad M, Belden, Jesse & Truscott, Tadd T 2019 Water entry of spheres with various contact angles. Journal of Fluid Mechanics 862, R3.
  * Truscott et al. (2014) Truscott, Tadd T, Epps, Brenden P & Belden, Jesse 2014 Water Entry of Projectiles. Annual Review of Fluid Mechanics 46 (1), 355–378.
  * Zhao et al. (2014) Zhao, Meng-Hua, Chen, Xiao-Peng & Wang, Qing 2014 Wetting failure of hydrophilic surfaces promoted by surface roughness. Scientific Reports 4 (1), 33–5.