3.1. Air film entrapment

As the disc approaches the water surface with a finite speed, there is an intervening air layer trapped between the two. The air in this layer is squeezed out such that its flow results in a radial pressure gradient. These pressure changes are sufficiently large to deform the water surface before the disc makes any contact with it. Thus the water surface is made aware of the approaching disc in advance; at the point that lies directly under the disc centre, the liquid surface is pushed down by the stagnation pressure in the air phase. Conversely, under the disc edge, a high hydrodynamic pressure in the air creates a suction which destabilises the water surface to be sucked upwards towards the disc (Jain Reference Jain2020; Jain et al. Reference Jain, Gauthier, Lohse and van der Meer2021a). As a result, at the first moment that the disc makes contact with the water surface, the water–air interface under the disc edge is vertically separated from that under the disc centre by approximately 200 ${\rm \mu}$m. This ‘air-cushioning’ effect on the free surface invariably creates conditions to trap an air layer under the disc at its first contact with the liquid surface, as is shown in figures 2 and 3.

Figure 2. The trapping of an air film on a water-entering disc is directly seen from below the water bath at $V = 0.06$ m s$^{-1}$ in (a), and at 0.1 m s$^{-1}$ in (b). The disc shown is 8 cm wide (as shown in figure 1b). The time labels are centred at 0 ms when the disc makes it first contact with water. The trapped air film contracts inwards due to surface tension of the water–air interface. The process is also shown in supplementary movies 1 and 2 available at https://doi.org/10.1017/jfm.2021.846. The film's retraction dynamics is discussed in Appendix A.1.

Figure 3. Snapshots of the free surface during the impact of a 12 cm wide disc at 0.5 m s$^{-1}$, recorded at 50 k f.p.s. using the TIR set-up, as shown in figure 1(a), imaging the free surface from below. The free surface is pushed down before the disc makes contact with it, which is evident from (a). The time labels shown at top left corner are centred about $t=0$ at the instant where the disc makes first contact with the deformed water surface. An air layer is trapped on the disc at impact. It collapses inwards as the disc plunges further into the liquid bath (b) and is finally punctured at the centre by the liquid (c). The puncturing process is shown in movie 4 and discussed in further detail in Appendix A.2. More examples using different parameters are shown in movies 5 and 6.

At low impact velocity $V$, such as shown in figure 2 at 0.06 and 0.1 m s$^{-1}$, after entrapment, the air layer merely contracts inwards on account of the high surface tension of the air–water interface. It is observed that, as the surrounding water displaces air, its perimeter stays remarkably stable and consistent in shape. This is due to the interface being stabilised by the Saffman–Taylor mechanism, which is only unstable to perturbations when a less viscous liquid displaces a more viscous one. Note how at a slightly higher $V$ of 0.1 m s$^{-1}$ in figure 2(b) the air film is briefly ruptured as it passes over the sensor at the disc edge, but then continues to contract inwards, preserving the interface shape to remain as a bubble at the disc centre. The inward retraction of the film is due to the high surface tension of the air–water interface, and occurs with a constant velocity, as is discussed in more detail in Appendix A.1). Our observations of how such a trapped air bubble behaves are consistent with the observations of Mayer & Krechetnikov (Reference Mayer and Krechetnikov2018), who used a flat rectangular plate as the impactor.

At a somewhat higher impact velocity $V = 0.5$ m s$^{-1}$ (figure 3) the trapped air film is shown at the first moment of impact in panel (a), using the TIR technique shown in figure 1(a). With this technique, all parts of the water surface that turn black indicate where the free surface has disappeared either due to wetting of the disc, or due to being ejected upward out of the initially quiescent liquid surface's plane. The moment when the disc makes first contact with the water surface can be identified when a section of the reflecting water surface disappears (i.e. turns black) by wetting the disc. In panel (b), it is observed that the air layer initially starts to collapse towards the disc centre while the disc plunges further into the liquid bath, until, in panel (c), the air bubble is punctured in the centre due to the high pressure that occurs in the stagnation point that has formed in the liquid. This particular observation is further addressed in Appendix A.2, where we present the measured puncture times. The current and additional examples of this process can be seen in movies 4–6.

For sufficiently high impact velocities, after the air layer is punctured, it is observed to be expelled from below the disc. This happens when the hydrodynamic forces that push the air outwards are stronger than the capillary forces that try to keep the air layer together.

Taking a look at the dimensionless numbers that may play a role during the impact, we note that the growth of the small cavity of thickness of order $100\ {\rm \mu}{\rm m}$, which is formed prior to impact on the free surface, is mainly driven by the stagnation pressure in the air phase at the disc's symmetry axis. The formation of such a cavity is resisted by both surface tension and inertia of the liquid. The Weber numbers $\textit {We} = \rho _{w} V^2 R/\sigma$ that concern both the entrapment of the air film, and the subsequent impact, range between approximately $1$ and $10^4$ (using $R=1\text {--}8$ cm and $V = 0.05\text {--}3$ m s$^{-1}$; $\rho_w$ and $\sigma$ are the density and surface tension of water). The Froude number $\textit {Fr} = V^2/gR$, with g the acceleration due to gravity, lies in the range between $0.006$ and $92$. Note that, in the pre-impact stage, the liquid inertia limits free surface deformation. In this particular stage prior to impact, the effective Froude number is $\textit {Fr}\rho _{{gas}}/\rho _{{water}}$, which is much smaller than the uncorrected Froude and Weber numbers at impact. Shortly after impact, when a large slamming force is created on the impactor, the liquid experiences much larger acceleration than gravity, making the Froude number irrelevant (Iafrati & Korobkin Reference Iafrati and Korobkin2004; Peters et al. Reference Peters, van der Meer and Gordillo2013; Mayer & Krechetnikov Reference Mayer and Krechetnikov2018).

In the next subsection, we will discuss the pressures that are measured during impact by the two pressure sensors in the $R=4.0$ cm steel disk of figure 1. Before starting this discussion, we want, however, to stress the difference in time scales in the buildup of the pressure on the one hand (${\ll }1$ ms), and that of the contraction (${\sim }100$ ms) and the puncturing (${>}1$ ms) and sometimes subsequent expulsion (${\sim }10$ ms) of the entrapped air layer on the other. This implies that, during pressure buildup, the air film can be viewed as largely stationary and covering most of the disc.

3.2. Pressure measurements

The effect of impact on pressure characteristics is found to be typical of solid–liquid impact events in general (Peregrine Reference Peregrine2003; Kapsenberg Reference Kapsenberg2011). The first contact between a transducer on the disc edge and water surface creates a large pressure spike (figure 4a,b), lasting for a short duration ${\sim }{O}(10^{-4}\,\text {s})$. Note from figure 4(a,b) how, in examples shown for both a low velocity impact at $V = 0.075$ m s$^{-1}$ and a larger velocity of 1 m s$^{-1}$, the pressure at the disc centre starts to grow prior to that at the edge. This is due to a stagnation pressure building up in the liquid at the disc centre, whose magnitude is of the order of $\rho _w V^2$. This can be expected, since the free surface under the disc responds to the pressure buildup before the disc comes into contact with the water. The magnitude of such a pressure buildup will exist in an approximate balance with the high pressures needed to accelerate the fluid via the cushioning layer. Another important observation from (a) is that, while the time taken for the air film to retract from the edge sensor to the one at the centre is ${\approx }185$ ms (see also figure 2, Appendix A.1, and supplementary movies 1 and 2), the time gap between the pressure peaks at the two sensors is of the order of 0.1 ms (also shown in the inset figure 4b). We can conclude that the both the pressure buildup and the attainment of the peak pressure at the centre are certainly mediated by the air cushion.

Figure 4. Time series of pressures measured at the two locations on an 80 mm wide disc (see figure 1b) when it impacts on a deep liquid bath at a velocity of (a) 0.075 m s$^{-1}$ and (b) 1.0 m s$^{-1}$. The signals are centred about $t=0$ at the instant when the central pressure reaches its peak. (c) Peak pressures measured at two locations on the same disc are plotted over a range of impact velocities. Error bars show the standard deviation of peak pressures over 3–6 repetitions of the experiment at each impact velocity. The peak pressures at the centre are found to scale quadratically with $V$, whereas those near the edge of the disc are found to be slightly below those at the centre. The impact peak is always attained first at the edge sensor, irrespective of where a higher peak pressure is attained. The inset of (b) highlights this by showing the time difference $\Delta t_{{c-e}} = t_{centre} - t_{edge}$ between the times $t_{centre}$ and $t_{edge}$ at which the peak pressure is attained at the centre and the edge, respectively.

While it is clear that the first contact at the disc's edge creates the first peak pressure, it is not immediately clear where the bubble's outer perimeter lies, and what shape it has when the peak pressure at the centre (a mere ${\sim }{O}(10^{-4}\,\text {s})$ later, inset figure 4b) is registered. The time delay $\Delta t_{c-e}$ between the peaks at the edge and centre is much shorter than the time span in which the trapped air retracts (${\sim }{O}(10^{-1}\,\text {s})$, also see figure 10), or over which it ruptures and fragments at higher velocities (${\sim }{O}(10^{-3}\,\text {s})$, also see figure 11). In addition, the impact velocity does not clearly affect $\Delta t_{c-e}$.

An oscillating component in the pressure signals with a period of ${\sim }4$ ms is visible at all times in (a), and after the impact in (b) in figure 4. This arises from structural oscillations in the disc–rod assembly, and is also present in the data from the load cell (figure 7). The natural frequency of this impactor assembly (with the masses of rod and disc combining to approximately 502 grams) was measured with the load cell to be $236.8\pm 30.5$ Hz, corresponding to a period of approximately 4.2 ms.

Peak pressures measured from impacts done at a range of velocities are plotted in figure 4(c). Multiple repetitions of the experiment at each velocity were performed to quantify reproducibility. At low velocities ($V \lessapprox 0.5$ m s$^{-1}$ in figure 4(c) for the present parameter $R=4$ cm), pressures at the edge of the disc were found to be higher than those at the disc's centre. The opposite, however, was observed in impacts at higher velocities. Thus, while at low velocities the impact pressures at the disc's centre are actively ‘cushioned’ by the trapped air film, for $V \gtrsim 0.5$ m s$^{-1}$, the air film appears to play a diminished role in reducing the peak pressure at the disc centre. Note that peak impact pressures at the disc centre are approximately 100 times larger than the stagnation pressure $\rho _w V^2$ in water and that they are found to scale quadratically with $V$, as expected for an inertial impact process (see figure 4c). Okada & Sumi (Reference Okada and Sumi2000) performed impact tests using a flat plate, and also found higher peak pressures closer to its centre. We thus designate impact velocities ${\gtrsim }0.5$ m s$^{-1}$ as ‘slamming’ velocities throughout the present work. At slamming velocities, the initially trapped air film does not persist – either getting punctured by the stagnation pressure (${\sim }{O} (10^{-3}\,\text {s})$ after impact, see Appendix A.2), or fragmenting prior to being violently expelled out from under the disc (${\sim }{O} (10^{-2}\,\text {s})$ after impact, movie 3). It is to be emphasised here that the threshold $V \approx 0.5$ m s$^{-1}$ (corresponding to $\textit {We} \approx 278$) is only intended to differentiate between low-speed impacts (where the entrapped air bubble contracts) and slamming impacts (where the entrapped air bubble is punctured and expelled) in the case of the $R=4.0$ cm disk.

3.3. Localised pressure impulses during impact

The most elementary measure of the intensity of an impact event is the peak pressure that is measured on the solid structure. However, in most circumstances, this peak pressure can have a large variance between tests performed at the same parameters. Since rise times of the initial impact peak can be very short ${\sim }{O}(10^{-5}\,\text {s})$, one reason for the variance in peak pressures is often due to the limited time resolution of the sensors. Another reason very often encountered is the inconsistent role played by the ambient air that is trapped at impact. Since the first such work by Bagnold (Reference Bagnold1939) studying the effect of trapped air on peak pressures, several later works (Denny Reference Denny1951; Chan, Melville & Gaster Reference Chan, Melville and Gaster1988; Peregrine Reference Peregrine2003) have also identified that the trapped air can play an important role during the production of the initial large pressure peak. Richert (Reference Richert1969), Partenscky (Reference Partenscky1989) and Hattori et al. (Reference Hattori, Arami and Yui1994) found from wave impacts on walls that the most severe pressures were produced when there were several small air bubbles, or a very thin ‘lens shaped’ air pocket trapped along the wall. It is a general consensus that pressure impulse, or the pressure integrated over duration of the peak, is a better predictor of the amount of local damage that is caused by an impact event on the solid structure (Kirkgöz Reference Kirkgöz1990; Hull & Müller Reference Hull and Müller2002; Peregrine Reference Peregrine2003; Bullock et al. Reference Bullock, Obhrai, Peregrine and Bredmose2007; Bredmose et al. Reference Bredmose, Peregrine and Bullock2009; Dias & Ghidaglia Reference Dias and Ghidaglia2018). Indeed, pressure impulse is also found to be a much more reproducible measure of the intensity of impact than the peak value in itself (Bagnold Reference Bagnold1939; Denny Reference Denny1951; Chan et al. Reference Chan, Melville and Gaster1988; Peregrine Reference Peregrine2003; Bullock et al. Reference Bullock, Obhrai, Peregrine and Bredmose2007; Dias & Ghidaglia Reference Dias and Ghidaglia2018).

Thus we calculate the pressure impulses $\varPi (t)$ from the measurements done at slamming velocities. They are computed by numerically integrating the pressure signal p from a time before it starts to rise $t_0$, to arbitrarily later times as

(3.1)\begin{equation} \varPi(t) = \int_{t_0}^{t} p(t')\,\textrm{d} t'. \end{equation}

Consistent with all the pressure measurements, the time coordinate of all pressure-impulse results are again centred about $t=0$, at the instant when the central pressure reaches its maximum. This way of presenting the results allows one to focus on the early-time rise in impulse until the trapped air film collapses at the disc's centre.

Pressure impulses measured at $V = 1.25$ and 2.5 m s$^{-1}$ are plotted in figure 5. At the disc centre, $\varPi (t)$ starts to rise earlier than at the disc's edge. This can be ascribed to an observation made earlier in figure 4(a), where it was seen that the pressure at the disc centre starts to rise before that at the edge. The difference in how the impulse at the edge grows compared with that at the centre occurs due to different behaviour of air flow in the two regions at impact. At the moment of impact, there is a trapped air film whose thickness is maximum under the disc centre. The opposite holds true at the disc edge, where the contact first happens. At slamming velocities, after the peak impact pressure is attained, the air film collapses. Immediately after this, the air that was initially trapped is fragmented and expelled from under the disc. As its escape happens from under the disc edge, fast air flow, followed by fast liquid flow, interfere with the pressure reading. That the impulse shown in figure 5 initially accumulates more smoothly at the centre than at the disc's edge suggests that impact at the disc centre is cushioned by the air layer. Indeed, the presence of air also prolongs the duration of the initial impact peak, which contributes to the impulse at the centre growing to larger values than that at disc edge (Iafrati & Korobkin Reference Iafrati and Korobkin2008).

Figure 5. Pressure impulses $\varPi$ measured from experiments at (a) $V = 1.25$ m s$^{-1}$ and (b) $V = 2.5$ m s$^{-1}$ are plotted as time series. Experiments were done with the sensor arrangement as shown in figure 1(b). As before, all time series are centred about $t=0$, the moment when the pressure at disc centre reaches its maximum.

The air film plays this role of a cushioning layer only in the earliest stages of impact before the peak pressure at disc centre reaches its maximum. Thus, whatever subsidiary effect is caused by it, such as mitigating pressure singularities, distributing the loading and prolonging the impact, lasts for a short duration of a few milliseconds prior to $t=0$ in figure 5. This stage is key to determining the final value of the impulse that is accumulated during the first impact peak. Noting that, inertially, pressure should scale as $\rho _w V^2$, and time as $R/V$, we expect the pressure impulse $\varPi (t)$ to scale as the product of the former. Using this inertial scaling, we non-dimensionalise the results in figure 5(a) together with data for other values of $V$ and re-plot them in figure 6. The growth of the pressure impulse at the centre prior to the peak at $t=0$ s is very well collapsed by the non-dimensionalisation. Thus, similar to the growth of stagnation pressure on the free surface under the disc's centre before impact, the air-mediated early-stage loading on the disc centre is an inertial process. This allows us to conclude that, at the disc centre, the impact impulse arises mainly due to sudden acceleration of a certain bulk of liquid by the impact. It leads us to expect that the impulse created by total force on the disc's entire surface should behave similarly to the pressure impulses as in figure 6. This indicates that early-time loading on the pressure sensor grows as it would have had the pressure sensor already been in touch with, or immersed in, the liquid. This finding is in keeping with those of Ross & Hicks (Reference Ross and Hicks2019) and Moore (Reference Moore2021), who found using a viscous air-cushioning analysis, that the initial (up to a time span of the order of $10^{-6}$ s, compared with an order of $10^{-4}$ s for our experiments) trapped volume of air moves with the same velocity as the impactor. In the added-mass sense for the surrounding (target) liquid, this trapped cushioning layer essentially acts as an extension of the solid body.

Figure 6. Non-dimensional pressure impulses $\hat {\varPi } = \varPi /V R \rho _w$ measured at the disc centre (from figure 5) are plotted against non-dimensional time $t V /R$. The convincing data collapse in the region of interest (below $t V/R=0$ in the early stages of impact loading) reveals that the accumulation of the pressure impulse at the disc centre, when it impacts at high velocities, can be described as an inertial process.

In fact, if for our disc impact experiment one estimates the time it takes for the entrapped air pocket to reach pressures of the order of the stagnation pressure in the liquid, after which the liquid will be able to start moving, one obtains values smaller than $10\,{\rm \mu}$s (see Appendix A.3). Since this pressurisation time is much shorter than the build-up time of the load, the latter will be dominated by what is happening in the liquid, which connects to the inertial scaling that was observed.

4.1. Force measurements and impulse calculation

We perform force measurements on the discs to check to what extent they are consistent with the pressure measurements. Some typical force measurements are shown in figure 7(a). The main features of interest around the impact peak are the same as were found in the pressure measurements. The first large peak results from the initial impact. Here, however, the forces measured by the load cell are averaged throughout the disc and rod assembly that is mounted on the linear motor. As a result, structural vibrations in the impactor dominate the measured signals shortly after $t=0$, and drown several weaker signals that might be of interest.

Figure 7. (a) Time evolution of the impact force measured using the disc shown in figure 1(b), impacted at various velocities. The time series are centred about $t=0$ at the time of registering peak impact force. The natural frequencies of the impacting structure used here were measured with the respective load cells. They ranged from $313.2 \pm 55.5$ Hz for the lightest disc, to $236.8 \pm 30.5$ Hz for the heaviest attachment. (b) Peak average pressure, $\tilde {P}_{{peak}} = F_{{peak}}/{\rm \pi} R^2$ computed from the force measured in experiments using discs of varying sizes, materials and impact speeds are plotted against their velocities of impact. Data shown using purple coloured markers are from experiments impacting steel discs, yellow from impacting 3-D printed plastic discs, and green markers from experiments using aluminium discs. Peak average pressures are found to deviate substantially from the previously found quadratic dependence on $V$ for the centre and (to a lesser extent) the side pressures presented in figure 4(c).

Force measurements were done using discs of different sizes and materials. The variation of peak forces with $V$ is shown in figure 7(b). As with pressure measurements, force measurements at each set of parameters were repeated a number of times to obtain variance in the measurements. They are found to vary substantially from the quadratic scaling that peak pressures were found to follow. Instead, peak forces appear to grow as $\sim V^{1.54}$ within the experimental range of errors. Since experiments done with discs made of steel, aluminium and plastic follow a consistent trend, we can conclude that the total forces produced on the discs indeed have a hydrodynamic origin. They do not depend on the material that a disc is made from. The peak pressure deviation from a $V^2$ scaling appears likely due to limited temporal resolution of the strain gauge load cell.

However, the challenge in resolving the peak in flat plate impacts is more indicative of the very short time scales involved in this particular impact scenario, rather than only the limited resolution of the equipment. For objects such as a wedge, a cone or a sphere entering water, the wetted surface expands in time. The impact force increases with the increase in wetted area. Space-averaged (Wagner Reference Wagner1932) and Logvinovich (Korobkin Reference Korobkin2004) models for the prediction of force on a water-entering object with a non-zero deadrise angle take into account the wetting rate to predict the impact force's time evolution. Force measurements by Vincent et al. (Reference Vincent, Xiao, Yohann, Jung and Kanso2018) show how increasing the deadrise angle of the wedge increases the duration of the impact peak. El Malki Alaoui et al. (Reference El Malki Alaoui, Nême, Tassin and Jacques2012) also find well-resolved force measurements in impacting cones with deadrise angles varying in the range $7\text {--}30^{\circ }$ while using an accelerometer with a natural frequency of 2500 Hz, and compared successfully with calculations by Shiffman & Spencer (Reference Shiffman and Spencer1951). Similarly impact drag coefficients measured by Moghisi & Squire (Reference Moghisi and Squire1981) in the liquid entry of a sphere are both well resolved and compared successfully with calculations by Shiffman & Spencer (Reference Shiffman and Spencer1945). Another indication of how the time scales at the first impact peak are affected by the curvature of the impacting face was provided by Ermanyuk & Gavrilov (Reference Ermanyuk and Gavrilov2011) by performing slamming experiments with discs with flat, convex and concave surfaces. Ermanyuk & Gavrilov (Reference Ermanyuk and Gavrilov2011) associate the greatest accelerations, and correspondingly the shortest time scales in the pressure time series, with the water impact of a disc with flat surface, followed by the convex and concave surfaces, respectively. Finally we comment that ignoring air-cushioning effects, a finite curvature of the impacting surface necessarily dilates the duration of the initial added-mass induced force peak.

However, coming back to the present experiments, and notwithstanding the limited resolution of load cell, we can numerically integrate the force measurements starting from the time when the first impact peak starts to rise. The force impulses

(4.1)\begin{equation} \mathcal{F}(t) = \int_{t_0}^{t} F(t')\,\textrm{d} t', \end{equation}

thus computed are non-dimensionalised as before using inertial length and time scales. The results are plotted in figure 8. As with pressure impulses, the growth of force impulses until the time of peak force (conveniently at $t=0$) is well collapsed by the re-scaling. Our expectation from previous section (based on figure 6) is upheld by force-impulse measurements.

Figure 8. Non-dimensional force impulses $\mathcal {F}/R^3 V \rho _w$ are computed from force measurements with a steel disc of radius $R = 4$ cm, impacting on water at various $V$. Each result is centred about $t=0$, the time when the corresponding impact force attains its peak value. The horizontal dashed line is drawn to mark the non-dimensional force-impulse value of 4/3. Inset: a force–time curve at $V=1$ m s$^{-1}$ is shown; the shaded area shows the time duration over which $F(t)$ is integrated to obtain $\mathcal {F}$.

After $t=0$, the structural oscillations (signals with a period of 4 ms, visible both in figures 4(a,b) and 7(a)) become very large, and depending on the impactor's extent of submergence in the bath, are affected to varying degrees by the bath's sloshing. Notice also in the inset of figure 8, that there is a secondary peak in the force, which cannot be easily ascribed to any hydrodynamic phenomenon. This explains why the later force signals and their resulting impulses do not follow hydrodynamic scales.

4.2. Impulse due to water impact of a disc

That, in our experimental set-up, both the pressure impulses, $\varPi (t)$, and force impulses, $\mathcal {F}(t)$, obey inertial scaling despite the presence of a cushioning air layer, is an important observation as regards to comparing the measurements with theory. Here, we start with a brief discussion of the well-known idea of pressure impulse in a fluid. The idea is useful when fluid boundaries, or a portion of the bulk, is subjected to a sudden acceleration of a large magnitude and short duration. Over this short duration, velocities of both the fluid and the boundary are small. Ignoring viscosity, Euler's equation, linearised in the velocity field $\boldsymbol {v}$, can be used to describe fluid motion (Cooker & Peregrine Reference Cooker and Peregrine1995)

(4.2)\begin{equation} \frac{\partial \boldsymbol{v}}{\partial t} ={-}\frac{1}{\rho_{{w}}} \boldsymbol{\nabla} p, \end{equation}

where both the terms are manifestly of large magnitude for a short interval of time, in which the acceleration term is dominant over the terms in Euler equation. Integrating both sides with time,

(4.3)\begin{equation} \boldsymbol{v}_{{after}} - \boldsymbol{v}_{{before}} ={-}\frac{1}{\rho_{{w}}} \boldsymbol{\nabla} \left(\int_{t_{before}}^{t_{after}} p(t')\,{\rm d}t'\right), \end{equation}

which can be integrated and re-written in terms of a scalar flow potential $\phi$ (with $\boldsymbol {v} = \boldsymbol {\nabla } \phi$) as

(4.4)\begin{equation} \phi_{after} - \phi_{before} = \frac{\varPi(t_{{before}}) - \varPi(t_{{after}})}{\rho_w}. \end{equation}

This expresses that, in a fluid that is initially stationary, any motion produced by sudden acceleration will, at least for a short time, be irrotational. In such a fluid domain, flow potentials throughout the bulk are uniquely determined by the normal component of velocities imposed at its boundaries. The pressure-impulse calculation that corresponds exactly to the present experiment with the impact of a disc shown in Batchelor (Reference Batchelor1967, § 6.10), yielding $\mathcal {F}_{{impact}} = 4/3R^3V\rho _{{w}}$. The same result was also arrived at in Lamb (Reference Lamb1945, § 102) by deriving the kinetic energy that is imparted to the surrounding fluid by an immersed disc that is impulsively accelerated from rest. Glasheen & McMahon (Reference Glasheen and McMahon1996) performed experiments impacting a disc on water, and measured the loss in its momentum from impact by measuring the sudden drop in its velocity (4.3). From the loss in the projectile's momentum, they estimated the added mass on the moving disc. The results were compared with the above analytical result (also in Birkhoff & Zarantonello Reference Birkhoff and Zarantonello1957), finding a good agreement.

Note that our method of computing the impact impulse differs significantly from that of Glasheen & McMahon (Reference Glasheen and McMahon1996), who could calculate the force impulse by measuring the sharp drop in projectile momentum. Their method would not be suited in our case due to the amount of control in our experiment that allows us to impose a constant velocity. In contrast, we numerically integrate the directly measured force and local pressure over the duration of the impact peak.

Although in essence both the methods to measure $F$ relate to the added-mass coefficient being measured, the difference between the two approaches can be understood by using the following expression for the inertial force on the disc in terms of the depth of penetration $h(t)$ into the liquid phase, as derived by Iafrati & Korobkin (Reference Iafrati and Korobkin2011),

(4.5)\begin{align} F &= m_a \rho_w R^3 \ddot{h} + \rho_w {R^2} \dot{h}^2\left[a_1\left(\frac{h}{R}\right)^{{-}1/3} + a_2 \log \left(\frac{h}{R}\right) + a_0 \right] + O(1)\nonumber\\ &\approx m_a \rho_w R^3 \ddot{h} + a_1\rho_w {R^2} \dot{h}^2 \left(\frac{h}{R}\right)^{{-}1/3}, \end{align}

where $\dot {h}$ and $\ddot {h}$ denote the first and second time derivatives of $h(t)$. Here, the first term represents the added mass (with $m_a = 4/3$) and the second term the build up of quadratic drag, between which the first contribution (with $a_1 \approx 2.7250$) is the dominant one for small $h$. Note that the quadratic drag term is diverging and therefore dominant for small $t$. Although (4.5) was explicitly derived for $t \geq 0$, such that $\dot {h} (0) = V$, we make the reasonable assumption that $\dot {h}(0^-) = 0$, and assuming the expressions’ integrability, time integrate the dominant contribution to (4.5) from $t=0^{-}$ just before impact, until some time $t=t_1$ after impact, yielding the impulse

(4.6)\begin{equation} \mathcal{F} = \int_{t = 0^-}^{t_1} F\,\textrm{d} t \approx \rho_w R^3 \left[m_a\dot{h}(t_1) + \frac{3}{2} a_1\int_{t = 0^-}^{t_1} \dot{h}\,\textrm{d}\left((h/R)^{2/3}\right)\right]. \end{equation}

One may estimate the second term by noting it is always positive and, since $\dot {h} \leq V$ we have

(4.7)\begin{equation} \frac{3}{2} a_1\int_{t = 0 ^ -}^{t_1} \dot{h}\,\textrm{d}\left((h/R)^{2/3}\right) \leq\frac{3}{2}a_1 V \int_{t = 0 ^ -}^{t_1}\,\textrm{d} \left((h/R)^{2/3}\right) = \frac{3}{2}a_1\left(\frac{h(t_1)}{R}\right)^{2/3}, \end{equation}

that is, in contrast to its dominance in the expression for the force for small $t$ (4.5), the contribution of the quadratic drag term to the impulse is vanishingly small for small $h$, and the impulse is dominated by the added-mass contribution, $\mathcal {F} \approx m_a \rho _wR^3 \dot {h}(t_1)$.

Clearly, when the velocity is kept constant, we find that $\mathcal {F} \approx m_a \rho _wR^3 V$ in the inertial approximation, and one would expect the impulse to quickly converge to $m_a \rho _wR^3 V$. Any deviation would be due to secondary effects, such as liquid compressibility, which accounts for the growth of the added mass, and takes place on a time scale $R/c_w$, with $c_w \approx 1.5\times 10^3$ m s$^{-1}$ the speed of sound in water. In our experiments, this time scale is of the order of several tens of microseconds.

However, when the velocity is not controlled, e.g. when the impactor has a mass $M$, then the impulse is not constant since $h(t)$ is the solution of the equation of $M\ddot {h} = - F + Mg$, where $F$ is given by (4.5). Even if we neglect the acceleration due to gravity $g$, this implies that $\dot {h}(t)$ is changing from its value upon impact ($V$) to its terminal value ($V/(1+m_a/M)$) on a time scale that is determined by the quadratic drag term, which is necessarily proportional to the inertial time $R/V$, which would be of the order of 10–100 ms for experiments similar to ours. During this time interval the force that the disc experiences is determined by the quadratic drag term in an indirect manner, namely by how it affects $h(t)$, and consequently $\dot {h}$ and $\ddot {h}$.

Thus, there is a crucial difference in free fall experiments reported before, and our experiments imposing a constant velocity. By keeping a constant velocity on the disc, we effectively suppress effects that cause variation of the added-mass terms in (4.5) and (4.6), thereby bringing the experiments closer to the classical theory. Additionally, our measurements extend the range of experimental parameters ($R^3 V \rho _{{w}}$) covered by Glasheen & McMahon (Reference Glasheen and McMahon1996) to a much larger range of disc momenta at impact.

We compute $\mathcal {F}_{{impact}}$ and $\varPi _{{impact}}$ by considering the signal up to the end of the first peak (as shown in the inset of figure 8). Here, $\varPi _{{impact}}$ values are multiplied by the disc area to make the two measures of impact impulses comparable. The comparison is shown in figure 9. Note that the range of data for $\mathcal {F}_{{impact}}$ is greater than that for ${\rm \pi} R^2 \varPi _{{impact}}$. This is due to the forces being measured with disc sizes up to $R = 8$ cm, while the pressures were measured with fixed disc size of $R=4$ cm. It is seen in figure 9 that force impulses from our measurements lie very close to the theoretical prediction. We find good agreement between our data and the $4/3$ slope for small impact momentum ($\rho _w R^3 V$). However, at larger momenta, discrepancies set in, which cannot be solely attributed to limited temporal resolution of the sensors. We also do not anticipate any effects from finite depth of the bath to have played in our measurements. Experiments by Ermanyuk & Gavrilov (Reference Ermanyuk and Gavrilov2011) showed the effects of finite bath approaching the asymptotic value of added mass coefficient (equalling 4/3, also see Chebakov Reference Chebakov1974) for a bath of depth $0.8 R$. Bath depths in the present work ranged between $3.75R-30 R$.

Figure 9. Impulses accumulated during the first impact peak (see inset of figure 8) are plotted against growth rate of the added mass of liquid at impact. Impulses calculated from pressure measurements ($\varPi _{{impact}}$) at the disc centre (shown in figure 6) were multiplied by the disc area so that they can be plotted directly on the same scale as $\mathcal {F}_{{impact}}$. Dashed line shows the theoretical calculation for the impulse transferred from an impulsively accelerated disc to an infinite half-space filled with inviscid fluid. Dotted line is the result from Peters et al. (Reference Peters, van der Meer and Gordillo2013) (4.8). The results in (b) are the collapsed version of the same in (a).

It is possible to at least approximately compute the influence of the entrapped air layer on $\mathcal {F}_{{impact}}$, as was done by Peters et al. (Reference Peters, van der Meer and Gordillo2013), who calculated the total force on a disc with the cushioning air layer. They estimated the change in momentum of a disc slamming at constant velocity as

(4.8)\begin{equation} \mathcal{F}_{{impact}} = 0.315 \rho_{w} {\rm \pi}R^3 V, \end{equation}

i.e. a reduction of the classical force impulse by approximately 25 %. This result is also compared with our measurements in figure 9. It lies closer to the area-integrated pressure impulses measured at the disc centre, but is consistently below the directly obtained force-impulse data from the load cell.