2.1. Set-up description

The cone used for experiments is 70 mm wide, and has a deadrise angle of 10$^{\circ }$. The wedge used in experiments has the same deadrise angle, an impacting cross-section that is 70 mm wide and a length of 140 mm. The overall dimensions are shown in figure 1(a). Both the wedge and cone house Kistler 601CAA dynamic piezoelectric transducers that are flush mounted with the impacting bottom surface. Vaseline is used to seal any gaps between the sensors and the mounting sites on the impactors. All the sensors had a sensing area 5.5 mm in diameter, and a peak acquisition rate of 200 kHz. Two sensors each are installed on either side of the keel such that the pressures measured on one side of the cone/wedge can be corroborated by measurements along the opposite side, a posteriori verifying the horizontality of impact. The first two sensors are installed such that the centres of their sensing area lie at a distance of 11 mm from the keel. The second set of sensors are installed a further 11 mm from the first pair of sensors. An illustration of the cone is shown in figure 1(b). The wedge is simply a two-dimensional (2-D) version of the cone with the same width: sensors on the wedge are installed at the same distance from the keel as in the cone. Note that, in addition to these flush-mounted transducers at the impacting surface, additional ‘placebo’ sensors are installed in both impactors to check that the pressures at impact are not affected by the impactor's change in acceleration. The reader is referred to Appendix A for discussion. The impactors are designed such that they are closed from the top to protect the sensors. Both were fabricated out of PET polyester to minimise the weight of the moving parts. This is done so as to maximise the velocity with which they can be translated. The mass of the wedge assembly was 795 g, and that of the cone was 611 g.

Figure 1. (a) Dimensions of the cone and wedge that are used for experiments. Pressure transducers are installed such that they are flush mounted with the surface on the impacting side. The impactor is closed on top such that the installed sensors installed are protected. (b) The cone's design is shown for illustration. Its bottom tip is called the keel. The location where its sloping surface sharply turns away into the vertical cylindrical surface is known as its knuckle.

The impactor is mounted on a linear motor using an aluminium rod. The assembly's position can be controlled with an accuracy of approximately 0.6 $\mathrm {\mu }$m. The degree of good velocity control is indicated by showing how repeatable the peak pressures are at a resolution of 200 kHz in figure 3. The acceleration that is generated by the linear motor in the course of its stroke is limited by the current it can draw. Thus, the linear motor's motion is programmed such that it achieves the specified velocity when in the middle of its available stroke. Concerning the wedge (or cone), the stroke is programmed such that it attains a constant velocity $V$ before it reaches the water free surface, and maintains so while the impactor plunges into the target pool until its submergence. We vary $V$ from 0.6 to 4.5 m s$^{-1}$. The target liquid bath consisted of de-mineralised water in a glass tank of area $50\,{\rm cm} \times 50\,{\rm cm}$, and a depth of 30 cm.

2.2. Typical pressure time series and repeatability

2.2.1. Cones

We denote the events that occur between the first touchdown of the cone's keel, until the time when flow separates from its knuckles, as events during impact. This stage in the experiments was recorded from a side view at 40 000 frames per second. Select key stages from an experiment at $V = 1$ m s$^{-1}$ are shown in figure 2. The two dotted lines in the figure mark the radial locations of sensors present on either side of the keel. During a time interval of 1.91 ms from the start of the cone's entry into the water (at $t = 0.87$ ms in figure 2), it moves a vertical distance of 1.91 mm into the liquid bulk, such that the cone ought to be submerged until the centre of the first sensor (location highlighted by orange dotted line). However, it can be seen from figure 2 that at this stage the rising liquid has already risen past the marked location. A similar observation can be made from the succeeding snapshot at $t = 2.77$ ms (3.82 ms from the start of cone's entry): when $V t$ would suggest the water level to have reached the blue dotted line, the liquid has risen well past the centre of the second sensor. The reason for this observation is as follows: as the cone enters into water, it displaces the water. Due to the liquid being impulsively displaced, it rises along the bottom surface, and emerges as a jet. This rising jet of liquid causes the cone's bottom surface to wet faster than the rate at which it would wet if the cone were to submerge quasi-statically. The liquid rising along the cone turns away from its contour, feeding mass to the emerging jet. This region of the liquid domain is known as the jet root region (see figure 5). The maximum impact pressure at a point occurs when it falls in the jet root region (Wagner Reference Wagner1932). Typical pressure measurements are shown in figure 3.

Figure 2. Snapshots showing the early stages of a cone entering water at 1.0 m s$^{-1}$. The time series are centred around at $t=0$ ms, which is defined as the moment when the pressure on the first sensor from the keel starts to rise. The orange and blue dotted lines show the locations of the pressure sensors whose centres are positioned 11 and 22 mm from the keel, respectively. At $t = {0.87}$ ms, the cone has travelled a distance at which a stationary water surface should reach the centre of the diaphragm of the first sensor. At $t = {2.77}$ ms, a stationary water surface would reach the centre of the second sensor's sensing area. At $t= {3.45}$ ms, the rising water is seen to separate from the cone. Without the local pile-up of water, the stage seen at $t = {3.45}$ ms, when the water separates from the cone's knuckle, would occur 5.13 ms after the start of cone's water entry.

Figure 3. Pressures time series recorded with the cone impacting a water surface at a controlled velocity of 4 m s$^{-1}$. The data are shifted in time such that at $t = 0$ ms, the water reaches the sensor(s) close to the keel (11 mm away along the cone contour), and the pressure here starts to rise. Approximately 0.34 ms later, the water reaches the second sensor, causing the pressure to rise there. Notice that immediately before the pressure starts to rise on the second sensors, it becomes negative. This transient under-pressure is created due to jet flow. A dashed line at $t = 0.88$ ms marks the time when the pressure reading on all the sensors starts to drop suddenly and simultaneously. This corresponds to the stage shown in the last panel of figure 2 when the rising water reaches the end of the contour, and separates at the cone's knuckles. Four realisations of the measurements are shown as an indicator of repeatability. Note that within each peak the agreement could even be further improved by performing a small time shift.

The final stage is reached when the rising liquid reaches the cone's knuckle, and separates from the cone's contour. As shown in the last panel of figure 2, this stage is reached 3.45 ms after the start of the cone's entry into water, while $V t$ would suggest this to occur at ${\rm \Delta} t = h_{{knuckle}}/V =6.17$ ms after impact, i.e. at $t = 5.13$ ms.

The pressure time series are shifted to $t=0$ at the point when the pressure on the first sensor from the keel starts to rise. A typical set of measurements from one experiment are shown in figure 3. A large $V$ example is shown in figure 3.

In figure 3 the respective data from sensor(s) are shown using the same colour-coding as their location highlights in figure 2. Before the readings on sensors at either location start to rise, the pressure becomes slightly negative. This feature is more prominently visible in the data from the second sensor(s) from the keel. This has been ascribed to jet flow of the liquid (Peseux et al. Reference Peseux, Gornet and Donguy2005). The jet precedes the oncoming thicker bulk of liquid which constitutes the ‘inner region’ (see figure 5). This pressure drop is regularly observed in experimental studies of wedge and cone impacts on water (Donguy et al. Reference Donguy, Peseux and Fontaine2001; Peseux et al. Reference Peseux, Gornet and Donguy2005; De Backer et al. Reference De Backer, Vantorre, Beels, De Pré, Victor, De Rouck, Blommaert and Van Paepegem2009; Lewis et al. Reference Lewis, Hudson, Turnock and Taunton2010; Tenzer et al. Reference Tenzer, Moctar and Schellin2015). However, to the best of the authors’ knowledge, its origin has not been systematically identified. Lewis et al. (Reference Lewis, Hudson, Turnock and Taunton2010) in particular suggested several sources for this pressure drop – from erring instruments to aerodynamic effects due to a ‘high-speed flow over the pressure sensor’. In § 2.3 we discuss the same finding. We are able to show (see § 2.3) that these pre-peak under-pressures in fact scale neatly with the dynamic pressure scale, but we are unable to conclusively understand their origin.

In the next stage, the pressure rises to its peak value, denoted as $P_{{peak}}$. Naturally, the sensors farther from the cone's keel register the peak at a later time than the first pair of sensors. After the peak, the pressure decreases towards a plateau value corresponding to the typical hydrodynamic pressure experienced by the sensors while the cone plunges further into the water bath. The final stage is reached when the readings on all the sensors near-simultaneously drop down from the value to which they appeared to be saturating. This point is highlighted in figure 3 using the grey dashed line, and corresponds precisely to the stage when the rising liquid jet reaches the cone's knuckle and separates from it. Peseux et al. (Reference Peseux, Gornet and Donguy2005) made the same observation regarding the sudden pressure drop. The small temporal inconsistencies that can be observed in measurements from different sensors at the same radial location can be attributed to an unavoidable, but small asymmetry in the angle with which the cone impacts.

2.2.2. Wedge impact pressures

Experiments with impacting wedges on water are done with the impactor as described in § 2.1. The flows caused by a wedge impacting on water can be approximated as a 2-D phenomenon, and be more readily described by an inviscid flow model. As with the experiments described in the previous section with a cone, we measure pressures along the wedge's impacting surface at two locations. Typical measurements are shown in figure 4. As with measurements from the cone (figure 3), the pressure readings become negative immediately prior to rising again, which again, is ascribed to the initial jet flow. Notice that compared with the similar under-pressure seen for cone impact from figure 3, the measurements from the wedge show both a more abrupt and a larger drop in pressure. This suggests that the jet and other flows created by the impact of a wedge are more violent than those due to a cone. Indeed it can be shown analytically that for a given deadrise angle, and at the same point in time, the rise of water is higher for a wedge than for a cone (Schmieden Reference Schmieden1953; Faltinsen & Zhao Reference Faltinsen and Zhao1998). From our measurements one can also notice that the peak pressures measured on a wedge are higher than those attained with a cone at the same radial locations, which we will discuss in more detail in § 2.6. It can be seen from both the experiments shown in figure 4 that the pressure drop prior to its rise occurs at nearly the same time at different distances from the wedge's keel. Compared with this, there is a significant time delay between when the impact pressure peaks are registered on sensors at varying distances. This is an indication that the initial liquid jet (which, according to Peseux et al. (Reference Peseux, Gornet and Donguy2005) and Lewis et al. (Reference Lewis, Hudson, Turnock and Taunton2010) causes the under-pressure), emerges much faster than the later rise of the jet root that causes the peak impact pressure.

Figure 4. Pressures time series recorded with the wedge impacting a water surface at controlled speeds of (a) 3.0 m s$^{-1}$ and (b) 4.0 m s$^{-1}$. The data are shifted in time such that at $t = 0$ ms, the water reaches the sensor(s) close to the keel (11 mm away along the cone contour), and the pressure here starts to rise. Notice that in both the plots, the pressures suddenly drop below zero for a short time before impact due to jet flow. The effect is substantially larger than in the case of the cone, owing to the wedge impact being a (quasi) 2-D process. In fact, the point at which the pressures suddenly drop is the time when the wedge starts to enter water (also see inset in figure 6a). The legend is shared between the two panels. The (dis-)similarities in pressures on the sensors on the two sides are indicative of the asymmetry of impact.

The successive stages show the pressure saturating to a steady value for a short time. The final stage is observed when the pressures suddenly drop (at approximately $t= 1.3$ ms in figure 4(a) and 1.2 ms in figure 4b) when the rising liquid reaches the wedge's knuckle and separates from its surface. Thus, it can be seen that until after the time of attainment of peak impact pressure, the contributions to pressure at a point on the wedge arise from different hydrodynamic sources. The model by Wagner (Reference Wagner1932) identifies and models the different contributions to pressure at a point on a wedge. This is discussed in the following section.

2.3. Comparisons of pressures from different models

Wagner (Reference Wagner1932) derives the impact pressures on a water-entering object due to resulting flows from first principles in the case of an ideal fluid. It has proven to be a particularly successful model to which numerous extensions and improvements could be made (e.g. Cointe & Armand Reference Cointe and Armand1987; Howison, Ockendon & Wilson Reference Howison, Ockendon and Wilson1991; Scolan & Korobkin Reference Scolan and Korobkin2001; Korobkin & Scolan Reference Korobkin and Scolan2006; Iafrati & Korobkin Reference Iafrati and Korobkin2008). Here we briefly mention the main results from the derivation of the pressure distribution on a water-entering wedge.

When the wedge enters water, a part of its surface is wetted. This wetted length of the wedge in the horizontal direction is treated as a plate. As the wedge further progresses in the liquid bulk, the wetted region, or length of the plate $2c(t)$, expands. A sketch of the situation is shown in figure 5. The resulting fluid flow problem posed in the model is one to calculate the flow potential due to an expanding plate at the fluid surface. The fluid is assumed to be inviscid and incompressible such that the flow potential $\phi$ satisfies Laplace's equation

(2.1)\begin{equation} \nabla^2 \phi = 0. \end{equation}

The pressure everywhere in the fluid is given by the unsteady Bernoulli equation

(2.2)\begin{equation} \frac{1}{\rho}p(x,y,t) ={-} \frac{\partial \phi}{\partial t} - \frac{1}{2} \lvert \boldsymbol{\nabla} \phi \rvert^2. \end{equation}

The liquid interface directly under the plate (region $x< c$) obeys the kinematic condition $(\partial \phi / \partial y )_{y=0} = -V$. Along the free surface (region $x>c$) away from the plate, ${\phi (x,0) = 0}$. Without repeating the complete derivation (Wagner Reference Wagner1932), $\phi$ may be written as the real part of the complex potential

(2.3)\begin{equation} \mathcal{W}(z) = {\rm i} V \left( z + \sqrt{z^2 - c^2} \right), \end{equation}

where $z = x + {\rm i}y$ and $c$ is the half-length of the wetted plate (shown in figure 5). The point $c$ lies at the jet root, approximately at the intersection of the inner and jet domains in figure 5. By computing

(2.4)\begin{equation} \frac{{\rm d}\mathcal{W}}{{\rm d} z} = \frac{\partial \phi}{\partial x} - {\rm i} \frac{\partial \phi}{\partial y} = u - {\rm i} v \end{equation}

on the $x$-axis, we find that, for $x< c$,

(2.5)\begin{equation} \left(\frac{\partial \phi}{\partial y} \right)_{y=0} ={-} V, \end{equation}

and

(2.6)\begin{equation} \frac{\partial \phi}{\partial x} (x,0) = V \frac{x}{\sqrt{c^2 - x^2}}. \end{equation}

Noting that $\phi (x,0) = \text {Re}[ \mathcal {W}(x,0) ] = - V \sqrt {c(t)^2 - x^2}$, the unsteady Bernoulli equation yields the pressure on the body as

(2.7)\begin{align} {p} &= \rho \frac{\partial}{\partial t} \left( V \sqrt{c^2 - x^2} \right) - \frac{\rho}{2} \left( V \frac{x}{\sqrt{c^2 - x^2}} \right)^2 \end{align}

(2.8)\begin{align} &= \rho\frac{{\rm d} V}{{\rm d} t}\sqrt{c^2 - x^2} + \rho \frac{V c}{\sqrt{c^2 - x^2}}\frac{{\rm d} c}{{\rm d} t} - \frac{\rho}{2} \frac{V^2 x^2}{c^2 - x^2}. \end{align}

The first term above becomes zero for constant $V$. The second term is known as the slamming pressure, where the dependence on the wetting rate ${\rm d} c/{\rm d} t$ plays a crucial role. It predominantly originates from the outer domain and is associated with accelerating the added mass of liquid. The final term is the jet pressure. Wagner's central contribution was to determine the wetting rate $c(t)$ for certain geometries. For a wedge with deadrise angle $\beta$, it is found to be (Wagner Reference Wagner1932)

(2.9)\begin{equation} c(t) = \frac{{\rm \pi} V t}{2\tan \beta}. \end{equation}

Dropping the first term from (2.8) for constant $V$, the expression for total pressure becomes

(2.10)\begin{equation} p(x,0) = \rho V^2 \left[\frac{\rm \pi}{2\tan \beta } \frac{c}{\sqrt{c^2 - x^2}} - \frac{1}{2} \frac{x^2}{c^2 - x^2} \right]\quad \text{for } (x< c). \end{equation}

Note that this equation can be written as

(2.11)\begin{equation} \frac{p}{\rho V^2} = f(x/c), \end{equation}

which suggests a dimensionless form introducing

(2.12a–c)\begin{equation} \tilde{p} = \frac{p}{\rho V^2},\quad \tilde{x} = \frac{x}{D}\quad\text{and}\quad \tilde{t} = \frac{Vt}{h_{{knuckle}}}, \end{equation}

with which $x/c = 4 \tilde {x}/{\rm \pi} \tilde {t}$. Accordingly, we non-dimensionalise pressure measurements from experiments done over a range of $V$ with the pressure scale $\rho V^2$, and plot them against time rescaled with $h_{{knuckle}}/V$ in figure 6. At each of the locations, the measurements are very convincingly collapsed by the rescaling described above. Note that the time series are shifted in time such that the impact on the first sensor is shown to occur at $t=0$ in figure 6, and the model calculations are shifted by the same amount.

Figure 5. Schematic and definitions of the different flow regions as used in the Wagner model for a wedge impacting in water. The inner domain is also variedly known as the jet root region.

Figure 6. For the wedge case, pressure measurements of the type shown in figure 4 for a range of impact velocities $V$ are non-dimensionalised and plotted from the first sensor in (a), and from the second sensor in (b). As in earlier plots, the experimental time series were shifted so that the point at which the reading on the first sensor starts to rise lies at $t = 0$ ms. The insets in (a) show zoomed-in regions of the data shown in both the panels in the vicinity of $t V / h_{{knuckle}}=0$. Dots and crosses in the inset represent the data from the first and second sensors, respectively. The data that start to rise at $t V / h_{{knuckle}} = 0$ are the readings from the sensor close to keel, the remainder at the data from the other sensor. The quantity $h_{{knuckle}} = D \tan \beta /2 \approx 6.17$ mm. The grey solid curves are the point-pressure computations from the composite solution ((2.13), also see row 5 and column 7 in table 1). The final four curves are the space-averaged pressures from the composite solution (SA COM, green line, (2.13)), the original Logvinovich model (SA OLM, blue-dashed, (2.19)), the modified Logvinovich model (SA MLM, yellow dashed line (2.20)) and the generalized Wagner model (SA GWM, red dotted line, (2.21)). Note that, of course, these theoretical time series are shifted in time by the same amount as the experimental ones.

Table 1. Values of $C_{p,{max}} (\equiv P_{{peak}}/\rho V^2)$ for several deadrise angles $\beta$ from the ordinary Logvinovich model (OLM), modified Logvinovich model (MLM) and generalized Wagner model (GWM) as described in Korobkin (Reference Korobkin2004) (columns 2–4), numerical solutions to Dobrovol'skaya (Reference Dobrovol'skaya1969) by Wang & Faltinsen (Reference Wang and Faltinsen2017) (EIM, column 5), the asymptotic solution from Wang & Faltinsen (Reference Wang and Faltinsen2017) (ASM, column 6) and the composite solution (COM) from Zhao & Faltinsen (Reference Zhao and Faltinsen1993) (column 7). For comparison, we have added the measured $C_{p,{max}}$ from figure 6 in columns 8 and 9, where one needs to realise that these values, other than those of the theoretical models, constitute pressures that are space-averaged over the sensor surface, and, therefore, considerably smaller than what is found in most models.

In the two insets of figure 6(a) we show that the under-pressure that occurs prior to the peak, also collapses when rescaled using inertial scales. Clearly the occurrence of this under-pressure is systematic, and the rescaling indicates a hydrodynamic origin, possibly connected to the jet region moving over the pressure sensor. This is also consistent with the observation that the pressure drops slightly earlier in the first sensor and subsequently in the second. The data collapse obtained with the inertial pressure and time scales is consistent with the theoretically expected self-similarity of the impact as a whole. However, since the expression (2.10) only concerns the region between the turnover points at the jet root, this under-pressure, although evidently also inertial, is not included in the above expression.

2.4. The singularity at $x = c$, and comparisons of measured pressures with models

2.4.1. Composite model from asymptotic matching by Zhao & Faltinsen (Reference Zhao and Faltinsen1993)

The singularity occurring in (2.10) for $x \rightarrow c$ hinders one from computing a peak impact pressure value that can be directly compared with experiments. Cointe & Armand (Reference Cointe and Armand1987), Wilson (Reference Wilson1989) and Howison et al. (Reference Howison, Ockendon and Wilson1991) described how to match the solutions in inner and outer domains using asymptotic expansions. The analysis yields an asymptotic jet thickness $\delta$. For a wedge with a small deadrise angle, one can show that $\delta = {\rm \pi}V^2 2c/(4\,{\rm d} c/{\rm d} t)^2$ (Wilson Reference Wilson1989; Zhao & Faltinsen Reference Zhao and Faltinsen1993). Zhao & Faltinsen (Reference Zhao and Faltinsen1993) took the idea forward and constructed a composite solution (COM) for the pressure that avoids the singularity. Without repeating the derivation, we write their result for pressure distribution on the wedge:

(2.13)\begin{equation} p = \rho V c \frac{{\rm d} c}{{\rm d} t} \frac{1}{\sqrt{c^2 - x^2}} - \rho V c \frac{{\rm d} c}{{\rm d} t} \frac{1}{\sqrt{2c (c-x)}} + 2\rho \left(\frac{{\rm d} c}{{\rm d} t}\right)^2 \lvert \tau \rvert^{1/2} \left(1+ \lvert\tau \rvert^{1/2}\right)^{{-}2}. \end{equation}

Here $\lvert \tau \rvert$ is a parameter that relates to $x$ via $\delta$ as

(2.14)\begin{equation} x - c = (\delta/{\rm \pi}) \left( - \ln\lvert \tau \rvert - 4 \lvert \tau \rvert^{1/2} - \lvert \tau \rvert + 5\right){,} \end{equation}

where the last terms from (2.13) and (2.14) are identical to expressions already derived by Wagner (more specifically, expressions (10) and (8) from Wagner (Reference Wagner1932), respectively). As before, the wetting rate $c(t)$ can be obtained from (2.9). The first term in (2.13) can again be identified from (2.10) as the slamming pressure, or the outer domain solution. The last term is the inner domain pressure. The second term is the common asymptotic factor in the jet root region and, hence, subtracted. The results from this composite solution are plotted as grey solid lines in figure 6. The rise in pressure from the model still exhibits near-singular behaviour, but reaches a finite maximum value. These values are listed for different deadrise angles $\beta$ in table 1 alongside the peak pressure coefficients obtained from other models discussed in § 2.4.2. The attainment of the impact peak is followed by the (computed) pressure saturating to a constant value (of approximately 10$\rho V^2$ for the deadrise angle studied here). The time delay between when pressure on the second sensor starts to rise after the first one is very well reproduced by the model. However, the main quantity of interest, the peak pressures, are still over-predicted by the analytical models. Note that the model (2.13) computes the pressure at a single location on the wedge. One important reason for the disagreement is therefore due to the finite size of the transducers’ sensing area. Indeed, it was already recognised that instead of the analytical Wagner profile, which is singular at $x = c$ (Takemoto Reference Takemoto1984; Faltinsen & Zhao Reference Faltinsen and Zhao1998), a finite size of the sensor in experiments ought to register a space-averaged pressure (Zhao, Faltinsen & Aarsnes Reference Zhao, Faltinsen and Aarsnes1996; Scolan Reference Scolan2014a). Such a quantity is better suited to a direct comparison with experimentally obtained peak pressures. The procedure for the space-averaging is described in Appendix C, and the space-averaged results are presented in figure 6. These will be discussed later together with the models presented in the next subsection.

2.4.2. Original logvinovich, modified logvinovich and generalized Wagner models

Clearly the composite asymptotic model discussed in the previous subsection is only one of many successful attempts to deal with the divergences present in the original Wagner (Reference Wagner1932) model. Here we also mention the exact self-similar solution in the form of an integral equation from Dobrovol'skaya (Reference Dobrovol'skaya1969) and several approximations that extend Wagner's formulation by incorporating higher-order terms in the Bernoulli equation (Korobkin Reference Korobkin2004).

As stated before, in (2.10) both the pressure contributions diverge as $x \rightarrow c$. Also the ratio of jet to slamming pressure, namely $-x^2 \tan \beta / ({\rm \pi} c \sqrt {c^2 - x^2})$, diverges to negative infinity in this limit. Generally, the result of the approach taken in Korobkin (Reference Korobkin2004) is that the region of interest is limited up to some point $a$, a short distance before $x=c$. Following Korobkin (Reference Korobkin2004) we write

(2.15a,b)\begin{equation} {a(t)= \xi c(t),\quad \xi = \sqrt{1-X^2},} \end{equation}

where $X$ has the following functional forms for the original Logvinovich model (OLM), modified Logvinovich model (MLM) and generalized Wagner model (GWM),

(2.16)\begin{gather} X = \frac{2 \tan \beta}{\rm \pi} \quad \text{(OLM)} , \end{gather}

(2.17)\begin{gather}X = \frac{\sin(2 \beta)}{{\rm \pi} \left[ 1+ \sqrt{1-4 {\rm \pi}^{{-}2} \sin^4 \beta} \right]} \quad \text{(MLM)} \text{ and} \end{gather}

(2.18)\begin{gather}X = \frac{\sin (2 \beta)}{{\rm \pi} \left[1+\sqrt{1-4{\rm \pi}^{{-}2} \sin ^2 \beta \left( \sin ^2 \beta + {\rm \pi}-2 \right)} \right]} \quad \text{(GWM)}, \end{gather}

respectively. The corresponding pressures are equal to

(2.19)\begin{gather} P(x,t ) = \frac{\rho V^2}{2} \left[ \frac{\rm \pi}{\tan \beta} \frac{c}{\sqrt{c^2-x^2}} - \frac{c^2}{c^2-x^2} \right] \quad \text{(OLM),} \end{gather}

(2.20)\begin{gather}P(x,t ) = \frac{\rho V^2}{2} \left[ \frac{\rm \pi}{\tan \beta} \frac{c}{\sqrt{c^2-x^2}} - \cos ^2 \beta \frac{c^2}{c^2 - x^2} - \sin ^2 \beta \right] \quad \text{(MLM)} \text{ and} \end{gather}

(2.21)\begin{gather}P(x,t) = \frac{\rho V^2}{2} \left[ \frac{\rm \pi}{\tan \beta} \frac{c}{\sqrt{c^2-x^2}} - \cos ^2 \beta \frac{c^2}{c^2 - x^2} - \sin ^2 \beta + 2 -{\rm \pi} \right] \quad \text{(GWM),} \end{gather}

respectively, where the expressions are understood to hold for $x \leq a(t)\ [< c(t)]$. The maximum (point-)pressure coefficients from these models, alongside the results from the self-similar integral equation by Dobrovol'skaya (Reference Dobrovol'skaya1969) (using the computed numerical results from Wang & Faltinsen Reference Wang and Faltinsen2017) are compared in table 1. Finally, the space-averaged pressure time series resulting from the three models presented above are added to figure 6. From that figure we conclude that, generally, all sensor-averaged models compare well to the experiments; the composite model (COM) reproduces the peak values slightly better, whereas the OLM and GWM models are closest to the experiment in the period after the peak, up to the moment that the inner region reaches the knuckle. Similarities and dissimilarities with observations in the literature are discussed in further detail in the next subsection.

2.5. Cone impact pressures

In this section we turn to experiments slamming a cone on water. As with the pressure measurements from the wedge (figure 6), the experimental data from cone impacts were obtained over a large range of velocities, and are very well collapsed by inertial pressure and time scales.

For a cone impact on water, the flow potential in the domain is known using an axisymmetric solution from Schmieden (Reference Schmieden1953) and Faltinsen & Zhao (Reference Faltinsen and Zhao1998): in place of the expanding plate, the no-penetration body condition was transferred to an expanding disc. Thus, the treatment remains analogous to the one for a wedge. The remaining boundary conditions were the same as before: $\phi = 0$ on the free surface, and $\partial \phi / \partial y = -V$ for $x< c$, where $x$ now designates the radial position on the cone. The resulting flow potential on the body along the $x$-axis becomes

(2.22)\begin{equation} \phi(x,0) ={-}\frac{2V}{\rm \pi} \sqrt{c^2 - x^2}\quad \text{for } x< c. \end{equation}

The Wagner condition for the cone was calculated to be

(2.23)\begin{equation} c(t) = \frac{4 V t}{{\rm \pi} \tan \beta}. \end{equation}

Shiffman & Spencer (Reference Shiffman and Spencer1951) found the same result in the outer domain using an elliptical contact line. Next, the axisymmetric solution in the outer domain was asymptotically matched with the 2-D solution in the jet domain to regularise over the singularity at $x=c$. The matching yielded a jet thickness $\delta = V^2 c /2{\rm \pi} ({\rm d} c/{\rm d} t)^2$. Note from (2.22) that with this treatment, only the outer domain solution is modified by a multiplicative factor of $2/{\rm \pi}$. Since in the jet domain the 2-D solution was used, its pressure contribution remains the same as in (2.13). The composite pressure solution for the cone thus becomes

(2.24)\begin{equation} p = \frac{2}{\rm \pi}\rho V c\frac{{\rm d} c}{{\rm d} t} \frac{1}{\sqrt{c^2 - x^2}} - \frac{\rho V }{\rm \pi}\frac{{\rm d} c}{{\rm d} t} \left( \frac{2c}{c-x}\right)^{1/2} + 2 \rho \left(\frac{{\rm d} c}{{\rm d} t}\right)^2 \frac{ \lvert \tau \rvert ^{1/2}}{ \left( 1 + \lvert \tau\rvert ^{1/2}\right)^{2}}, \end{equation}

where $\tau$ is related to $(x-c)$ and $\delta$ as in (2.14) (Faltinsen & Zhao Reference Faltinsen and Zhao1998). The results from (2.24) are compared with non-dimensionalised measurements in figure 7.

Figure 7. Pressure measurements from the impacting cone at different impacting velocities $V$ are non-dimensionalised and plotted as a function of dimensionless time $t V / h_{{knuckle}}$. Results from the first sensor are found in (a), while those from the second sensor in (b). The quantity $h_{{knuckle}} = D \tan \beta /2 \approx 6.17$ mm. As in earlier plots, the time series were shifted so that the point at which the reading on the first sensor starts to rise lies at $t = 0$ ms. The grey solid curves are computations from the composite solution (COM) for the cone (2.24), while the green solid curves are the same composite solution, space-averaged over the area of the pressure transducers (SA COM).

The origin of the time series in figure 7 were shifted such that $t = 0$ corresponds to the moment at which the reading on the first sensor from the keel starts to rise. Accordingly, the time delay between when the peak pressures are attained on the two sensors is again very well reproduced by the modified Wagner condition for a cone.

After reaching the peak pressure, the measures at the given locations reduce and approach a saturated value shown by the plateau. At later times ($t V / h_{{knuckle}} \approx 0.66$), the sudden drop in experimental measurements at both locations corresponds to the liquid jet reaching the cone's knuckles and detaching completely.

The space-averaged pressures from the composite solution for a cone (2.24) are computed for the two locations where the sensors were installed in the experiment. They are compared with the non-dimensionalised measurements in figure 7. As with the wedge, the agreement is found to be remarkably good at both the sensors. However, compared with the results for a wedge from figure 6, on the cone, the peak pressure is slightly under-predicted, while the rise time is somewhat over-predicted.

2.6. Comparing peak pressures on cones and wedges

The most notable progress in modelling water-entry pressures has been made using 2-D models (Wagner Reference Wagner1932; Logvinovich Reference Logvinovich1969; Howison et al. Reference Howison, Ockendon and Wilson1991; Zhao & Faltinsen Reference Zhao and Faltinsen1993) and their extensions in three dimensions or cylindrical coordinates (Scolan & Korobkin Reference Scolan and Korobkin2001; Oliver Reference Oliver2002; Faltinsen & Chezhian Reference Faltinsen and Chezhian2005; Moore et al. Reference Moore, Howison, Ockendon and Oliver2012; Moore Reference Moore2014; Moore & Oliver Reference Moore and Oliver2014). However, in practice, all impact processes are three dimensional. As such, it is important to know the limits of how the Wagner treatment is extended to model the wetting rate in a 3-D system. The most straightforward extension to three dimensions is made by considering an axisymmetric impact. In this context, the impact of a cone represents an important test case of how the Wagner condition on a wedge is modified to include 3-D flow effects. It can be seen from (2.13) and (2.24) that the rate of local rise-up of water along the impactor body has a crucial contribution to the impact pressure.

The maximum pressure from the model at any given location along the impactor is simply $\rho {({\rm d} c/{\rm d} t)}^2/2$. This is clearly larger than what any finite-sized sensors would measure (figures 6 and 7). However, the relative magnitudes of peak pressures between a wedge and cone from $\rho {({\rm d} c/{\rm d} t)}^2/2$ may be compared with those measured in experiments. A direct comparison of peak pressures between the cone and wedge from sensors at the same horizontal locations is made in figure 8.

Figure 8. The peak pressures from wedge and cone are directly compared at both sensors. The peak pressures measured on the cone are found to be $64/{\rm \pi} ^4$ times to those measured on the wedge across the whole range of $V$ used. The measurements are compared with the analytical result (2.25) finding excellent agreement.

Recall that for a wedge, $c(t) = {\rm \pi}V t /2 \tan \beta$ (2.9). While for a cone, $c(t) = 4Vt/{\rm \pi} \tan \beta$. The ratio of peak pressures is therefore expected to be

(2.25)\begin{equation} \frac{p_{{peak}}^{{cone}} } { p_{{peak}}^{{wedge}}} = \frac{64}{{\rm \pi}^4} {\approx 0.66}. \end{equation}

This ratio is plotted as the dashed line in figure 8, and it agrees very well with the experimental data. This is an independent measure, corroborating the inclusion of 3-D effects only in the outer domain, by modifying the wetting rate $c(t)$. Note that, Chuang (Reference Chuang1969) had found the same ratio of peak pressures to be $0.75$. Although peak pressure data for wedge and cone impacts are certainly present in Takemoto (Reference Takemoto1984), unlike in the current work, it was not analysed beyond the documentation of the pressure peaks.

Finally, we turn to the decay of the pressure after the occurrence of the peak in figures 6 and 7, that is, after the root of the jet has passed the pressure sensor. While, for wedges, the experimental pressure consistently decays faster than that of the models, this is not the case for our pressure measurements on the cones, where experiments are coinciding or even slightly above the theoretical prediction. Whereas the origin of this observation is unknown, it does appear that the deviations from the models in both cases become smaller with larger impact velocities. We therefore speculate that they could well find their origin in (viscous) interaction of the liquid with the impactor body, in both the outer and the inner region. This is also consistent with the fact that, for the wedge impacts, the kink (at $t V /h_{{knuckle}} \approx 0.5$ for wedges and ${\approx }0.6$ for cones) that corresponds to the jet root passing the knuckle becomes more and more pronounced for increasing impact velocities.

3.1. Measuring the free-surface deflections using TIR-deflectometry

The free surface of the water in the tank is observed using a TIR configuration as shown in figure 9. A fixed pattern ($O$) placed on one side of the tank is reflected at the water surface, and recorded by a camera on the other side. The image appears to come from ‘behind’ the reflecting surface ($O'$). When the water surface moves from its stationary position, the mirror in the present optical set-up moves, and distorts the pattern seen by the camera. When an object is present at the water surface, it obstructs the mirror such that the water surface at all such locations does not reflect the pattern.

Some examples of the free surface being visualised in such a set-up in an experiment with a cone impacting on water are shown in figure 10 and movies 1–3, wherein the first contact occurs at $t=0$. In particular, it can be seen at times before impact that the pattern is distorted, already indicating that the water surface has undergone some deformation before the cone has come into contact with it.

Such movements of the pattern are recorded using high-speed imaging, and quantified using a particle image velocimetry algorithm. The stacks of displacement fields thus obtained correlate to the local spatial gradients of the water surface. The displacement fields are integrated in a least-squares sense to obtain the instant-wise reconstruction of the deformation of the water surface. The details of this reconstruction are described in Jain et al. (Reference Jain, Gauthier and van der Meer2021b). The fully reconstructed free surface is also shown in movie 4.

In addition to the surface reconstruction before impact, the extent to which the cone is wetted after impact is also easily measured using the TIR view. Measurements of the wetted region of a cone with $\beta =10^{\circ }$ at different impact velocities are shown in figure 10(e). They compare very favourably with the Wagner condition for cone (2.23). Results from similar measurements with cones with $\beta = 5^{\circ }, 20^{\circ }$ and 30$^{\circ }$ are available in the supplementary material.

3.2. Two-fluid BI simulations

We use an in-house BI code to simulate the interface's response to the pressure build up in the air layer between the cone and liquid. The implementation has been described in detail in earlier works (Oguz & Prosperetti Reference Oguz and Prosperetti1993; Bergmann et al. Reference Bergmann, van der Meer, Gekle, van der Bos and Lohse2009; Gekle & Gordillo Reference Gekle and Gordillo2011), and here will only be briefly discussed. We first describe the basic principles.

The implementation is based on the flow being inviscid and incompressible. Thus, a scalar flow potential describes flow in the bulk, and the fluid(s) satisfy Laplace's equation. As is done in boundary element methods, Laplace's equation is transformed using Green's third identity to obtain a so-called BI equation. This equation consists of only surface integrals wherein flow potentials are defined only along the boundaries of the concerned fluid domain. From this integral, flow information in the fluid bulk can be deduced. The boundaries of each domain are populated by ‘nodes’ at which the flow potentials are defined. We use an axisymmetric formulation of such a code, which further reduces the surface integrals in the BI equation to line integrals. The surface then is initiated at the first constituent node of this curve, which in our set-up lies on the symmetry axis (at $r=0$). In the far field all surface potentials decay to zero in the formulation. Since we are only concerned with the region nearer to $r=0$, and contribution of far regions to the flow potentials would be negligible, each fluid surface in the code can be cut off at some sufficiently large $r$. Finally, the BI implementation can be made a fully two-phase coupled formulation by incorporating a Laplace pressure jump across the interface, and imposing a balance of normal velocities at the fluids’ interface (Rodríguez-Rodríguez, Gordillo & Martínez-Bazán Reference Rodríguez-Rodríguez, Gordillo and Martínez-Bazán2006; Gordillo, Sevilla & Martínez-Bazán Reference Gordillo, Sevilla and Martínez-Bazán2007; Gekle & Gordillo Reference Gekle and Gordillo2011).

The present simulations are performed with the fully coupled, two-phase BI code, that has previously been used by Peters, van der Meer & Gordillo (Reference Peters, van der Meer and Gordillo2013) and Jain et al. (Reference Jain, Gauthier, Lohse and van der Meer2021a). Here the two fluid phases are water and air – both of which are characterised by their respective densities and interfacial tension. The cone is defined as a region bounded by solid curves along which the normal flow potential in surrounding air (in respective frame of reference) is set to zero.

The node distribution along the interface is initialised such that there are two regions of high node density: the node density gets linearly denser closer to $r=0$, and in the vicinity of $r=D/2$. At the initialisation, the minimum inter-node distance used in the simulations here (and in the supplementary material) are typically 0.0005 and 0.001 of the representative length scale (for these simulations, $D/2$). Furthermore, the node density evolves in time based on interface curvature and gas velocity, such that it is allowed to grow past the maximum resolution defined at initialisation. However, we impose that the number of nodes between each time step do not grow faster than a multiplicative rate of 1.1. Due to a lack of natural damping in BI simulations, numerical instabilities may accumulate. This is handled by regridding the entire curve every few time steps, and making the new nodes lie exactly halfway in between previous ones. However, some accumulation of numerical instabilities at $r=0$ at later times is not avoided despite the regridding. More details may be found in Gekle & Gordillo (Reference Gekle and Gordillo2011).

The simulation is initiated at a non-dimensional time which is determined as the instance in experiments when the interface starts to move downward steadily (see $\tau \approx 0.052$ s in movies 4 and 5 for the $\beta =10^{\circ }$ cone with $V=1$ m s$^{-1}$). At the determined starting time, the interface in the experiment is already in the process of deforming (since even prior to the action of the air-cushioning layer, there were unavoidable minuscule fluctuations of the interface), whereas in the simulation, the interface position is initiated at zero. This is the reason for the differences between experiments and BI at early times in figure 12(b–d), which for the present discussion, we denote as the start-up discrepancy. The discrepancy becomes smaller as the process progresses in time – this is further discussed in the next subsection.

3.4. Inertial vs viscous air cushioning

When compared with the relevant length and time scales conveyed by the viscous air-cushioning models (Ross & Hicks Reference Ross and Hicks2019) for the same impact geometry, our air-cushioning results present a very different picture. The model in Ross & Hicks (Reference Ross and Hicks2019) explicitly starts with the assumption that gas pressure build up in the intervening layer does not influence the liquid until the vertical separation between the solid body and the liquid free surface is of thickness $\varepsilon ^2 D$, where $\varepsilon \equiv ( \mu _g/(\rho _l V D) )^{(1/3)}$, $\mu _g = 1.81\times 10^{-5}$ Pa s is the gas dynamic viscosity, $\rho _l = 998$ kg m$^{-3}$ the liquid density and $V$ the approach velocity. In our experiments, $D = 0.07$ m. Using $V=0.1$ m s$^{-1}$, this separation $\varepsilon ^2 D$ is 13.2 microns, while for 1 m s$^{-1}$, we obtain $\varepsilon ^2 D=2.84\,\mathrm {\mu }$m. These body-liquid separations are much smaller than those of our observations in figure 12(b–d), where the body-liquid separation is approximately 5 cm when the interaction is clearly established. Similar results are shown in the supplementary material for cones with deadrise angles $\beta = 5^{\circ }, 20^{\circ }$ and 30$^{\circ }$. Assuming that viscous pressurisation of the gas layer indeed begins when $\tau = \varepsilon ^2 D/V$, it would imply that, of the distance of $0.7 - 0.8D$ (approximately 5 cm) that the $10^{\circ }$ cone travels at $V = 1$ m s$^{-1}$ whilst interacting with the liquid surface via the air layer, the viscous cushioning is active for only 0.0057 % of the initial gap distance. Thus, the deformations caused by viscous air cushioning will be created on an interface that has already been substantially deformed into a cavity by the earlier inertial air cushioning. Another important aspect in which the two will differ is the actual volume of the created cavity. While potential flow creates an interface cavity of the width of the impactor, the lubrication-pressure distribution would result in interface deformations that are much more localised around $r=0$. The natural limits to when potential or viscous air cushioning would hold for a spherical projectile, were addressed using both numerics and theory by Bouwhuis et al. (Reference Bouwhuis, Hendrix, van der Meer and Snoeijer2015).

In Appendix D we use the results from Bouwhuis et al. (Reference Bouwhuis, Hendrix, van der Meer and Snoeijer2015) to approximate the crossover air-layer thickness where the type of air cushioning, in our experiments, is expected to switch from inertial to viscous. Note that the crossover air-layer thickness derived from such a balance, approximately $106\,\mathrm {\mu }$m, is still around 40 times larger than the air-layer thickness for which the cushioning problem is initiated by the non-dimensionlisation of Ross & Hicks (Reference Ross and Hicks2019) and Moore (Reference Moore2021).