3.1 Force measurements

Figure 2(a–c) shows the force evolution recorded by the force sensor for three wedges of opening angles $\unicode[STIX]{x1D6FC}=120^{\circ }$ , $\unicode[STIX]{x1D6FC}=90^{\circ }$ and $\unicode[STIX]{x1D6FC}=60^{\circ }$ , respectively, and for various entry velocities. As in figure 1(b), the force quickly rises from 0 to a positive value after impact. For $\unicode[STIX]{x1D6FC}=120^{\circ }$ and $\unicode[STIX]{x1D6FC}=90^{\circ }$ , the force peaks after entry, with peak values ranging from $10$ N to $40$ N, then decreases to a quasi-steady value once the wedge is fully submerged. For $\unicode[STIX]{x1D6FC}=60^{\circ }$ , the force gradually increases from 0 to a terminal value; no prominent peak force is observed. We call the first pattern ‘impactful entry’ and the second ‘smooth entry’.

Given the inertial flow regime ( $Re>10^{4}$ ), we scale force by $(\unicode[STIX]{x1D70C}SV^{2})/2$ , where $S=L\times d$ is the projected area of the wedge, and introduce the non-dimensional force coefficient $C=F/(\unicode[STIX]{x1D70C}SV^{2})/2$ . We also introduce a non-dimensional time $t^{\ast }=t/(h/V)$ by scaling time by the inertial time scale $h/V$ , with $t^{\ast }=0$ being the time of impact and $t^{\ast }=1$ corresponding to the instant when the wedge is fully submerged with respect to the undisturbed water surface. The dimensionless force coefficient is depicted in figure 2(d–f). For each wedge, the data of different entry velocities collapse for $t^{\ast }\leqslant 1$ , confirming that the force and time scales are appropriate. From these graphs, we extract a single value of the maximum force $C_{max}$ . We also mark the time $t_{max}^{\ast }$ when the maximum force happens: $t_{max}^{\ast }=0.66,\,0.73,\,1$ for $\unicode[STIX]{x1D6FC}=120^{\circ },\,90^{\circ },\,60^{\circ }$ , respectively. For $t^{\ast }>1$ , the force measurements do not collapse. We observe larger force coefficients at smaller entry velocities, particularly obvious for $\unicode[STIX]{x1D6FC}=120^{\circ }$ , which can be attributed to hydrostatic forces becoming significant as the Froude number approaches unity.

Figure 3.

(a) The maximum force coefficient increases sharply with wedge angle $\unicode[STIX]{x1D6FC}$ , and follows theoretical predictions by Logvinovich (Reference Logvinovich1972) (solid line); excellent agreement is observed when Logvinovich’s two-dimensional theory is corrected for finite aspect ratio $L/d=4.2$ (dashed line). (b) For $\unicode[STIX]{x1D6FC}\leqslant 70^{\circ }$ , the maximum force happens roughly when the wedge is fully submerged ( $t_{max}^{\ast }\approx 1$ ), while for $\unicode[STIX]{x1D6FC}>70^{\circ }$ , $t_{max}^{\ast }$ drops sharply and converges to the naive prediction from Wagner’s theory $t_{max}^{\ast }=2/\unicode[STIX]{x03C0}$ . (c,d) The theoretical pressure profile can be used to improve the prediction for $t_{max}^{\ast }$ . Assuming the maximum force occurs when the negative pressure area completely escapes the wedge ( $a=d/2$ , circles) leads to an analytical, refined estimation (dash-dotted line in panel b). In reality, the pressure integral (force $F$ ) on the wedge reaches a maximum when $b=d/2$ (triangles), leading to a second numerical prediction (dashed line in panel b), providing the same trend as the experimental measurements. Transition to smooth entry is expected when the peak force occurs after the wedge’s submersion, that is, $t\geqslant 1$ (arrow in b). In (a,b), red square data points denote smooth entry, defined as the absence of prominent peak.

These measurements are repeated with wedge angles $\unicode[STIX]{x1D6FC}=60^{\circ }$ , $75^{\circ }$ , $80^{\circ }$ , $90^{\circ }$ , $105^{\circ }$ and $120^{\circ }$ , leading to the values of $C_{max}$ and $t_{max}^{\ast }$ shown in figure 3(a,b) and table 1. Rather intuitively, $C_{max}$ increases sharply with $\unicode[STIX]{x1D6FC}$ , starting at $C_{max}=0.65$ for $\unicode[STIX]{x1D6FC}=45^{\circ }$ and reaching $C_{max}=4.2$ for $\unicode[STIX]{x1D6FC}=120^{\circ }$ , while $t_{max}^{\ast }$ decreases with $\unicode[STIX]{x1D6FC}$ , starting at $t_{max}^{\ast }=1.08$ for $\unicode[STIX]{x1D6FC}=45^{\circ }$ and plummeting to $t_{max}^{\ast }=0.66$ for $\unicode[STIX]{x1D6FC}=120^{\circ }$ . No prominent peak force was observed for wedges with $\unicode[STIX]{x1D6FC}\lesssim 70^{\circ }$ .

3.2 Force model

We show that (i) our experimental data compares favourably with Logvinovich’s model once three-dimensional effects are taken into account and (ii) the transition between smooth and impactful entry, occurring at $\unicode[STIX]{x1D6FC}\approx 70^{\circ }$ , can be anticipated using the same model.

Table 1.

Measured and theoretical force coefficient $C_{max}$ and time ratio $t_{max}^{\ast }$ (occurrence of maximum force).

Following Logvinovich (Reference Logvinovich1972), we consider the diving of a two-dimensional wedge of opening angle $\unicode[STIX]{x1D6FC}$ in the $(x,z)$ plane, with the wedge crossing the interface at a constant velocity $V$ . For convenience, we define the deadrise angle $\unicode[STIX]{x1D6FD}=(\unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC})/2$ . The hydrodynamic pressure everywhere in the fluid domain is given by the unsteady Bernoulli equation,

(3.1) $$\begin{eqnarray}p(x,z,t)=-\unicode[STIX]{x1D70C}\left(\frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}t}+\frac{1}{2}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}|^{2}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D711}$ is the velocity potential due to a horizontally expanding flat plate (plate of non-constant length to emulate the diving wedge) located on the free surface $z=0$ (see, for example, Howison, Hockendon & Wilson Reference Howison, Hockendon and Wilson1991):

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D711}=-V\sqrt{c(t)^{2}-x^{2}}.\end{eqnarray}$$

Here, $c$ is the location of the jet root defined as the extremity of the wetted surface (see figure 1 a), and is given by the so-called ‘Wagner condition’:

(3.3) $$\begin{eqnarray}c(t)=\unicode[STIX]{x03C0}Vt/(2\tan \unicode[STIX]{x1D6FD}).\end{eqnarray}$$

Relation (3.2) is valid for $|x|<c(t)$ , everywhere below the wedge’s wetted surface. The prediction of $c(t)$ is well supported by experimental measurements, as we show in § 4.1. The pressure on the contact region is given by (Korobkin Reference Korobkin2004)

(3.4) $$\begin{eqnarray}P(x,t)=-\unicode[STIX]{x1D70C}\left[\unicode[STIX]{x1D719}_{t}+{\textstyle \frac{1}{2}}V\sin (2\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D719}_{x}+{\textstyle \frac{1}{2}}\cos (\unicode[STIX]{x1D6FD})^{2}(\unicode[STIX]{x1D719}_{x}^{2}-V^{2})\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D719}$ is the velocity potential on the wedge’s surface, related to $\unicode[STIX]{x1D711}$ via the approximation

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D719}(x,t)\approx \unicode[STIX]{x1D711}(x,0,t)-V^{2}t=-V\sqrt{c(t)^{2}-x^{2}}+V^{2}t.\end{eqnarray}$$

For $\unicode[STIX]{x1D6FD}\ll 1$ , (3.4) becomes

(3.6) $$\begin{eqnarray}P(x,t)=-\unicode[STIX]{x1D70C}\left[\unicode[STIX]{x1D719}_{t}+{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D719}_{x}^{2}-V^{2})\right].\end{eqnarray}$$

Substituting (3.5) into (3.6) leads to the approximate pressure distribution along the wedge:

(3.7) $$\begin{eqnarray}P(x,t)=\frac{1}{2}\unicode[STIX]{x1D70C}V^{2}\left[\frac{\unicode[STIX]{x03C0}}{\tan \unicode[STIX]{x1D6FD}}\frac{c}{\sqrt{c^{2}-x^{2}}}-\frac{c^{2}}{c^{2}-x^{2}}\right].\end{eqnarray}$$

The first term in (3.7), called the ‘slamming pressure’, comes from the unsteady pressure contribution in Bernoulli’s equation and is associated with the rate of change of momentum of the added mass (Faltinsen Reference Faltinsen2005). The second term comes from the quadratic term and represents the dynamic pressure contribution. Figure 3(c) shows the non-dimensional pressure $P(x,t)/(\unicode[STIX]{x1D70C}V^{2})/2$ as a function of $x/c$ , as well as the two contributions given by the first and second terms on the right-hand side of (3.7). The slamming pressure increases with distance from the wedge’s keel and diverges at $x=c$ ; it also depends on the deadrise angle $\unicode[STIX]{x1D6FD}$ of the wedge. In contrast, the dynamic pressure is maximum at the keel and decreases with $x$ , tending to $-\infty$ as $x\rightarrow c$ , and is independent of the wedge’s angle. The total pressure is positive on most of the wetted region but becomes negative close to the jet root due to the dynamic pressure term. The sign change happens at $a/c=\sqrt{1-(\tan \unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0})^{2}}$ (marked by circular symbols in figure 3 c). This singularity is common to most modified Wagner models. It is regularized using the ansatz that only positive pressure matters (Logvinovich Reference Logvinovich1972; Korobkin Reference Korobkin2004): pressure should be integrated from $x=0$ to $x=a$ , which yields (accounting for both sides of the wedge)

(3.8) $$\begin{eqnarray}F(t)=2\int _{0}^{a(t)}P(x,t)\,\text{d}x=\unicode[STIX]{x1D70C}V^{2}c(t)\left[\frac{\unicode[STIX]{x03C0}^{2}}{2\tan \unicode[STIX]{x1D6FD}}-K(\unicode[STIX]{x1D6FD})\right],\end{eqnarray}$$

where

(3.9) $$\begin{eqnarray}K(\unicode[STIX]{x1D6FD})=\frac{\unicode[STIX]{x03C0}}{\tan \unicode[STIX]{x1D6FD}}\left[\frac{\unicode[STIX]{x03C0}}{2}-\arcsin (a/c)\right]-\frac{1}{2}\ln \left[\frac{1+a/c}{1-a/c}\right].\end{eqnarray}$$

To predict the maximum force, we assume that it is reached when the wetted length $c$ equals the wedge’s half-width  $d/2$ , that is to say, when the splash root escapes the wedge. From (3.3), we get that $t_{max}^{\ast }=2/\unicode[STIX]{x03C0}\approx 0.636$ , independently of $\unicode[STIX]{x1D6FC}$ . Substituting this prediction for $t_{max}^{\ast }$ into (3.8), (3.9) leads to

(3.10) $$\begin{eqnarray}F_{max}=\frac{1}{2}\unicode[STIX]{x1D70C}SV^{2}\left[\frac{\unicode[STIX]{x03C0}^{2}}{2\tan \unicode[STIX]{x1D6FD}}-K(\unicode[STIX]{x1D6FD})\right].\end{eqnarray}$$

In figure 3(a,b), we compare $F_{max}/(\unicode[STIX]{x1D70C}SV^{2})/2$ and $t_{max}^{\ast }$ (solid lines) to experimental data. The prediction for $t_{max}^{\ast }$ matches roughly with the experimental data for large $\unicode[STIX]{x1D6FC}$ , but the discrepancy increases sharply as $\unicode[STIX]{x1D6FC}$ decreases. The force prediction $F_{max}/(\unicode[STIX]{x1D70C}SV^{2})/2$ is consistently larger (by approximately 15 %) than the force data for $\unicode[STIX]{x1D6FC}>90^{\circ }$ . One reason for this discrepancy stems from the fact that the prediction is based on two-dimensional theory while our wedges are not infinitely long ( $L/d=4.2$ ); it is thus necessary to consider three-dimensional effects. Following the suggestion of Zhao et al. (Reference Zhao, Faltinsen and Aarsnes1996), we correct Logvinovich’s two-dimensional prediction using results of Meyerhoff (Reference Meyerhoff1970), who calculated the added mass coefficients of rectangular plates of various length-to-width ratio. His calculations show that in order to accurately represent the effect of finite aspect ratio in situations where added mass plays an important role, two-dimensional predictions have to be corrected by a factor approximately equal to $1-d/(2L)$ . This results in a 12 % decrease in maximum force (dashed line in figure 3 a), which is in good agreement with the experimental data for moderate and large wedge angles.

The values of $F_{max}$ and $t_{max}$ are based on the assumption that the maximum force happens when $c=d/2$ . Yet, the theoretical pressure profile along the wetted length (figure 3 c) is negative from $x=a$ to $x=c$ . To refine the prediction of $t_{max}$ , we consider the maximum force to happen when the region of negative pressure has completely escaped the wedge (that is, when $a=d/2$ ), which yields $t_{max}^{\ast }=(2/\unicode[STIX]{x03C0})(c/a)=(2/\unicode[STIX]{x03C0})/\sqrt{1-(2\tan \unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0})^{2}}$ . This improved prediction, represented in figure 3(b) as a dash-dotted line, increases sharply as $\unicode[STIX]{x1D6FC}$ decreases, but underestimates the experimental results, especially for moderate to large wedge angles. To refine the prediction further, we note that the integral of the pressure profile on the wedge continues to increase until $b=d/2$ , as shown in figure 3(c,d) (triangles). Numerical computation of this integral leads to a refined estimate of $t_{max}^{\ast }$ , represented as a dashed line in figure 3(b). This estimate is now in reasonable agreement with the experimental values for wedge angles down to $\unicode[STIX]{x1D6FC}\approx 80^{\circ }$ .

We now use the model to predict the transition from smooth and impactful entry. To this end, recall that the pressure on the wedge consists of two components: the ‘slamming’ pressure, which sharply increases with $\unicode[STIX]{x1D6FC}$ , and the dynamic pressure, which is independent of $\unicode[STIX]{x1D6FC}$ . For $\unicode[STIX]{x1D6FC}=120^{\circ }$ , the slamming pressure dominates, inducing a prominent peak in the pressure profile in figure 3(c). Consequently, the maximum force (indicated by triangles in figure 3 c) occurs immediately after the jet root leaves the wedge at $t_{max}^{\ast }\approx 2/\unicode[STIX]{x03C0}\simeq 0.636$ . This peak force before wedge submersion is characteristic of the impactful regime (see figure 2). As the wedge becomes sharper, the relative importance of the dynamic pressure increases and the pressure profile becomes flatter (figure 3 c). As a result, the time to reach the maximum force progressively increases. As it passes $t_{max}^{\ast }=1$ (wedge fully submerged), the entry is expected to be smooth: no peak before wedge submersion. This transition criterion gives $\unicode[STIX]{x1D6FC}\simeq 73^{\circ }$ (arrow in figure 3 b), in good agreement with the observed transition, between $60^{\circ }$ and $75^{\circ }$ . The important conclusion here is that the transition between the smooth and impactful regimes is expected to happen when the unsteady ‘slamming’ pressure has an effect comparable to the dynamic pressure, that is to say, when the rate of change of added mass is no longer the dominant effect. See supplementary movie 1, available at https://doi.org/10.1017/jfm.2018.273, for an illustration of the entry force in the smooth and impactful regimes.

Finally, it is interesting to compare the peak force with the drag force of a completely submerged wedge (figure 3 a). For large wedge angles, the peak force is significantly higher than the drag force. However, the peak force decreases sharply with the opening angle and, for $\unicode[STIX]{x1D6FC}<80^{\circ }$ , the measured maximum force becomes comparable to the drag coefficient of an immersed wedge. As $\unicode[STIX]{x1D6FC}$ decreases further, the peak force continues to decrease, in agreement with the impact force theory, to approximately half of the submerged drag force. This low value reflects that the drag force acting on diving wedges is smaller than we would expect for an immersed wedge, as we will explain in greater detail in the next section.

3.3 Drag force with a cavity

After submersion, the wedge reaches a quasi-steady regime characterized by a relatively small vertical force (figure 4). The measured force (dash-dotted line in figure 4) is significantly lower than the drag force on an immersed wedge of the same angle. The obvious difference between the two cases is the presence of the cavity in the wake of the diving wedge; see figure 5. The cavity has a notable effect on the wake of the wedge. There is no systematic model of how transient cavities affect drag on translating bodies; however, the influence of cavities on drag has been studied extensively in the context of cavitation (Knapp et al. Reference Knapp, Daily and Hammitt1970). Cavitation studies show that the presence of the cavity induces a dramatic reduction in drag, in the range two to three times. The reduction depends on the cavitation number $K=(p_{\infty }-p_{vap})/(\unicode[STIX]{x1D70C}V^{2})/2$ , which compares the drop in Bernoulli pressure $(\unicode[STIX]{x1D70C}V^{2})/2$ with the pressure drop needed to reach vapour pressure $p_{\infty }-p_{vap}$ in order for cavitation to happen. Here, $p_{\infty }$ is the pressure far away from the object. Typically, complete cavitation is reported for values up to $K\approx 0.4$ , and partial cavitation happens beyond this value (Knapp et al. Reference Knapp, Daily and Hammitt1970). Because the cavities in the present work are fully formed, a sensible range of value for $K$ is between 0 and 0.4. According to Knapp et al. (Reference Knapp, Daily and Hammitt1970) steady drag coefficients for supercavitating wedges can be expressed as $C_{D}=C_{D_{o}}(1+K)$ , with $C_{D_{o}}$ ranging from $0.489$ to $0.745$ for $\unicode[STIX]{x1D6FC}=60^{\circ }$ and $\unicode[STIX]{x1D6FC}=120^{\circ }$ . The corresponding ranges of $C_{D}$ , with $K$ varying from 0 to 0.4, are reported as a light khaki zone in figure 4 for each wedge angle. The experimental force measurements exhibit residual oscillations due to the drop mechanism. Oscillations notwithstanding, the predictions proved to be in good agreement with experimental data. It indicates that transient (inertial) cavities affect drag in the same way as cavitation.

Figure 4.

Force history on wedges with opening angles $\unicode[STIX]{x1D6FC}=60^{\circ },90^{\circ },120^{\circ }$ , in the limit of small hydrostatic effects ( $Fr>2.8$ ). The non-dimensional drag force on the entering wedge is significantly smaller than the corresponding drag coefficients of an immersed wedge (solid grey line); it matches the drag force measured on cavitating wedges (dash-dotted line and green overlay).

Figure 5.

Cavity and drag reduction. The presence of a transient cavity induces striking changes in the wedge’s wake and dramatically reduces the drag on the wedge. $\unicode[STIX]{x1D6FC}=90^{\circ }$ , $V\simeq 1~\text{m}~\text{s}^{-1}$ . Equivalent exposure for both pictures is $\unicode[STIX]{x0394}t=12$  ms.

4.1 Splash ejection: empirical observations

Using high-speed video recordings (see figure 7 a for an example), we systematically measure three relevant parameters of the splash kinematics at short times: the wetted length $c$ , jet speed $V_{so}$ and jet angle $\unicode[STIX]{x1D703}_{o}$ .

Figure 7.

Jet root expansion and jet velocity, measured from high-speed video (a); the vertical white dashed line shows the location of the jet root; the black arrow shows the jet tip. (b) Splash tip velocity $V_{so}$ is proportional to entry velocity $V$ . (c) The coefficient of proportionality increases with $\unicode[STIX]{x1D6FC}$ , and is measured to be $1.55\times {\dot{c}}/V$ . (d) The initial jet angle is independent of $V$ . (e) Before submersion, the shooting velocity and pitch angle are constant in time, as imposed by the non-penetration condition.

Raw measurements of $V_{so}$ as a function of entry velocity $V$ are shown in figure 7(b). Clearly, $V_{so}$ is linearly proportional to $V$ , even at low entry speeds, and the coefficient of proportionality $V_{so}/V$ is an increasing function of $\unicode[STIX]{x1D6FC}$ . In figure 7(c), we plot $V_{so}/V$ versus $\unicode[STIX]{x1D6FC}$ and compare it to the expansion speed of the wetted length ${\dot{c}}/V$ obtained experimentally (purple squares) and theoretically (solid line) based on equation (3.3):

(4.1) $$\begin{eqnarray}{\displaystyle \frac{V_{so}}{V}}=1.55\left({\displaystyle \frac{{\dot{c}}}{V}}\right)={\displaystyle \frac{1.55\unicode[STIX]{x03C0}}{2\tan \unicode[STIX]{x1D6FD}}}.\end{eqnarray}$$

The jet angle $\unicode[STIX]{x1D703}_{o}$ between the trajectory of the water particles in the jet and the horizontal axis is reported in figure 7(d). The schematic in figure 7(e) highlights the theoretical predictions of $\unicode[STIX]{x1D703}_{o}$ and $V_{so}/V$ assuming that the jet stays in contact with the wedge’s surface, leading to the implicit equation for $\unicode[STIX]{x1D703}_{o}$ ,

(4.2) $$\begin{eqnarray}\frac{\tan \unicode[STIX]{x1D6FC}}{\cos \unicode[STIX]{x1D703}_{o}-\sin \unicode[STIX]{x1D703}_{o}\tan \unicode[STIX]{x1D6FC}}={\displaystyle \frac{V_{so}}{V}}.\end{eqnarray}$$

Figure 7(d) shows that the theoretical predictions are in good agreement with the experimental measurements. Taken together, these observations imply that Wagner’s approach allows us to predict both the initial jet tip velocity $V_{so}$ and jet ejection angle $\unicode[STIX]{x1D703}_{o}$ , given a single corrective parameter (the 1.55 constant) that is independent of $\unicode[STIX]{x1D6FC}$ . We are now equipped to tackle the long-term evolution of the splash shape past the initial submersion of the wedge.

4.2 Splash model: kinematics and initial conditions

We develop a discrete fluid particle model of the splash based on the idea that the splash is primarily ballistic. The splash is represented by a succession of discrete particles $P_{j}$ , $j=1,2,3,\ldots \in \mathbb{N}$ . Particle $j$ is located at $\boldsymbol{r}_{j}=x(P_{j},t)\boldsymbol{e}_{1}+z(P_{j},t)\boldsymbol{e}_{3}$ , where $P_{j}$ is a Lagrangian label of particle $j$ , and moves at a velocity $\boldsymbol{v}_{j}=v_{x}(P_{j},t)\boldsymbol{e}_{1}+v_{z}(P_{j},t)\boldsymbol{e}_{3}$ under the influence of gravity, surface tension, and aerodynamic forces.

Figure 8.

Initial conditions applied to shooting particles. (a) For times before wedge submersion, particles are shot with constant velocity $V_{so}$ and angle $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{0}$ . (b) After submersion, $\unicode[STIX]{x1D703}$ is adjusted to maintain continuity with the cavity shape, and the shooting velocity $V_{s}$ decreases exponentially.

Figure 9.

(a) Overview of the model and physical processes. The pitch angle is tied to the cavity local slope, and the shooting velocity decreases exponentially in time. Particles are subject to weight, drag, surface tension, and Venturi-induced suction. (b–d) Schematic representation of forces on an individual particle: (b) anywhere along the splash, and (c) at the tip, where capillary retraction takes place. (d) Initial stages of capillary retraction, showing the first particle merging. A straightforward merging routine allows the retraction speed to quickly converges to the Taylor–Culick velocity $V_{\unicode[STIX]{x1D70E}}=\sqrt{2\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}e}$ , independently of the merging threshold.

Particles are initially ejected from the water surface ( $z=0$ ) at a shooting point $x(P_{j},t_{j})$ , where $t_{j}$ represents the time of ejection of particle $j$ . Our goal in this section is to determine the initial conditions, shooting position and velocity, for each particle. To this end, it is convenient to represent the velocity $\boldsymbol{v}_{j}$ of particle $j$ by its speed $V_{s}(P_{j},t)$ and pitch angle $\unicode[STIX]{x1D703}(P_{j},t)$ measured from the horizontal.

Before the wedge is fully submerged, we make the assumption that the shooting point is located at the intersection of the undisturbed water surface with the wedge given by $x=Vt_{j}\tan (\unicode[STIX]{x1D6FC})$ , and all particles are assumed to eject with the same velocity $V_{so}$ and angle $\unicode[STIX]{x1D703}_{o}$ (figure 8 a). That is to say, for ejection time $t_{j}\leqslant h/V$ , the initial conditions of particle $P_{j}$ are given by

(4.3a-c ) $$\begin{eqnarray}V_{s}(P_{j},t_{j})=V_{so},\quad \unicode[STIX]{x1D703}(P_{j},t_{j})=\unicode[STIX]{x1D703}_{o},\quad x(P_{j},t_{j})=Vt_{j}\tan (\unicode[STIX]{x1D6FC}).\end{eqnarray}$$

This assumption is in agreement with the actual interface shape below the wedge (see figure 7) in that we consider the fluid pile-up as part of the splash.

After submersion, $t_{j}\geqslant h/V$ , the shooting point is located at the cavity wall $w(z=0)=w_{o}$ , where $w$ is the cavity half-width and $w_{o}$ is the value of $w$ at the free water surface ( $z=0$ ); see figure 8(b). The initial shooting conditions are now dependent on the time of ejection $t_{j}$ . Namely, the shooting velocity decreases with time and eventually decays to 0 while the base of the splash becomes steeper with time. We consider the ejection velocity to decrease exponentially such that

(4.4) $$\begin{eqnarray}V_{s}(P_{j},t_{j})=V_{so}\text{e}^{-(t_{j}-h/V)/\unicode[STIX]{x1D70F}_{s}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{s}$ is a parameter that we set to $\unicode[STIX]{x1D70F}_{s}=0.025/V$ regardless of the wedge angle.

An implicit expression for the initial ejection angle $\unicode[STIX]{x1D703}(P_{j},t_{j})$ can be obtained by considering continuity between the slope of the splash and the slope of the cavity $w_{o}^{\prime }=\text{d}w/\text{d}z|_{z=0}$ ,

(4.5) $$\begin{eqnarray}{\displaystyle \frac{V_{s}(P_{j},t_{j})\sin (\unicode[STIX]{x1D703}(P_{j},t_{j}))}{V_{s}(P_{j},t_{j})\cos (\unicode[STIX]{x1D703}(P_{j},t_{j}))-\dot{w_{o}}(t_{j})}}=\tan (w_{o}^{\prime }(t_{j})).\end{eqnarray}$$

Here, $\dot{w_{o}}(t_{j})$ and $w_{o}^{\prime }(t_{j})$ are, respectively, the horizontal speed and the slope of the cavity at $z=0$ and $t=t_{j}$ .

To close the system in (4.4) and (4.5), we need a model for the time evolution of the cavity wall $w$ . Duclaux et al. (Reference Duclaux, Caille, Duez, Ybert, Bocquet and Clanet2007) applied a slice-averaged model to the transient dynamics of axisymmetric cavities created by spherical or cylindrical bodies. Using an unsteady potential flow model per slice, Duclaux et al. (Reference Duclaux, Caille, Duez, Ybert, Bocquet and Clanet2007) found that the evolution of the cavity radius $R$ can be described at each value of $z$ by the Rayleigh–Plesset equation, $R\ddot{R}+(3/2){\dot{R}}^{2}=\unicode[STIX]{x0394}p$ , where $\unicode[STIX]{x0394}p=-gz$ is the pressure difference between the inside and the outside of the cavity at depth $z$ , assuming atmospheric pressure inside the cavity. This equation is not directly applicable to wedges – one would have to derive a new analytic expression for the slice-averaged flow potential assuming the wedge is infinitely long. However, because our wedges have a relatively short aspect ratio $L/d$ , we make the assumption that the equation derived in Duclaux et al. (Reference Duclaux, Caille, Duez, Ybert, Bocquet and Clanet2007) applies in a modified form. Namely, we postulate that

(4.6) $$\begin{eqnarray}w\ddot{w}+{\textstyle \frac{3}{2}}C_{c}{\dot{w}}^{2}=-gz,\end{eqnarray}$$

where $C_{c}$ is an ad hoc parameter (equal to 1 for axisymmetric cavities). By definition, $C_{c}$ is independent of $V$ , but we expect it to depend on the geometry of the wedge, represented by the aspect ratio $L/d$ and opening angle $\unicode[STIX]{x1D6FC}$ . The value of $C_{c}$ is adjusted for each wedge, by matching the cavity dynamics to high-speed experimental images. To obtain the cavity dynamics, we solve (4.6) per slice for all slices from the top of the submerged wedge to the undisturbed water surface subject to initial conditions $w(t=0)=d/2$ and ${\dot{w}}(t=0)=V\tan \unicode[STIX]{x1D6FC}$ .

Put together, equations (4.4), (4.5) and (4.6) form a closed set of equations that determine the initial conditions $V_{s}(P_{j},t_{j})$ , $\unicode[STIX]{x1D703}(P_{j},t_{j})$ and $x(P_{j},t_{j})$ for the ejection of particle $P_{j}$ for $t_{j}\geqslant h/V$ .

4.3 Splash model: force balance

Figure 9(a) shows an overview of the physical processes affecting the splash and the cavity evolution during water entry. Particles are shot at regular intervals of time in such a way to be initially separated by a constant distance $d\ell _{o}$ . The mass per unit length attributed to each particle is $\unicode[STIX]{x1D6FF}m=\unicode[STIX]{x1D70C}e_{o}d\ell _{o}$ , where $e_{o}$ is the initial thickness of the splash sheet, assumed to be constant during splash ejection. Each particle is subject to the following forces: weight $\unicode[STIX]{x1D6FF}m\,\boldsymbol{g}$ , drag $\boldsymbol{F}_{\boldsymbol{D}}$ , surface tension $\boldsymbol{F}_{\unicode[STIX]{x1D748}}$ , and a Venturi-induced suction $\boldsymbol{F}_{\boldsymbol{V}}$ . All forces are expressed per unit length of the splash sheet unless otherwise stated. The force balance on particle $j$ is given by

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}m\frac{\text{d}\boldsymbol{v}_{j}}{\text{d}t}=\unicode[STIX]{x1D6FF}m\,\boldsymbol{g}+\boldsymbol{F}_{\boldsymbol{D}}+\boldsymbol{F}_{\unicode[STIX]{x1D748}}+\boldsymbol{F}_{\boldsymbol{V}}.\end{eqnarray}$$

Drag $\boldsymbol{F}_{\boldsymbol{D}}$ on particle $j$ is calculated by considering the splash segment $\unicode[STIX]{x0394}\boldsymbol{r}_{j}=\boldsymbol{r}_{j-1}-\boldsymbol{r}_{j+1}$ of length $d\ell _{j}=\Vert \unicode[STIX]{x0394}\boldsymbol{r}_{j}\Vert$ , moving with velocity $\boldsymbol{v}_{j}$ at an angle of attack $\unicode[STIX]{x1D713}_{j}$ defined as the angle between $\boldsymbol{v}_{j}$ and $\unicode[STIX]{x0394}\boldsymbol{r}_{j}$ . This leads to (Andersen, Pesavento & Wang Reference Andersen, Pesavento and Wang2005)

(4.8) $$\begin{eqnarray}\boldsymbol{F}_{\boldsymbol{D}}=-{\textstyle \frac{1}{2}}C_{D}\sin ^{2}(\unicode[STIX]{x1D713}_{j})\unicode[STIX]{x1D70C}_{a}d\ell _{j}\Vert \boldsymbol{v}_{j}\Vert \boldsymbol{v}_{j}.\end{eqnarray}$$

Here, $C_{D}$ designates the drag coefficient for when the element is perpendicular to the incoming flow, which we fix at $C_{D}=2.5$ for all experiments. The drag force is calculated using the absolute travelling velocity of the fluid particle, and not its velocity relative to the ambient air. The effect of the air flow is considered independently in the Venturi-suction force made explicit later on in this section.

Surface tension is accounted for in the most straightforward way: each particle experiences a longitudinal traction of magnitude $\unicode[STIX]{x1D70E}$ from its closest neighbours, as represented in figure 9(b). For $j\neq 1$ , one has

(4.9) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D748}}=\unicode[STIX]{x1D70E}\left(\frac{\boldsymbol{r}_{j-1}-\boldsymbol{r}_{j}}{\Vert \boldsymbol{r}_{j-1}-\boldsymbol{r}_{j}\Vert }+\frac{\boldsymbol{r}_{j}-\boldsymbol{r}_{j+1}}{\Vert \boldsymbol{r}_{j}-\boldsymbol{r}_{j+1}\Vert }\right).\end{eqnarray}$$

This force is normal to the local tangent $\unicode[STIX]{x0394}\boldsymbol{r}_{j}$ , and acts as a restoring force: it tends to cancel any shape curvature. The particle at the tip ( $j=1$ ) is a particular case: surface tension results in only one longitudinal force directed towards its next neighbour, leading to the retraction of the particle at the tip (figure 9 c). This retraction is observed in systems such as free sheets and ligaments (Marmottant & Villermaux Reference Marmottant and Villermaux2004; Lhuissier & Villermaux Reference Lhuissier and Villermaux2009). The speed of retraction, relative to the sheet, is constant and known as the Taylor–Culick velocity $V_{\unicode[STIX]{x1D70E}}=\sqrt{2\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}e}$ (Taylor Reference Taylor1959; Culick Reference Culick1960). To account for capillary retraction in our model, the receding particle of initial mass $\unicode[STIX]{x1D6FF}m$ has to merge with its successive closer neighbours, as shown in figure 9(d). When merging happens, (i) the new particle’s mass increases by $\unicode[STIX]{x1D6FF}m$ , (ii) the new particle is positioned at the centre of mass of the two former particles, and (iii) the new particle’s momentum is the sum of the momentum of the two former ones. Given these conditions, we observe that the retraction speed of the tip particle quickly converged to $V_{\unicode[STIX]{x1D70E}}$ , independently of the merging criteria (interparticle distance).

Figure 10.

(a) Average axial velocity into the chamber is determined using volume conservation. (b) Typical calculated axial flow field (purple) and resulting suction force on the sheet (blue).

The Venturi-induced suction force $\boldsymbol{F}_{\boldsymbol{V}}$ has a significant effect on the late-stage development of the sheet. This suction happens because the motion of the splash dome induces an axial flow of air rushing in to fill the expanding chamber beneath it, as represented in figure 10. The air is accelerated through the gap between the undisturbed water surface and the lower part of the splash, causing the pressure to drop and generating a downward suction. To calculate this suction force, we begin by evaluating the volume $\unicode[STIX]{x1D6FA}(x,t)$ of the chamber under the splash

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}(x,t)=\int _{x_{o}}^{x}Z(\tilde{x},t)\,\text{d}\tilde{x},\end{eqnarray}$$

where $x_{o}$ is the position of the base of the splash and $Z(x,t)$ refers to the local height of the splash above the free water surface. Assuming the chamber is two-dimensional – that is, no flow in the direction perpendicular to the drawing plane – the average volume change $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}(x,t)=\unicode[STIX]{x1D6FA}(x,t+\unicode[STIX]{x0394}t)-\unicode[STIX]{x1D6FA}(x,t)$ of the splash is related to the average flow $U(x,t)$ passing through the section $\bar{Z}=([Z(x,t)+Z(x,t+\unicode[STIX]{x0394}t)])/2$ , yielding

(4.11) $$\begin{eqnarray}U={\displaystyle \frac{1}{\bar{Z}}}{\displaystyle \frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x0394}t}}.\end{eqnarray}$$

Using Bernoulli’s principle between a position far away from the splash and the bottom of the sheet, we can estimate the pressure drop as $\unicode[STIX]{x0394}p_{v}=-(\unicode[STIX]{x1D70C}_{a}(kU)^{2})/2$ , where $kU$ is the corrected axial velocity. The correction factor $k$ accounts for the difference between the real system and the idealized model, including non-uniformity of this velocity in the vertical direction and three-dimensional effects. We expect $k$ to depend on $\unicode[STIX]{x1D6FC}$ : $k$ converges to $1$ as $\unicode[STIX]{x1D6FC}$ approaches $180^{\circ }$ because the splash evolves closer to the water surface and the lubrication approximation becomes more justified. We set $k=1.5$ for all computations presented in § 5 ( $\unicode[STIX]{x1D6FC}=120^{\circ }$ ). The magnitude of the suction force $\boldsymbol{F}_{\boldsymbol{V}}$ at particle $P_{j}$ is given by $\unicode[STIX]{x0394}p_{v}d\ell _{j}$ , where $\unicode[STIX]{x0394}p_{v}$ is evaluated at $x=x(P_{j},t)$ and $d\ell _{j}=\Vert \unicode[STIX]{x0394}\boldsymbol{r}_{j}\Vert$ , as previously defined. The direction of $\boldsymbol{F}_{\boldsymbol{V}}$ is along the local normal $\unicode[STIX]{x0394}\boldsymbol{r}_{j}^{\bot }/\Vert \unicode[STIX]{x0394}\boldsymbol{r}_{j}\Vert$ to the sheet, namely,

(4.12) $$\begin{eqnarray}\boldsymbol{F}_{\boldsymbol{V}}=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{a}(kU)^{2}\unicode[STIX]{x0394}\boldsymbol{r}_{j}^{\bot }.\end{eqnarray}$$

We substitute expressions (4.8), (4.9) and (4.12) for $\boldsymbol{F}_{\boldsymbol{D}}$ , $\boldsymbol{F}_{\unicode[STIX]{x1D748}}$ and $\boldsymbol{F}_{\boldsymbol{V}}$ into (4.7). The resulting equations are integrated using an explicit forward Euler method with typical time step $dt=10^{-6}~\text{s}$ . The initial distance between particles $d\ell _{o}=100~\unicode[STIX]{x03BC}\text{m}$ is chosen to be of the order of the sheet thickness.

5.1 Splash depression

In order to elucidate the nature of the vertical force causing the depression in the splash sheet, we examine the dynamics of the dip. In figure 13(a,b), we measure the vertical acceleration $\unicode[STIX]{x1D6FE}$ of a marker located at the local minima in the dip, designated by a red dot. The measurements are taken shortly after the downward dip becomes clear, in the time interval between the first two snapshots of figure 13. We find that acceleration is an increasing function of the splash velocity $V_{s}$ and, consequently, of $V$ , with values well above the gravitational acceleration: approximately $10g$ for the slowest velocity and up to $90g$ for $V_{s}=2.35~\text{m}~\text{s}^{-1}$ . Importantly, we find that for $V_{s}>7$ , $\unicode[STIX]{x1D6FE}$ is proportional to $V_{s}^{2}$ , suggesting the downward force has an aerodynamic origin.

We push the analysis further in order to determine the local thickness $e_{1}$ (and, thus, the initial thickness $e_{o}$ ) and to derive a criterion for the growth of the dip. To this end, we write the force balance in the vertical direction on a small fluid element of length $d\ell$ , width $L$ and thickness $e_{1}$ . The forces are drag $\boldsymbol{F}_{D}$ , surface tension $\boldsymbol{F}_{\unicode[STIX]{x1D70E}}$ acting upwards, and an aerodynamically related downward suction $\boldsymbol{F}_{V}=(\unicode[STIX]{x1D70C}_{a}d\ell LV_{s}^{2})/2$ . We ignore $\boldsymbol{F}_{D}$ because the vertical velocity of the marker is at least five times smaller than $V_{s}$ . We are left with the force balance in the vertical direction $\unicode[STIX]{x1D70C}e_{1}Ld\ell \unicode[STIX]{x1D6FE}=\boldsymbol{F}_{\unicode[STIX]{x1D70E}}-\boldsymbol{F}_{V}$ . Dividing throughout by $Ld\ell$ , we get the balance law in terms of the pressure difference between a capillary contribution $\unicode[STIX]{x0394}p_{\unicode[STIX]{x1D70E}}$ and an aerodynamic contribution $\unicode[STIX]{x0394}p_{V}=(\unicode[STIX]{x1D70C}_{a}V_{s}^{2})/2$ ,

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D70C}e_{1}\unicode[STIX]{x1D6FE}=-\unicode[STIX]{x0394}p_{\unicode[STIX]{x1D70E}}+\unicode[STIX]{x0394}p_{V}.\end{eqnarray}$$

From (5.1), we expect the growth of the dip to be inhibited when the restoring effect of surface tension overcomes the destabilizing effect of the suction pressure.

To derive a criterion for predicting the dip growth, we assume that the local radius of curvature of the splash scales as the wedge’s lateral size $d$ , in qualitative agreement with observation. The capillary pressure contribution is thus $\unicode[STIX]{x0394}p_{\unicode[STIX]{x1D70E}}\approx 2\unicode[STIX]{x1D70E}/d$ across the two interfaces. Substituting into (5.1), we get the following criteria for the growth of the dip: $(\unicode[STIX]{x1D70C}_{a}V_{so}^{2}>2\unicode[STIX]{x1D70E}/d)/2$ . For $d=36$  mm and $\unicode[STIX]{x1D70C}_{a}=1.25~\text{Kg}~\text{m}^{-3}$ , the criteria yield $V_{so}>2.5~\text{m}~\text{s}^{-1}$ . Our experimental trials correspond to $V_{so}$ between 4 and $10~\text{m}~\text{s}^{-1}$ , for which we always see dip growth, as expected. This criterion corresponds to $We_{a}>4$ when rewritten in terms of a modified Weber number $We_{a}=\unicode[STIX]{x1D70C}_{a}V_{so}^{2}d/\unicode[STIX]{x1D70E}$ . Unlike the Weber number commonly used in impact problems, $We_{a}$ depends on the density of air instead of water.

Equation (5.1) also yields an estimate of the local thickness $e_{1}=((\unicode[STIX]{x1D70C}_{a}V_{so}^{2}-2\unicode[STIX]{x1D70E}/d)/2)/\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}$ . A value of $e_{1}\approx 75~\unicode[STIX]{x03BC}\text{m}$ is obtained based on this expression; see inset of figure 13(b). We estimate $e_{1}/e_{0}\simeq 0.5$ and obtain $e_{0}\approx 150~\unicode[STIX]{x03BC}\text{m}$ , which is the value we adopted for all computational results presented in this work.

Figure 13(c) shows the flow visualization below the splash and validates the presence of a strong inward flow, at least of the same magnitude as the splash tip velocity. The relative variation of the horizontal magnitude is in good agreement with our model based on mass conservation, showing in particular a maximum below the dip.

5.2 Effect of non-dimensional parameters on splash shape

The modified Weber number $We_{a}$ reflects the competition between surface tension and aerodynamic suction. Figure 14(a) shows a schematic of the expected splash shapes for various $We_{a}$ . For low $We_{a}$ , the splash is shorter due to capillary retraction, and flatter because surface tension prevents the bending of the liquid sheet. For larger Weber numbers, the bending becomes more pronounced and the splash shapes reach further.

There are, however, two other effects that the splash shape depends on: drag and gravity. The primary effect of drag is to slow down the splash, in a fairly uniform fashion. One way to estimate the effect of drag on the splash shape is to calculate the relative deceleration of a particle along its trajectory. Let us consider the motion of a slice of splash, of mass $\unicode[STIX]{x1D6FF}m=\unicode[STIX]{x1D70C}e_{o}d\ell$ , travelling at velocity $v$ , over a distance proportional $d$ , and subject to a drag force $\boldsymbol{F}_{D}=(C_{D}\unicode[STIX]{x1D70C}_{a}d\ell v^{2})/2$ . According to Newton’s second law of motion, the deceleration can be approximated by $\unicode[STIX]{x1D70C}e_{o}\unicode[STIX]{x0394}v/\unicode[STIX]{x0394}t=(C_{D}\unicode[STIX]{x1D70C}_{a}v^{2})/2$ , which leads to, upon substituting $\unicode[STIX]{x0394}t\approx d/v$ ,

(5.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x0394}v}{v}\approx \frac{1}{2}C_{D}\frac{\unicode[STIX]{x1D70C}_{a}}{\unicode[STIX]{x1D70C}}\frac{d}{e_{o}}=\frac{I}{2}.\end{eqnarray}$$

The quantity $I$ is independent of $v$ and thus of the entry velocity $V$ , emphasizing that drag is not responsible for the various degrees of splash bending observed for different entry velocities $V$ . However, the shape of the splash is affected by changing $I$ , either by varying the air–water densities or by changing the dimensions of the wedge. Figure 14(b) shows the effect of $I$ on the splash shape. Smaller $I$ (thinner or lighter sheet) produces shorter splashes, without affecting the shape of the splash itself.

Lastly, we redefine a Froude number based on the splash velocity, $Fr_{s}=V_{so}/\sqrt{gd}$ , which reflects the competition between inertia and gravity. Figure 14(c) shows that decreasing $Fr_{s}$ (increasing gravity) induces a global downward motion and tilting of the whole splash. Significant differences are expected starting at $Fr_{s}=4$ . The experiments presented before are in the range $Fr_{s}=7.7{-}15$ , confirming gravity has indeed negligible impact on the splash shapes considered here.

Figure 14.

Snapshots of the predicted shapes taken at $t^{\ast }=2.5$ showing the effect of the three non-dimensional parameters affecting the splash shape: (a)  $We_{a}=\unicode[STIX]{x1D70C}_{a}V_{so}^{2}d/\unicode[STIX]{x1D70E}$ , (b)  $I=C_{D}(\unicode[STIX]{x1D70C}_{a}/\unicode[STIX]{x1D70C})d/e_{o}$ and (c)  $Fr_{s}=V_{so}/\sqrt{gd}$ .

5.3 Sheet fragmentation

The strong downward pull locally stretches the water sheet. To estimate the stretching ratio, given the initial sheet thickness that we estimated in the previous section ( $e_{o}=150~\unicode[STIX]{x03BC}\text{m}$ ), we need an estimate of the sheet thickness $e_{min}$ of the splash sheet at its lowest and thinnest point. Fortunately, we have access to $e_{min}$ by observing a spontaneous puncture of the sheet. Figure 15 presents the time sequence of the expansion of a hole in the bottom of the sheet at $t^{\ast }\approx 2.5$ . The hole expands in all directions at a velocity $V_{\unicode[STIX]{x1D70E}}=\sqrt{2\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}e_{min}}$ , known as the Taylor–Culick velocity (Taylor Reference Taylor1959; Culick Reference Culick1960). We measure $V_{\unicode[STIX]{x1D70E}}=5.25\pm 0.05~\text{m}~\text{s}^{-1}$ and calculate $e_{min}=5.1\pm 0.3~\unicode[STIX]{x03BC}\text{m}$ . This is approximately 30 times smaller than the estimated $e_{o}$ , in agreement with the maximum stretching ratio of last computable shapes of figure 12(c) ( $e_{min}/e_{o}\approx 50$ ). More importantly, it underlines the dramatic stretching induced by the downward suction. These calculations are the first steps towards developing a low-order model of fragmentation to be pursued in future work.

Figure 15.

(a,b) Front view of the splash at $t^{\ast }\approx 0.85$ for $V=1.15~\text{m}~\text{s}^{-1}$ showing the spontaneous opening of a hole on the thinnest part of the dip. (c) The hole expansion velocity is $2V_{\unicode[STIX]{x1D70E}}$ , where $V_{\unicode[STIX]{x1D70E}}=\sqrt{2\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}_{a}e_{min}}$ is the Taylor–Culick velocity, and provides an indirect measurement of the local sheet thickness $e_{min}\approx 5~\unicode[STIX]{x03BC}\text{m}$ , $30$ times smaller than the estimated initial sheet thickness $e_{o}$ .