2.1 Facility and instrumentation

Experiments were performed at the high-speed impact facility designed and built at CNR-INSEAN within the SMAES project. The facility is composed of a guide, 64 m long, suspended over a water basin, 13.5 m wide, by five bridges. The guide can be rotated, thus allowing $V/U$ ratios in the range 0.03–0.05 to be achieved. The trolley carrying the box structure with the acquisition system and the specimen to be tested is accelerated by a set of elastic cords from rest up to the final speed, and it is left to run freely along the guide during the impact phase. Eventually, the trolley is decelerated by the strong hydrodynamic loads exerted on the front of the box structure when the leading edge of the plate falls below the undisturbed water level.

The plates were instrumented with a total of 18 pressure probes, Kulite XTL 123B, full-scale range 300 Psi absolute, and six biaxial strain gauges. The positions of the pressure probes are shown in figure 1. Accelerations in the three directions were measured through Kistler M101 and M301 resistive accelerometers located at three different positions around the frame and at the centre of the plate. A total of six acceleration channels were acquired. The loads acting in the direction normal to the plate were measured by four Kistler type 9343 piezoelectric load cells with full-scale range of 70 kN each. Sensor signals were acquired on-board by four Sirius and one Dewe43 Dewesoft modules, controlled by a shock protected computer which was operated as a remote desktop in wireless mode. The sampling rate for the pressure signals was 200 kilosamples per second ( $\text{kS}~\text{s}^{-1}$ ), whereas all other channels were sampled at $20~\text{kS}~\text{s}^{-1}$ . The velocity at the initial impact time was measured by a high-speed camera located at the side. The camera was operated at 5000 f.p.s. and the pixel resolution on the measuring plane was 6.2 mm. The impact velocity was computed by using the time taken by the plate to cross a given pixel line, resulting in an uncertainty of $\pm 0.32~\text{m}~\text{s}^{-1}$ . The velocity reduction during the test was computed as the integral of the acceleration. A more detailed description of the facility and the set-up is provided in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), where the test repeatability and the inherent uncertainties are assessed as well.

2.2 Test conditions

The test conditions were chosen to be representative of aircraft ditching, and variations of single parameters were considered in order to investigate their influence on the results and on the physical phenomena (table 1). The horizontal velocity and the approach angle were decided in order to achieve a vertical velocity of $1.5~\text{m}~\text{s}^{-1}$ , which is the value to be used in airframe certification as requested by the air-worthiness authorities. Only for the test conditions at $4^{\circ }$ and $V/U=0.0375$ were horizontal velocities below $40~\text{m}~\text{s}^{-1}$ considered, which led to reduced vertical velocity components. As is discussed in the following, tests that give rise to a reduced loading allow investigation of the influence of the structural deformation of the guide on the measurements.

It is worth observing that in table 1 the nominal horizontal velocity $U$ is provided, whereas the actual horizontal velocity at the impact was within $\pm 0.5~\text{m}~\text{s}^{-1}$ of the target value (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015). In the analyses presented in the following, the actual velocities derived from the high-speed camera images just before the impact are used in the computation of the scaling factors.

The plate used for the tests was made of an aluminium alloy, AL2024 T351, and it was 15 mm thick, 1000 mm long and 500 mm wide. The plate was clamped to a much thicker aluminium frame over a region 75 mm wide all around, thus leaving an unsupported region in the middle of 850 mm by 350 mm (figure 1). The aluminium frame was connected to the box structure containing the acquisition system and then to the trolley. Although strains were measured, the structural aspects are not discussed here. On the basis of the data, it is estimated that the maximum out-of-plane deformation of the plate was of the order of 0.5 mm, which is rather small to affect the hydrodynamics.

Table 1 shows that several repeats were performed for all cases except for those at reduced vertical velocity. It is worth remarking that tests were performed in a new facility for which there was no preliminary information on test repeatability or on the uncertainties in the measurements. Hence, as discussed in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), it was decided to perform 10 repeats for the tests at $U=40~\text{m}~\text{s}^{-1}$ at the two limiting pitch angles, i.e. $10^{\circ }$ and $4^{\circ }$ . At $10^{\circ }$ the large number of repeats provides the uncertainty related to the facility itself, which is mainly in the measuring system, initial impact velocity and guide oscillations, whereas the repeats at $4^{\circ }$ give an estimate of the increased air-cushion effect. The analysis conducted in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015) indicates that for the $10^{\circ }$ case the strains and pressures of the different repeats are well overlapped. For the $4^{\circ }$ condition, the scatter of the data is still very small for the strains, but it is much larger for the pressure peak values, particularly for the probes located in the first two rows (figure 1). In this case the uncertainty in the data is not associated with the facility but with the air-cushion effects, which are well known to occur when the angle between the plate and the still water level is below $5^{\circ }$ (Chuang Reference Chuang1967; Okada & Sumi Reference Okada and Sumi2000). Independently of the test conditions, a large scatter of the data is observed for the total loads. In Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015) it is shown that the scatter of the data for the tests at $4^{\circ }$ and $10^{\circ }$ is quantitatively similar, but in the former case they appear to be relatively larger because of the lower average value. The independence of the scatter in the amplitude from the specific test conditions indicates that it is mostly associated with oscillations of the guide-rail system.

In the following, depending on the type of analysis, the data are presented either as single repeats or as ensemble averages. In all cases the raw data are provided in order to avoid the introduction of any spurious effect associated with the use of low-pass filters.

3.1 Theoretical solutions of the two-dimensional problem

In order to understand the fluid dynamics of ditching and to support the discussion of the experimental data, an approximate solution is derived. By following classical approaches to similar water entry problems, the mathematical model is formulated for an inviscid and constant-density fluid, with negligible effects caused by surface tension, gravity and air (Semenov & Iafrati Reference Semenov and Iafrati2006; Faltinsen & Semenov Reference Faltinsen and Semenov2008). It is assumed that both the horizontal ( $U$ ) and vertical ( $V$ ) velocity components of the plate remain constant during the impact and that the plate is semi-infinite in the longitudinal direction, with the trailing edge of the plate located at $x=Ut$ and $y=-Vt$ , where $x$ and $y$ denote the horizontal and vertical axes of an Earth-fixed frame of reference with origin at the initial contact point, $y=0$ being the undisturbed water level. The flow is approximated as two-dimensional, takes place on the $(x,y)$ plane and, on the basis of the above considerations, can be described in terms of the velocity potential ${\it\phi}(x,y)$ .

As there are no length scales, the problem is self-similar and can be formulated in terms of the new variables

With the definitions (3.1), the trailing edge of the plate is located at ${\it\xi}=1$ and ${\it\eta}=-V/U$ .

Figure 2. (a) The free-surface shape obtained by the fully nonlinear potential flow model. The solution refers to the conditions $V/U=0.0375$ , ${\it\alpha}=10^{\circ }$ . The dashed line drawn on the back of the plate represents the pressure distribution (not to scale). (b) A close-up view of the spray region. The dashed lines represent the projection on the plate of the pressure peak and spray root and are used to identify the parameters ${\it\sigma}_{{\it\psi}}$ and ${\it\sigma}_{R}$ .

By following the approach developed in Iafrati & Korobkin (Reference Iafrati and Korobkin2004, Reference Iafrati and Korobkin2008), the initial boundary value problem is transformed into a time-independent boundary value problem, which is solved by the iterative method briefly recalled in appendix A. The free-surface shape for a reference case is shown in figure 2 together with the corresponding non-dimensional pressure distribution

plotted (not to scale) on the back of the plate. The pressure distribution is drawn in figure 3 as a function of the abscissa along the body, ${\it\sigma}$ , the origin of which is at the trailing edge.

Figure 2 shows that, on approaching the plate from the right, the free-surface level rises, takes a sharp curvature at the root, and develops a thin and long spray. The angle formed by the free surface with the plate at the tip of the spray, located at ${\it\sigma}={\it\sigma}_{T}$ , is very small but finite. The pressure starts from zero at the trailing edge, increases moving towards the spray root, reaches a peak just behind it, drops abruptly and approaches zero pressure in the spray region. For this reason, in figure 3 the pressure is shown only in a reduced portion of the wetted surface. It is worth noticing that, due to the self-similarity, the farther a point is from the origin, the higher the speed is. As both the pressure peak, spray root and spray tip are located at ${\it\xi}>1$ , they are all faster than the plate. In the rear part, the free surface leaves the plate tangentially, rises up to a maximum which is reached just behind the initial contact point ${\it\xi}=0$ and gradually approaches the undisturbed water level ${\it\eta}=0$ as ${\it\xi}\rightarrow -\infty$ .

Figure 3. The non-dimensional pressure distribution on the plate. It should be noted that the wetted surface is much longer but the pressure is essentially zero in the spray. The solution refers to the case $V/U=0.0375$ , ${\it\alpha}=10^{\circ }$ .

In order to compare solutions for different impact conditions, some basic parameters are identified in figure 2 and figure 3 and reported in table 2. Some of the parameters denote the relevant points of the free-surface shape in terms of the abscissa along the plate, ${\it\sigma}$ . The most relevant for the analysis to be carried out in the following are the geometric intersection point between the plate and the still water level, located at ${\it\sigma}={\it\sigma}_{G}$ , where ${\it\sigma}_{G}=V/(U\sin {\it\alpha})$ , the spray root, located at ${\it\sigma}={\it\sigma}_{R}$ , and the spray tip, located at ${\it\sigma}={\it\sigma}_{T}$ . Other parameters are the non-dimensional coordinates $({\it\xi}_{I},{\it\eta}_{I})$ of the inversion point, the pressure peak, which is located at ${\it\sigma}={\it\sigma}_{{\it\psi}}$ where it takes a non-dimensional value ${\it\psi}={\it\psi}_{M}$ , and, finally, the non-dimensional value of the normal load acting on the plate per unit of transverse length, denoted by ${\mathcal{F}}$ . The latter is related to the dimensional term, $F(t)$ , by the following relation:

where $b$ is the breadth of the plate, which is $b=1$ in the two-dimensional assumption.

The data in table 2 show that ${\it\psi}_{M}$ varies substantially with the test conditions. As ${\it\psi}_{M}$ is derived from (3.2), such large variation indicates that $U^{2}$ is not the proper scaling factor for the pressure peak. By recalling the basic features of water entry flows (Cointe & Armand Reference Cointe and Armand1987; Howison, Ockendon & Wilson Reference Howison, Ockendon and Wilson1991), the pressure peak should correlate to the square of the velocity in the $x$ -direction of the spray root, $U(1+{\it\sigma}_{R}\cos {\it\alpha})$ , or to the square of the propagation velocity of the pressure peak which is approximately the same, $U(1+{\it\sigma}_{{\it\psi}}\cos {\it\alpha})$ . The above hypothesis is indeed confirmed by the data provided in figure 4(a), which show that the peak pressure coefficient

is nearly one for all impact conditions.

Figure 4. (a) The peak values of the pressure coefficient given by (3.4) for the different conditions. (b) The pressure distributions obtained for different pitch angles are drawn by using the peak intensity ${\it\psi}_{M}$ and its abscissa ${\it\sigma}_{{\it\psi}}$ to scale the vertical and horizontal axes respectively. The data indicate that the scaled pressure distribution has a larger area, i.e. a higher load, on increasing the pitch angle, whereas the effect of the $V/U$ ratio is very small.

Although the square of the horizontal velocity of the spray root is representative of the peak value, the pressure distribution along the plate also depends on the normal velocity of the relative flow, $U_{n}$ , which varies with both the $V/U$ ratio and the pitch angle as

where ${\it\beta}=\arctan V/U$ is the inclination angle of the guide. Such dependence can be easily recognized by the scaled pressure distributions reported in figure 4(b), which display a significant variation with the pitch angle, whereas the $V/U$ velocity ratio plays a less relevant role.

The solution and results derived above are used in the following as a basis for the discussion of the experimental data. However, several important assumptions are made in the derivation of the model which have to be accounted for when establishing comparisons. According to Korobkin & Pukhnachov (Reference Korobkin and Pukhnachov1988), the effects of liquid compressibility are negligible in the water entry of pointed bodies, which is the case considered here. Viscous effects might have an effect in the spray region where strong velocity gradients occur, but considering the quite high Reynolds number of the flow, $\mathit{Re}=U_{R}L_{R}/{\it\nu}\simeq 4\times 10^{7}$ , ${\it\nu}$ being the kinematic viscosity of water, the impact loads are essentially independent of them (Moghisi & Squire Reference Moghisi and Squire1981). The dynamics of the spray as well as its stability are also influenced by surface tension effects, as is briefly discussed at the beginning of the next section, but a detailed description of the spray region is out of the scope of the present work. For the problem under consideration, gravity is rather small compared to the inertia, $\mathit{Fr}=U_{R}/\sqrt{gL_{R}}\simeq 13$ , and there is experimental evidence that in such conditions there are no significant effects on the loads and on the spray root dynamics (Zhao, Faltinsen & Aarsnes Reference Zhao, Faltinsen and Aarsnes1996), although the dynamics of the spray may be affected to some extent. Finally, comparisons between experimental data and numerical results for the water entry of finite wedges (Zhao et al. Reference Zhao, Faltinsen and Aarsnes1996; Zhao, Faltinsen & Haslum Reference Zhao, Faltinsen and Haslum1997; Iafrati & Battistin Reference Iafrati and Battistin2003) indicate that the assumption of a semi-infinite plate, which is essential for the self-similarity of the solution, is acceptable at least up to the time when the spray root reaches the leading edge of the plate.

Because of the peculiarities of the facility and of the experimental set-up and conditions, other assumptions deserve more careful consideration. In particular, it is important to evaluate the role played by the air, by the variation of the test velocity due to both deceleration of the trolley and oscillation of the guide as well as by the third dimension. The latter especially is expected to have a significant influence as the transverse breadth of the plate is only half the plate length and thus the possibility for fluid to escape from the side may affect the spray root propagation velocity, the pressure distribution and, consequently, the loads. The effect of the three-dimensionality on the loads is theoretically addressed in Korobkin & Scolan (Reference Korobkin and Scolan2006), whereas the strong changes in the spray root propagation velocity, pressure distribution and loads when passing from a two-dimensional wedge or cylinder to the corresponding axisymmetric shape, cone or sphere, are clearly highlighted in Battistin & Iafrati (Reference Battistin and Iafrati2003). The above aspects are discussed in the following with the aim of explaining the deviation of the experimental results from the theoretical model.

3.2 Experimental results for the pressure peak propagation velocity

On the basis of what has been discussed so far, as soon as the plate touches the water surface, a thin spray develops and propagates along the plate, becoming longer and longer as the plate penetrates. The spray formation and propagation can be seen from the sequences of underwater images provided in figures 5(b,d,f,h,j) and 6(b,d,f). In order to help in the interpretation of the images, dashed lines are drawn along the sides of the plate and a solid line is drawn at the spray root. The plate is moving from left to right and the spray develops on the right of the spray root. The underwater images show that the spray root is not exactly straight but develops a curved shape as a consequence of outflow at the sides, as is discussed below. A preliminary discussion on the spray root shape and on the differences observed for the $4^{\circ }$ and $10^{\circ }$ pitch angles is provided in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), whereas a much deeper and more quantitative analysis of the spray root is presented in the next section.

Figure 5. Impact sequence for test conditions of $45~\text{m}~\text{s}^{-1}$ , $10^{\circ }$ . Images from the side and underwater cameras are shown in (a,c,e,g,i) and (b,d,f,h,j) respectively. The plate is moving from left to right and the time interval between two successive images in the sequence is 0.005 s. In (a,c,e,g,i), the white solid lines are used to highlight the tip of the splash released on the side as a result of three-dimensional effects. In (b,d,f,h,j), the white solid lines are used to locate the spray root, whereas the dashed lines are drawn about the contour of the plate.

Figure 6. Images from the side and underwater cameras taken at the same penetration depth for different pitch angles: (a,b), (c,d) and (e,f) show the $10^{\circ }$ , $6^{\circ }$ and $4^{\circ }$ conditions. The images refer to $45~\text{m}~\text{s}^{-1}$ horizontal velocity. The white lines have the same meaning as in figure 5.

Just to give an estimate of the tip velocity, from the underwater images in figure 5(b,d,f,h) it is seen that, for the selected test conditions, the spray tip reaches the leading edge of the plate approximately 0.015 s after the first contact. Since the plate is 1 m long, the propagation velocity of the spray tip along the plate is of the order of $66~\text{m}~\text{s}^{-1}$ , which corresponds to an absolute horizontal velocity of the order of $110~\text{m}~\text{s}^{-1}$ . Because of such high speed and the interaction with the air and with the plate surface, the thinnest part of the spray is fragmented into tiny droplets, the size of which is governed by surface tension (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015).

The images from the camera at the side provided in figure 5(a,c,e,g,i) show the development of two water splashes at the sides of the plate. A white solid line is drawn to highlight the tip of the splash located on the side closest to the camera and to distinguish it from the one located on the opposite side. Such splashes are a consequence of the flow outgoing from the side of the plate due to the finite breadth, which in fact is a three-dimensional effect. The splashes generated for different pitch angles are shown in figure 6(a,c,e). The three images, which are taken at the same penetration depth, indicate that the splash is stronger for larger pitch angle. The more intense outflow generated for larger pitch angles is associated with the higher value of the normal velocity component (3.5) and, consequently, with the higher mean pressure value on the plate (figure 4 b). The above result supports the expectation that three-dimensional effects are more important for larger pitch angles.

In the previous section it was explained that the pressure peak is strongly correlated with the spray root propagation velocity. Moreover, it was seen that from the theoretical solution, the propagation velocity of the spray root is very close to that of the pressure peak, and indeed the latter was used for the computation of the corresponding pressure coefficient from (3.4). Even though the result is well known and accepted in the field of water entry (Cointe & Armand Reference Cointe and Armand1987; Howison et al. Reference Howison, Ockendon and Wilson1991), it is worth verifying whether a similar relation holds for the present experimental data.

Unfortunately, as shown in figures 5 and 6, due to the too poor spatial and temporal resolution, the underwater images do not allow reliable estimates to be derived for the spray root propagation velocity and the only possibility is to exploit the pressure measurements and use the pressure peak propagation velocity for the scaling. To this purpose, the time interval taken by the pressure peak to reach different probes located at the same transverse position is computed first. The probe P4, located in the middle of the plate on the first sensor row $r1$ (figure 1) is used to define the reference for the time. The time delay ${\it\tau}_{j}$ taken by the pressure peak to reach the $j$ th probe is computed as the time shift leading to the maximum correlation between the pressure history recorded at P4 and that at P $j$ (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015).

The time delays for the test conditions at 40 and $45~\text{m}~\text{s}^{-1}$ are shown in figure 7. The vertical coordinate $s_{P}$ denotes the longitudinal position of the pressure peak that coincides with that of the $j$ th pressure probe when $t={\it\tau}_{j}$ . It should be noted that hereafter $s$ indicates the abscissa along the plate, with origin at the trailing edge. For all impact conditions the data follow nearly linear trends, with the propagation velocity of the pressure peak along the plate being somewhat higher than that of the geometric intersection, $U{\it\sigma}_{G}$ or $V/\text{sin}({\it\alpha})$ .

Figure 7. Time delays of the probes for the different test conditions. The data in (a) and (b) refer to the tests at $40~\text{m}~\text{s}^{-1}$ and $45~\text{m}~\text{s}^{-1}$ respectively. The vertical coordinate $s_{P}$ denotes the distance of the probe from the trailing edge measured along the plate. The straight lines represent the position along the plate of the geometric intersection point computed as $Vt/\sin {\it\alpha}$ , where $V$ is the nominal value of the vertical velocity.

Starting from the longitudinal positions of the probes and the time delays ${\it\tau}_{j}$ , the propagation velocity of the pressure peak along the plate is computed by a second-order finite difference scheme, and its projection on the $x$ -axis is denoted by $u_{P}$ . The propagation velocity can be computed by using the data of probes located at the same transverse position. Hence, this is done for probes located along the midline of the plate, i.e. P4, P8, P12, P16, P18, which are referred to as M, and for those located at the side, i.e. P1, P6, P9, P14, P17, which are referred to as O (figure 1). The propagation velocities computed for the tests at 40 and $45~\text{m}~\text{s}^{-1}$ and for the different pitch angles are provided in figure 8. In order to facilitate the comparison between the different test conditions, the horizontal component of the pressure peak propagation velocity is scaled by the horizontal velocity of the geometric intersection point, $u_{G}=U{\it\sigma}_{G}\cos {\it\alpha}=V/\tan ({\it\alpha})$ . The data indicate that the ratio $u_{P}/u_{G}$ diminishes as time elapses and the plate penetrates into the water, with maximum values in the range 1.3–1.6 and minimum values in the range 0.90–1.05.

Figure 8. The propagation velocity of the spray root, $u_{P}$ , scaled by the propagation velocity of the geometric intersection point, $u_{G}$ . The results in (a,c,e) and (b,d,f) refer to tests at $40~\text{m}~\text{s}^{-1}$ and $45~\text{m}~\text{s}^{-1}$ respectively. The data for all test repeats are drawn in order to provide an estimate of the related uncertainty.

The variation of the propagation velocity with time already indicates that the behaviour is not self-similar. In order to establish more quantitative comparisons with the theoretical solutions, it is worth recalling that, in terms of the notation introduced in the previous section, the horizontal component of the pressure peak propagation velocity along the plate is $U{\it\sigma}_{{\it\psi}}\cos {\it\alpha}$ , $U$ being the nominal test speed, and hence $u_{P}/u_{G}$ should be compared with ${\it\sigma}_{{\it\psi}}/{\it\sigma}_{G}$ . The data provided in table 2 show that this ratio is between 2 and 3, with lower values obtained for smaller pitch angles. In all cases the theoretical values are much higher than the experimental values.

Similar conclusions may be derived by the analysis of the experimental data reported in Smiley (Reference Smiley1951). At least for the test condition referred to as Run 8 in the report, the times of the pressure peak occurrence at the different probes, and thus the corresponding propagation velocities, can be retrieved from the curves provided in figure 2, on p. 29 of the report. The test conditions are $U=19.05~\text{m}~\text{s}^{-1}$ , $V=1.585~\text{m}~\text{s}^{-1}$ , $12^{\circ }$ pitch angle. It should be noted that in Smiley (Reference Smiley1951) the aspect ratio of the plate is 5 instead of 2 for the present test conditions. The extracted data, which are reported in figure 9, exhibit the same behaviour as the present experiments even though the $u_{P}/u_{G}$ velocity ratio displays a much stronger reduction, with the propagation velocity dropping below one third of the propagation velocity of the geometric intersection point.

Figure 9. (a) The time delays of the probes as derived from the data of figure 2 in Smiley (Reference Smiley1951). The corresponding values of the propagation velocity scaled by the velocity of the geometric intersection point are given in (b).

As anticipated at the end of the previous section, in order to better understand the reasons for the large reduction in the propagation velocity of the pressure peak with respect to that of the geometric intersection point, the role played by some of the assumptions made in the derivation of the theoretical model has to be investigated. In particular, bearing in mind the basic features of the solution discussed in the previous section, it is difficult to justify a propagation velocity of the pressure peak below that of the geometric intersection point, i.e. $u_{P}/u_{G}<1$ , as occurs at the end of the curves in figure 8.

A first point deserving consideration is the value of $u_{G}$ adopted for the scaling. The curves in figure 8 are drawn by using the nominal value of $u_{G}$ computed as $V/\text{tan}{\it\alpha}$ , and thus the deceleration of the trolley during the tests is not taken into account. For instance, the data of Run 8 provided in table II of Smiley (Reference Smiley1951) indicate that during the impact the vertical velocity undergoes a reduction by a factor of $1/3.5$ , and thus the nominal value of $u_{G}$ greatly overestimates the propagation velocity of the geometric intersection point, which explains most of the reduction observed in figure 9(b).

Figure 10. The time histories of the $u_{P}/u_{G}$ ratio for the tests at $10^{\circ }$ at $40~\text{m}~\text{s}^{-1}$ (a) and $45~\text{m}~\text{s}^{-1}$ (b). The deceleration of the trolley is accounted for in $u_{G}^{a}$ .

In order to derive an improved estimate of the propagation velocity of the geometric intersection point, the velocity reduction during the impact phase is computed as the time integral of the acceleration, and the actual horizontal velocity of the trolley, $U_{a}(t)$ , is obtained by subtracting the velocity reduction from the initial impact velocity. For the present experimental campaign, thanks to the quite high mass of the trolley, the velocity reduction is very small, being of the order of $0.5~\text{m}~\text{s}^{-1}$ in the $4^{\circ }$ case and $2~\text{m}~\text{s}^{-1}$ in the $10^{\circ }$ case (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015). By using the instantaneous entry velocity, the actual propagation velocity of the geometric intersection point is computed as $u_{G}^{a}=U_{a}(t){\it\sigma}_{G}\cos ({\it\alpha})$ . The data for $u_{P}/u_{G}^{a}$ are provided in figure 10 for the tests at $10^{\circ }$ , which are characterized by the largest velocity reduction. Apart from a small increase compared with $u_{P}/u_{G}$ that can be observed at the end of the curves, $u_{P}/u_{G}^{a}<1$ still holds. Hence, the deceleration of the trolley is not sufficient to explain the propagation velocity of the pressure peak being below that of the geometric intersection.

In addition to the effect of the deceleration on the horizontal velocity, the vertical velocity of the plate may also vary as a consequence of the structural deformation of the guide when exposed to the high loads generated during the impact. Such variations have direct consequences on the propagation velocity of both the pressure peak and the geometric intersection point.

Figure 11. Effect of the test speed and of the guide inclination on the propagation speed. In both cases the data at the end of the tests indicate that higher values of $u_{P}/u_{G}^{a}$ are obtained for lower horizontal speed.

Quantitative estimates of the vertical displacement and velocity measured at one point of the guide are provided in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015) for one of the tests performed at $40~\text{m}~\text{s}^{-1}$ , $10^{\circ }$ . In the early phase, the vertical velocity displays oscillations about a non-zero value with a period of approximately 0.0125 s and maximum upwards values of the order of $0.3~\text{m}~\text{s}^{-1}$ . In the next stage, the period rises to approximately 0.025 s and the amplitude of the oscillations grows: the vertical velocity of the guide changes from upwards to downwards and takes peak values of approximately $0.5~\text{m}~\text{s}^{-1}$ . In the very late stage of the entry process, just before the spray root leaves the plate, the velocity reaches a peak value of approximately $0.7~\text{m}~\text{s}^{-1}$ upwards. Of course the occurrence of upwards velocity leads to reductions in the pressure peak propagation velocity, in the pressure values and in the total loading. In particular, the upward velocity of $0.7~\text{m}~\text{s}^{-1}$ , being approximately half of the nominal value, is expected to have relevant effects. However, as explained, such a high value occurs in a very late stage of the entry process lasting approximately 0.01 s, which is well beyond the stage considered in the above discussion. Furthermore, for the tests at $10^{\circ }$ the impact phase lasts approximately 0.1 s, and considering the period of oscillations, it appears to be more reasonable to filter them out by computing an average vertical velocity based on the displacement. By using this approach it is estimated that during the impact phase the guide has an average upwards velocity of approximately $0.076~\text{m}~\text{s}^{-1}$ , which is approximately 5 % of the nominal value. As the structural deformation of the guide is not accounted for in the evaluation of $u_{G}^{a}$ , the reduction in the average vertical velocity and the corresponding reduction in the propagation velocity of the geometric intersection point explain the occurrence of $u_{P}/u_{G}^{a}<1$ in some conditions.

Detailed data on the guide bending are not available for all test conditions, but it is reasonable to expect that the structural deformations are smaller for lower hydrodynamic loads. In order to further investigate this aspect, in figure 11 two comparisons are established. In figure 11(a) the data for three tests performed at the same $V/U$ and pitch angle but at different horizontal speeds are provided. As is discussed in the following, the loads scale as the square of the normal velocity component, and thus the loads generated during the tests at $34.5~\text{m}~\text{s}^{-1}$ are approximately 30 % smaller than those at $40~\text{m}~\text{s}^{-1}$ (figure 24). Despite such significant reduction in the loading, no substantial differences are found in terms of $u_{P}/u_{G}^{a}$ (figure 11 a). Another comparison is presented in figure 11(b). Again, the horizontal speed is varied, but in this case both the vertical velocity and the pitch angle are kept constant. From the data provided in figure 25 it is seen that in the tests at $30~\text{m}~\text{s}^{-1}$ the normal loading on the plate is approximately 45 % of that measured at $40~\text{m}~\text{s}^{-1}$ and even smaller compared with the $45~\text{m}~\text{s}^{-1}$ condition. Nonetheless, again, in spite of the large reduction in the loading, a limited increase of approximately 10 % is observed at the end of the curves. Such an increase is, however, found to always have $u_{P}/u_{G}^{a}>1$ for the $30~\text{m}~\text{s}^{-1}$ condition.

From the above considerations, it is follows that, even accounting for the reduction of the deceleration of the trolley and for the upwards velocity induced by the structural deformation of the guide, the propagation velocity of the pressure peak remains much lower than that predicted by the two-dimensional approximation, and in any case it diminishes substantially as the plate penetrates, gradually approaching that of the geometric intersection point.

It is speculated here that both the lower value of the pressure peak propagation velocity and its reduction during the impact are consequences of the three-dimensionality of the problem. Such a statement could only be confirmed through additional tests performed on plates of different breadth or by numerical simulations of the full three-dimensional flow. However, important arguments to support the above conjecture are provided in Battistin & Iafrati (Reference Battistin and Iafrati2003), where the free-surface profiles generated during the water entry of a wedge and a cylinder are compared with the profiles obtained for the corresponding cone and sphere. Whereas the propagation velocity of the geometric intersection point is the same, the results clearly show that in the axisymmetric cases, when the fluid can move in all directions, the propagation velocity of the pressure peak, or of the spray root, is much smaller than that in the two-dimensional configuration. For the rectangular plate considered here, the ratio between the longitudinal and transverse dimensions of the wetted area grows in time, and the two quantities become comparable in approximately the middle of the impact phase. Hence, in the first half of the entry process the free-surface flow undergoes a transition from quasi-two-dimensional to a fully three-dimensional one, and that is in fact the phase during which the sharp reduction in the $u_{P}/u_{G}$ ratio is observed.

3.3 Pressure peak line and spray root shapes

The underwater images in figures 5 and 6 show that the spray root is characterized by a backward curvature, with the foremost point reached at the middle of the plate. Here and in the following, the curvature is intended in an average sense to indicate the amount of bending of the spray root line. As already anticipated in the previous section, the curvature of the spray root is a consequence of the three-dimensionality and of the possibility for the fluid to exit from the sides. Therefore, a quantitative estimate of the spray root shape provides important insights into the hydrodynamics and into the effects of the third dimension and the pitch angle. Being very sensitive to the flow features, the spray root shape is also very useful for the validation of computational approaches.

As mentioned above, the quality of the underwater images in terms of pixel resolution, sharpness and contrast is too poor and does not allow the spray root shape to be retrieved accurately. In § 3.1 it has been explained that the spray root is just ahead of the pressure peak location (see figure 2(b) and data in table 2) and, thus, instead of the spray root shape, the pressure peak line is derived here starting from the pressure measurements, as is explained in the following.

The high sampling rate of the measurements provides very accurate information on the time at which the pressure peak crosses each probe. Therefore, the shape of the pressure peak line can be evaluated from the times at which the pressure peak reaches the probes located at the same longitudinal position, i.e. located along the same sensor row. If all of the probes located along the same sensor row are reached by the pressure peak at the same time, the pressure peak line is just straight. Alternatively, the shape of the pressure peak line is estimated by multiplying the time differences between the probes by the pressure peak propagation velocity computed in the previous section. Of course there is a test-to-test dispersion of the data in terms of either the time differences or the propagation velocity. Some data concerning the dispersion of the time delays as well as some considerations about the symmetry of the data are presented in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), whereas the scatter in the pressure peak propagation velocity can be estimated from the data shown in the previous section. In the following the results are presented in terms of the ensemble average over all of the repeats performed at the same test conditions.

Figure 12. Ensemble averages of the time delays of the probes located in the five rows of pressure probes computed with respect to the probe located in the same row at the middle of the plate ( $z=0$ ). The longitudinal coordinate of the row grows from top to bottom (figure 1). The data in (a,c,e,g,i) and (b,d,f,h,j) refer to tests at $40~\text{m}~\text{s}^{-1}$ and $45~\text{m}~\text{s}^{-1}$ respectively. It should be noted that lines are only drawn with the purpose of connecting points belonging to the same test condition.

The time delays with respect to the probe lying in the same row at the middle of the plate are shown in figure 12 for the different test conditions. The data are presented for the right side of the plate, i.e. $z\leqslant 0$ , where most of the probes are located (figure 1). The data in figure 12(a,b) show that for the $4^{\circ }$ condition, the distribution of the time delays in the first sensor row $r1$ is quite different from the other cases and, furthermore, negative time delays are also found. Negative time delays mean that the pressure peak reaches the probes located at the side earlier than that at the middle. This phenomenon, already discussed in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), is associated with the air-cushion effect which is relevant for the $4^{\circ }$ case, whereas it is much less important for larger pitch angles due to the increased possibility for air to escape from the side (Chuang Reference Chuang1967; Okada & Sumi Reference Okada and Sumi2000).

A quantitative estimate of the free-surface deformation caused by the air-cushion effect can be derived by computing the stagnation pressure of the airflow with respect to the plate. For the $4^{\circ }$ case at $45~\text{m}~\text{s}^{-1}$ , the normal velocity component of the relative airflow is $V\cos {\it\alpha}+U\sin {\it\alpha}=4.64~\text{m}~\text{s}^{-1}$ , and therefore, for an air density of $1.25~\text{kg}~\text{m}^{-3}$ , the stagnation pressure is $P_{s}=13.46$ Pa, which corresponds to a water column of 1.37 mm. For a vertical entry velocity of $1.5~\text{m}~\text{s}^{-1}$ , this implies that the lowest part of the free surface is touched by the plate approximately $915~{\rm\mu}\text{s}$ after the plate edges. Such an estimate is consistent with what was found from the analysis of the underwater movie, not presented here, which shows that it takes approximately 3 frames, or $1000~{\rm\mu}\text{s}$ , for the trailing edge to be fully wetted. As soon as the spray root starts to move along the plate, the initial effects of the air cushion are compensated by the three-dimensionality of the flow and, as shown in figure 12(a,b), already at the first sensor row, the time delays are an order of magnitude smaller.

Moving forward along the plate, as already observed in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015) on the basis of the data for the two test conditions at $40~\text{m}~\text{s}^{-1}$ , $4^{\circ }$ and $10^{\circ }$ pitch angles, there is a growth in the time delays of the probes located away from the midline. The time delays measured in the tests at $40~\text{m}~\text{s}^{-1}$ are generally larger than those measured in the tests at $45~\text{m}~\text{s}^{-1}$ , which can be explained by the higher propagation velocity of the pressure peak for higher horizontal velocities.

As explained above, the shape of the pressure peak line is obtained by multiplying the time delays provided in figure 12 by the pressure peak propagation velocity as computed in the previous section. Since the propagation velocity of the pressure peak is obtained by numerical differentiation, it may introduce additional errors to the results. Moreover, the available data do not allow any estimate concerning the variation of the propagation velocity in the transverse direction to be extracted, and therefore the pressure peak lines are computed by assuming the propagation velocity to be uniform in the transverse direction and equal to the value it takes in the middle of the plate for each sensor row.

Figure 13. The shapes of the pressure peak lines computed as the product of the time delays of the different probes by their propagation velocities. The results in (a,c,e) and (b,d,f) refer to the tests performed at $40~\text{m}~\text{s}^{-1}$ and $45~\text{m}~\text{s}^{-1}$ respectively. As in figure 12, data are obtained from the ensemble average over the different repeats and are computed for the right side of the plate, $z\leqslant 0$ , where most of the probes are located (figure 1). In the keys, $r1$ – $r5$ indicate the five sensor rows and the lines are drawn with the only purpose of connecting points belonging to the same row.

Figure 14. Underwater images of the tests at $10^{\circ }$ (a,c) and $4^{\circ }$ (b,d). The dashed lines are drawn around the edges of the plate. The trailing edge is in the lower part of the images and the plate is moving upwards. The white solid lines highlight the spray root. The images in (a,b) refer to a time T1 when the spray root is at sensor row $r2$ , i.e. 0.40 m ahead of the trailing edge (figure 1). The images in (c,d) refer to a time T2 when the spray is approximately 0.65 and 0.80 m from the trailing edge for (c) and (d) respectively.

The data for the pressure peak line are presented in terms of the ensemble average of the shapes computed for the different repeats of the same test conditions (figure 13). The results show that the curvature of the line grows moving forward along the plate. However, there are also cases in which the curvature at row $r2$ seems to be larger than that at row $r3$ (e.g. figure 13 a–c). On the other hand, the data in figure 12(c,e,f) indicate that the time delays are still growing and, thus, the larger curvature observed at row $r2$ compared with that at row $r3$ can be related either to the errors in the computation of the propagation velocity or to the oscillations in the vertical velocity induced by the structural deformations of the guide.

Another conclusion that can be drawn from the data is that for the $10^{\circ }$ case the pressure peak line quickly develops its final shape and the curvature at row $r2$ is approximately the same as that at row $r5$ . On the contrary, for smaller pitch angles the growth of the curvature is slower and more regular. The interesting aspect is that at row $r5$ the ${\rm\Delta}s$ between P18 and P17 is essentially the same for all test conditions. With regard to the latter aspect, from the data in figure 12 it is seen that the time delays at row $r5$ are generally larger for larger pitch angles. This result means that the difference in time is compensated by the difference in the pressure peak propagation velocity, which is lower for larger pitch angles.

The faster growth of the curvature at larger pitch angles and the similar shapes of the pressure peak lines occurring at row $r5$ are also observed for the spray root lines displayed in the underwater pictures (figure 14). Figure 14(a) and (b) for the $10^{\circ }$ and $4^{\circ }$ cases respectively refer to a time T1 at which the spray roots are approximately 0.40 m ahead of the trailing edge of the plate, which means at row $r3$ . By comparing them, it can be noticed that for the $10^{\circ }$ case the spray root exhibits a noticeable curvature already, whereas for the $4^{\circ }$ case the curvature is less pronounced. A similar comparison is established in figure 14(c,d), for a later stage. The two images correspond to a time T2 at which the spray is approximately 0.65 and 0.80 m ahead of the trailing edge for $10^{\circ }$ and $4^{\circ }$ respectively. The 0.80 m distance from the trailing edge means that the spray is at sensor row $r5$ . Due to the different inclination of the plate, for the $10^{\circ }$ condition the spray reaches sensor row $r5$ outside the field of view of the underwater camera, and thus a shorter distance is considered. Even though the light level is quite low, it is observed that in the $10^{\circ }$ case the shapes of the spray root at the two times are comparable, whereas in the $4^{\circ }$ case there is a noticeable increase in the curvature. Moreover, the curvature in figure 14(d) looks quantitatively similar to that in figure 14(a), in agreement with what has been found in terms of the pressure peak line.

All of the above considerations are supported by the data shown in figure 15, where the spray root lines drawn on the underwater images of figure 14 are digitalized and plotted together with the data available for the pressure peak lines. The spray root is provided in terms of both the digitalized points and their Bezier interpolation. From the comparison of the spray root lines obtained at the two times for $10^{\circ }$ it is seen that the shapes are essentially unchanged, confirming that for the large pitch angle the spray root develops its final shape rapidly. On the contrary, for the $4^{\circ }$ case, the curves display a more regular growth, in agreement with what has been observed on the basis of the pressure peak lines. Bearing in mind the limits in the pixel resolution, which can be argued by the distributions of the symbols, the data in figure 15 also support the fact that, in a later stage, the spray root approaches the same shape independently of the pitch angle. Finally, a quite satisfactory quantitative agreement is found between the spray root and the pressure peak lines.

Figure 15. Comparison between the spray root (SR) derived from the underwater images in figure 14 and the data for the pressure peak lines (PL) provided in figure 13. For the sake of clarity, the symbols for the SR are shown for $z>0$ only, whereas the symbols for $z\leqslant 0$ refer to the pressure peak lines. The data refer to the tests performed at $45~\text{m}~\text{s}^{-1}$ for the $4^{\circ }$ (a) and $10^{\circ }$ (b) conditions.

3.4 Pressure distributions and scaling

In Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015) some basic features of the pressure acting on the ditching plate have been discussed. In the following a more quantitative analysis of the data is presented, aimed at achieving a better comprehension of the role played by the different parameters.

The time histories of the pressure recorded during the tests at 40 and $45~\text{m}~\text{s}^{-1}$ by probes located along the midline of the plate for the three different pitch angles are shown in figure 16. All of the curves are characterized by a quite similar behaviour: a sharp rise, a pressure peak and a gentle decay. Moreover, a widening of the area underneath the curves can be observed moving towards the leading edge of the plate. Such features are consistent with the theoretical predictions discussed above. However, aside from some fluctuations, more evident at $4^{\circ }$ due to the air cushion and to the bubble entrainment (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), the data display a general reduction of the pressure peak moving forward along the plate. As already discussed at the end of § 3.2, considering the increasing role played by the finiteness of the transverse dimension during the plate penetration and the changes in the pressure distribution observed in Battistin & Iafrati (Reference Battistin and Iafrati2003) when passing from a cylinder to a sphere, such reduction in the pressure peak seems to be associated with three-dimensional effects.

It is worth noticing that the times of the sharp rises of the pressure in figure 16(a,c,e) and (b,d,f) are essentially the same. As observed when commenting on figure 7, this result indicates that the propagation along the plate is not greatly dependent on the horizontal velocity but rather on the vertical velocity and on the pitch angle, which, in the end, define the propagation velocity of the geometric intersection point.

Figure 16. Time histories of the pressure measured by probes located along the midline of the plate during one single test performed at $40~\text{m}~\text{s}^{-1}$ (a,c,e) and $45~\text{m}~\text{s}^{-1}$ (b,d,f) horizontal velocity. In (a,b), (c,d) and (e,f) the data refer to $4^{\circ }$ , $6^{\circ }$ and $10^{\circ }$ conditions respectively. Within each panel, the data from probes P4, P8, P12, P16 and P18 can be recognized by the position of the peak, which occurs earlier for probes located closer to the trailing edge.

Figure 17. Pressure coefficients computed as in (3.6). From (a–c) the data refer to tests at $4^{\circ }$ , $6^{\circ }$ and $10^{\circ }$ respectively. All repeats carried out at the same test conditions are shown.

The theoretical analysis presented in § 3.1 and, more specifically, the results in figure 4(a) have shown that the pressure peak equals the stagnation pressure associated with the horizontal velocity of the peak or, say, its absolute propagation velocity. Therefore, the reduction of the pressure peak observed in figure 16 moving forward along the plate might be simply explained by the reduction of the pressure peak propagation velocity (figure 8). In order to verify the above hypothesis, the coefficient

is computed for the pressure peaks $p^{P}$ measured at the probes located along the midline of the plate in all test repeats, and results are reported in figure 17. On the basis of the positions of the probes given in figure 1, the data at $s=0.125$ m refer to probe P4, those at $s=0.25$ m to probe P8, and so forth, up to P18 at $s=0.8$ m.

Aside from some scatter in the data, the results indicate that for all tests and for all probes the coefficient $C_{p}^{P}$ is nearly one, confirming what has been discussed in § 3.1 for the theoretical solution. The origin of the scatter is in both the pressure peak and the propagation velocity. For the $4^{\circ }$ case, and for P4 in particular, it is mainly related to the scatter in the pressure peak as a consequence of the air-cushion effect and bubble entrainment (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015). This also explains why the coefficient is greatly below one in some test repeats. In the $10^{\circ }$ case, the close-up view of the pressure peak region provided in figure 18(a) displays a very good overlapping of the data for the 10 repeats carried out for the test condition at $40~\text{m}~\text{s}^{-1}$ . The maximum variation of the pressure peak is of the order of 7 %, whereas the scatter in the $C_{p}^{P}$ coefficient is of the order of 20 %. Hence, it is conjectured here that the additional variation is associated with the scatter in the propagation velocity observed in figure 10(a,b) for $t=0$  s.

Some considerations are deserved by the double peak occurring in figure 18(a), the one on the left being quite sharp. The origin of the double peak in the pressure measurements is discussed in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), where it is shown that it is related to the interaction between the flow and the probe. This spurious first peak is responsible for the quite large pressures, with $C_{p}^{P}\simeq 1.3$ , observed in the tests at $45~\text{m}~\text{s}^{-1}$ and $6^{\circ }$ (figure 17 b). The close-up view of the pressure peak region for probe P18, displayed in figure 18(b), shows that at $45~\text{m}~\text{s}^{-1}$ the first peak is much higher than that occurring at $40~\text{m}~\text{s}^{-1}$ . This difference can be explained by the fact that the probes were removed and reinstalled between the two test series. It is believed that the differences in the first peak are related to the different positioning of the probe with respect to the plate surface. Nevertheless, these effects are characterized by a very short duration and have no influence on the next phase.

Figure 18. Close-up views of pressure peaks. (a) The pressure peak recorded at probe P4 in the 10 repeats of the tests at $10^{\circ }$ , $40~\text{m}~\text{s}^{-1}$ conditions. (b) A comparison of the pressure peaks measured at P18 during the three repeats of the tests at $6^{\circ }$ for the 40 and $45~\text{m}~\text{s}^{-1}$ conditions.

As said above, the fact that the pressure peak equals the stagnation pressure associated with its propagation velocity in the horizontal direction is indeed in agreement with the theory. However, since the propagation velocity is far below the theoretical value, the pressure peak intensity is correspondingly much lower than the value predicted by the self-similar solution. From table 2, for the $10^{\circ }$ and $V/U=0.0375$ conditions, the non-dimensional value of the pressure peak is ${\it\psi}^{M}=1.261$ , which corresponds to a peak value of approximately 2020 kPa. From figure 18(a) it is seen that at the probe P4, which is in the first sensor row where the propagation velocity has the maximum values among those observed along the plate, the pressure peak is in a range between 1450 and 1550 kPa, far below the theoretical estimate. Moving forward along the plate, with the reduction of the propagation velocity the agreement is even worse, and at probe P18 the pressure peak drops to a range between 1080 and 1124 kPa.

The relations between the pressure peak and its propagation velocity or the propagation velocity of the geometric intersection point were already considered in Smiley (Reference Smiley1951). As shown in figure 5 of Smiley (Reference Smiley1951), for most of the data for the tests at pitch angle lower than $12^{\circ }$ the pressure coefficients were far below one, which is apparently in contrast to the present data and to the classical results on water entry (Cointe & Armand Reference Cointe and Armand1987). It is believed that the too low sampling frequency and maybe the large size of the probes filtered the peak out (Van Nuffel et al. Reference Van Nuffel, Vepa, De Baere, Degrieck, De Rouck and Van Paepegem2013).

Figure 19. Pressure distributions: pressures measured in all test repeats by the probes located along the midline of the plate at the time when the peak arrives at P18. The horizontal and vertical axes are scaled by the abscissa ( $s^{P}$ ) and intensity ( $p^{P}$ ) of the pressure peak. The solid line represents the fully nonlinear self-similar theoretical solution.

Besides the pressure peak, it is interesting to establish comparisons in terms of the pressure distributions. A few points of the pressure profiles can be extracted from the time histories of the pressures measured by the five probes located at the middle of the plate at the time when the pressure peak is at P18. The data for tests performed at $40~\text{m}~\text{s}^{-1}$ horizontal velocity, which correspond to $V/U=0.0375$ , for $4^{\circ }$ , $6^{\circ }$ and $10^{\circ }$ pitch angle are shown in figure 19. For the $10^{\circ }$ case the results at $30~\text{m}~\text{s}^{-1}$ , or $V/U=0.05$ , are shown as well. The pressure values are scaled by the pressure peak measured at P18 and by its abscissa, $s^{18}=0.800$  m.

As a general comment, the experimental data exhibit a drop on the left of the peak which is, on average, not far from the theoretical prediction. A quite good agreement is found for tests at $40~\text{m}~\text{s}^{-1}$ , $6^{\circ }$ and at $30~\text{m}~\text{s}^{-1}$ , $10^{\circ }$ . For the two tests at $40~\text{m}~\text{s}^{-1}$ , $4^{\circ }$ and $10^{\circ }$ , the data at probe P16 are far above and far below the theoretical prediction respectively. With regard to the differences between the tests at $10^{\circ }$ performed at 40 and $30~\text{m}~\text{s}^{-1}$ , it is not clear whether they are due to a different role played by the three-dimensional effects at 30 and $40~\text{m}~\text{s}^{-1}$ or to the different bending of the guide.

Figure 20. The transverse distribution of the pressure peaks at the different sensor rows: $r1$ (▫), $r2$ (○), $r3$ (▵), $r4$ (▿), $r5$ (♢).

In order to examine the role played by the sides and the effect of the flow in the transverse direction, the pressure peaks measured by the probes in different test conditions are drawn in figure 20 versus the transverse coordinate. The data are provided in terms of the ensemble averages over the repeats carried out in the same test conditions. It should be noted that, due some technical problems with the probe, the data for P3 are missing (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015). Moreover, due to the problem with the first peak discussed above, the data for P18 are not reported for the test case at $6^{\circ }$ and $45~\text{m}~\text{s}^{-1}$ . As the propagation velocity is not available at all probes, the pressure coefficient is computed on the basis of the horizontal velocity of the geometric intersection point, that is

In general, the data in figure 20 are characterized by quite large variations, which have different origins. For the $4^{\circ }$ case, the variation is primarily related to the effect of the bubble entrainment and to the air cushion. Indeed, the pressure peaks recorded at the first row are far below those measured at the other rows. Moving forward, the pressure peaks rise and approach $C_{p}^{G}\simeq 1$ , which is consistent with the values $u_{P}/u_{G}\simeq 1$ found at sensor row $r5$ (figure 8). For the test at $10^{\circ }$ , the highest peaks are measured at the first row, and they gradually diminish moving forward. In this case the variation is only related to the reduction of the pressure peak propagation velocity.

Maybe because of the effects of the air cushion, for the $4^{\circ }$ and $6^{\circ }$ cases it is not possible to identify a clear dependence of the pressure peak on the transverse $z$ -coordinate. For instance, in figure 20(a) the profile at row $r4$ display a maximum in the middle and a minimum at the side, but at row $r5$ probes P17 and P18 take approximately the same peak value. For the $10^{\circ }$ case the transverse profiles are more regular. However, although the data at row $r1$ in figure 20(e) would suggest a distribution with the maximum at the middle, the profiles at the next sensor rows become more uniform. A nearly uniform distribution is also found in figure 20(f) for the tests at $45~\text{m}~\text{s}^{-1}$ .

From the above considerations, it follows that the available data do not allow any conclusions to be derived on the side effect, which is in contrast with what is shown in figure 6(a,b) of Smiley (Reference Smiley1951), where transverse distributions with peaks located a short distance from the edges appear for the $6^{\circ }$ and $15^{\circ }$ cases. Such a distribution is indeed consistent with what would be obtained by applying a $2D+t$ representation based on the pressure distribution generated during the water entry of a flat plate (Iafrati & Korobkin Reference Iafrati and Korobkin2008). However, it seems that the profiles proposed in Smiley (Reference Smiley1951) are rather fortuitous as they are based on a single test measurement and on the data from a single sensor row with four probes. According to what is shown in Iafrati & Korobkin (Reference Iafrati and Korobkin2008), the peak in the transverse distribution moves inwards as the plate penetrates, and thus it appears coincidental that the peak occurs at the same probe for $6^{\circ }$ and $15^{\circ }$ , also taking into account the different velocity reductions that characterize the two tests.

Nonetheless, a deeper investigation of the transverse distribution would be interesting. In particular, it would be important to understand to what extent the transverse distribution is affected by the differences in the set-up and in the test conditions adopted in the present experiments and in Smiley (Reference Smiley1951). Among the most relevant differences, it is worth mentioning the aspect ratio of the plate and the corresponding three-dimensionality of the flow, the probe size and sampling rate and the variation of the entry velocity during the tests. Considering the quite high complexity of the test campaign, it is believed that such an investigation can only be conducted by using highly resolved numerical simulations of the three-dimensional problem.

3.5 Hydrodynamic loads and scaling

The loads acting on the plate are measured by a total of six piezoelectric cells, four used to measure the normal loads and two to measure the loads acting parallel to the plate, in the longitudinal direction. The latter are not relevant during the impact phase, that is the phase during which the spray root is propagating along the plate, even though they become quite large in the next phase when the leading edge of the plate falls below the still water level and help to stop the trolley before it reaches the end of the guide (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015). For the purpose of the present study the normal loads are far more interesting and deserve consideration.

Figure 21. Comparison between the time histories of the normal loads measured in tests at $40~\text{m}~\text{s}^{-1}$ (a) and $45~\text{m}~\text{s}^{-1}$ (b) at $4^{\circ }$ (solid), $6^{\circ }$ (dashed) and $10^{\circ }$ (dash–dot).

Figure 22. (a) The time histories of the normal loads measured at the three pitch angles and at the two horizontal speeds. All of the curves approach a $(t-t_{I})^{1/2}$ behaviour for $t-t_{I}>0.01$ s. (b) The normal loads measured in the tests at $40~\text{m}~\text{s}^{-1}$ scaled as indicated in (3.3) and compared with the linear trend predicted by the self-similar two-dimensional model.

A preliminary analysis of the loads acting on the plates for the tests at $40~\text{m}~\text{s}^{-1}$ , $4^{\circ }$ and $10^{\circ }$ pitch angles is presented in Iafrati et al. (Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015), where the ensemble averages and the standard deviation of the data are provided. The normal loads measured in the different test conditions considered here are shown in figure 21. For each test condition the data are given in terms of the ensemble averages over the different repeats, and a time shift $t_{I}$ is applied in order to have them synchronized at the initial contact. In all cases, soon after the initial contact, the data display a quasi-linear rise lasting a short time interval. Subsequently, the loads continue to increase but the growth rate diminishes. A sharp change in the slope occurs at approximately $t{-}t_{I}\simeq 0.01$  s, for all test conditions. The fact that the change in the slope is independent of the horizontal velocity and the pitch angle indicates that it is related to the penetration depth, the vertical velocity being the same for all test conditions.

The data show that for both the 40 and $45~\text{m}~\text{s}^{-1}$ test conditions the curves display a quite similar initial trend and the slope in the next phase is also comparable. The differences in the maximum loads at the end of the entry process are a consequence of the different pitch angle, as shown later on. For a better estimate of the growth rate, the time histories of the normal loads are drawn in logarithmic scale in figure 22(a), and it is seen that for $t>0.006$  s all curves approach a $t^{1/2}$ behaviour.

The above results differ considerably from the theoretical predictions. Of course, the theory predicts a linear variation, in contrast to the square-root variation exhibited by the measurements in the later stage of the water entry. Moreover, according to the data in table 2, the fully nonlinear model predicts a substantial dependence of the force coefficient on the pitch angle, the values for $6^{\circ }$ and $4^{\circ }$ being approximately 15 % and 30 % larger than that at $10^{\circ }$ respectively, which is not observed in the experimental measurements. More relevant, however, is that the experimental data are far below the theoretical estimates. This is clearly shown in figure 22(b), where the forces measured in the tests at $40~\text{m}~\text{s}^{-1}$ are scaled as in (3.3) and comparisons with corresponding linear trends based on the force coefficients ${\mathcal{F}}$ provided in table 2 are established. Quantitatively, for the $40~\text{m}~\text{s}^{-1}$ , $10^{\circ }$ conditions, the force coefficient provided by the theoretical model is 0.200. For a plate breadth of 0.5 m, from (3.3) 128 kN is obtained at $t-t_{I}=0.02$ s, three times larger than the measured value of approximately 40 kN.

As the difference appears from the very beginning, it seems to be associated, mostly at least, with three-dimensional effects. In the previous sections reference has been already made to the comparison of the pressure distributions for a cylinder and a sphere provided in Battistin & Iafrati (Reference Battistin and Iafrati2003). The results provided therein clearly show the substantial reduction of the pressure and of the wetted area, both effects contributing to a reduction of the loads occurring when the flow changes from a pure two-dimensional case to a pure three-dimensional one, which is what happens here in the first half of the entry phase. There could be, of course, some other effects related to the deceleration of the trolley or to the structural deformations of the guide, but they seem to be minor as the loads behave similarly in all test conditions (figure 22 a).

The uncertainty in the data and the small changes in the values recorded for the different test conditions make the identification of a parametric dependence of the loads rather challenging. Furthermore, the comparison in terms of time histories is not fair as it is established among loads acting on different surface portions. On the basis of the data available from the experiments, there are two possible ways of comparing loads acting on the same surface. One way is to compare the loads at the time when the pressure peak arrives at the same sensor row. This approach is rather accurate, but it provides the data on five points only. Another option is to use the position of the geometric intersection point to retrieve the wetted length. Of course, such an approach is inherently approximate as the pressure peak propagation velocity is generally faster (figure 10), but it gives information also for the phase when the spray root is beyond sensor row $r5$ , which is the phase when the highest loads are generated. For the purpose of the study, the latter approach is preferred. The time axis is multiplied by $V/\sin {\it\alpha}$ , which is the velocity at which the geometric intersection point moves along the plate.

Results reported in figure 23(a,b) show that in all cases the lines at which the pressure peak arrives at the leading edge are located at approximately 0.9 m, and not at 1 m as they should be. This is of course a consequence of the assumption made about the propagation velocity, and confirms that, at least in the early phase, the pressure peak propagates faster than the geometric intersection point. Aside from such mismatch between the computed and the actual wetted surface, the representation in terms of the geometric wetted length clearly highlights the increase of the loads acting on the plate for larger pitch angles ${\it\alpha}$ , as a consequence of the higher value of the normal velocity component (3.5). This is confirmed by the data drawn in figure 23(c,d), where the loads are scaled by the square of the normal velocity component, $U_{n}^{2}$ .

In order to distinguish the role played by the velocity from that of the pitch angle and guide inclination, a comparison is established in figure 24 between tests performed at different horizontal velocities but at the same ${\it\alpha}$ and ${\it\beta}$ . As the measurements of single repeats are used for the tests at 34.5 and $38.5~\text{m}~\text{s}^{-1}$ , the corresponding data are characterized by larger oscillations compared with those exhibited in the $40~\text{m}~\text{s}^{-1}$ case for which the ensemble average is shown. However, on average, the three curves of the loads scaled by the square of the normal velocity exhibit a quite good overlapping. Since the two angles ${\it\alpha}$ and ${\it\beta}$ are the same for the three tests, the results indicate that the loads scale with the square of the impact velocity.

Nevertheless, by comparing the results in figures 21(a) and 21(b) it is seen that the effect of the horizontal velocity is more relevant for large pitch angles because of the larger contribution to the normal velocity. This is made clearer in figure 25, where the loads measured during the tests performed at three different horizontal velocities but at the same vertical velocity and pitch angle are compared. Due to the large pitch angle, the loads display a large variation with the horizontal velocity. The data scaled by the square of the normal velocity exhibit a quite good overlapping, at least up to approximately half of the plate. Further, the data measured in the $30~\text{m}~\text{s}^{-1}$ condition are somewhat beneath the other two solutions.

On the basis of the above considerations, it can be concluded that the parameters of the problem, i.e. horizontal velocity, vertical to horizontal velocity ratio and pitch angle, affect the loads mainly through their effect on the normal velocity component. Moreover, the results indicate that the pressure peak, being rather sharp, has only a limited effect on the loads.

Figure 23. The loads acting on the plates as a function of the submerged length. The data refer to tests at $40~\text{m}~\text{s}^{-1}$ (a,c) and $45~\text{m}~\text{s}^{-1}$ (b,d) at $4^{\circ }$ (solid), $6^{\circ }$ (dashed) and $10^{\circ }$ (dash–dot).

Figure 24. Comparison between loads measured during tests performed at different velocities, keeping the pitch angle ( $4^{\circ }$ ) and the inclination of the guide ( $1.5/40$ ) constant.

Figure 25. Comparison between loads measured during the tests at different horizontal velocities and guide inclinations, keeping the pitch angle ( $10^{\circ }$ ) and the vertical velocity ( $V=1.5~\text{m}~\text{s}^{-1}$ ) constant.

Before closing the section, it is worth providing some data on the position of the centre of the loads $s_{c}$ , which can be retrieved by exploiting the loads measured by the two couples of load cells located at the rear and ahead of the box structure carrying the plates (Iafrati et al. Reference Iafrati, Grizzi, Siemann and Benítez Montañés2015). The data for the three pitch angles and different horizontal velocities are shown in figure 26, together with the lines that identify the time of initial contact and that at which the pressure peak reaches the leading edge. In the very early stage, the loads measured by the cells are of small amplitude and the oscillations are such that unrealistic locations of the centre of the loads are obtained. These values are not shown in the graphs. In all cases, the coordinate $s_{c}$ is characterized by a sharp initial growth up to $t\simeq 0.01$ s and follows a linear rise afterwards. The data indicate that there is no significant effect of the horizontal velocity. It is interesting to note that in all cases the maximum value is taken at the time when the pressure peak reaches the leading edge, when $s_{c}\simeq 0.8$ m. In the next stage, the pressure peak leaves the plate surface and the pressure is more uniform along the plate, which is the reason why the centre of the loads moves backwards.

Figure 26. Time histories of the centre of the loads for the tests at $40~\text{m}~\text{s}^{-1}$ (solid) and $45~\text{m}~\text{s}^{-1}$ (dashed). From (a–c) the data refer to tests at $4^{\circ }$ , $6^{\circ }$ and $10^{\circ }$ respectively.