High-speed water impacts of flat plates in different ditching configuration through a Riemann-ALE SPH model

The violent water entry of flat plates is investigated using a Riemann-arbitrary Eulerian-Lagrangian (ALE) smoothed particle hydrodynamics (SPH) model. The test conditions are of interest for problems related to aircraft and helicopter emergency landing in water. Three main parameters are considered: the horizontal velocity, the approach angle (i.e., vertical to horizontal velocity ratio) and the pitch angle, α. Regarding the latter, small angles are considered in this study. As described in the theoretical work by Zhao and Faltinsen (1993), for small α a very thin, high-speed jet of water is formed, and the time-spatial gradients of the pressure field are extremely high. These test conditions are very challenging for numerical solvers. In the present study an enhanced SPH model is firstly tested on a purely vertical impact with deadrise angle α = 4°. An in-depth validation against analytical solutions and experimental results is carried out, highlighting the several critical aspects of the numerical modelling of this kind of flow, especially when pressure peaks are to be captured. A discussion on the main difficulties when comparing to model scale experiments is also provided. Then, the more realistic case of a plate with both horizontal and vertical velocity components is discussed and compared to ditching experiments recently carried out at CNR-INSEAN. In the latter case both 2-D and 3-D simulations are considered and the importance of 3-D effects on the pressure peak is discussed for α = 4° and α = 10°.

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Adopted SPH scheme
In the present work Euler equations for compressible fluids are solved. Indeed, since the Reynolds number of the flow is quite high and only the impact stage is simulated (short time-range regime) viscous effects can be considered negligible. The weakly-compressible model is adopted, the fluid is, therefore, assumed to be barotropic and a classical stiffened state equation is used can be reached, and in this case the pressure peak intensity becomes proportional to 1/Ma . On the other hand, in such a condition an incompressible constraint can induce singularities on the pressure field (see Ref. [6] for a discussion on the difference between these two models in impact situations). This is linked to the fact that for this kind of impacts the presence of the air phase is generally crucial and the single-phase approach can lead to incorrect pressure evaluations under the incompressible/weakly-compressible hypothesis (for a deeper discussion see also Ref. [4]). Being aware of these limits of the single-phase model, the results obtained in this paper have been produced considering a possible Mach dependency.
In the present work the Riemann-based solver described in Ref. [7] is adopted. In that work the ALE formalism is used allowing for maintaining a regular particle spatial distribution and smooth pressure fields while preserving the whole scheme conservation and consistency of the classical SPH scheme. The introduction of the Riemann-based solver in the SPH scheme leads to an increased stability and robustness of the scheme with respect to the standard SPH formulation. The formalism proposed by Ref. [8], Thanks to the introduction of Riemann-solvers the fluxes between particles are upwind oriented and the resulting scheme is characterized by good stability properties. The discrete Euler equations are written as follows: where E  , E P and E v are the solutions of the Riemann problem at the interface = ( between particles i and j . The particle transport velocity 0 v is obtained as the summation of the particle velocity plus a small perturbation which helps to preserve a regular particle distribution (details about the adopted model can be found in Ref. [7]).

Vertical water entry of a flat panel
In this first section the vertical impact of a flat plate is numerically investigated and validated. Specifically, results of the 2-D single-phase simulations are described and compared to the experimental data from a wet drop test performed in Ref. [9]. In that work a flat panel (panel length L equal to 0.64 m) impacting with a deadrise angle of o 4 and a vertical impact velocity U of 6.0 m/s is studied. Measures of pressures at several positions along the plate are taken, allowing for a detailed control of the pressure peak repeatability and for possible 3-D effects.
As described in the theoretical work by Ref. [10], for these small deadrise angles a very thin, high-speed jet of water is formed, and the time-spatial gradients of the pressure field are extremely high. This makes the test conditions very demanding for numerical solvers. From the potential flow theory by Ref. [10], the jet thickness at model scale is about 0.1 mm. It is worth noting that the theory in Ref. [10] is formulated for symmetric wedge impacts whereas in the present case the impact of an inclined single plate is considered. More details about the reference solution to be adopted are given in Section 2.2.
On the base of this theoretical data, the 2-D simulation has been conducted using a very high spatial resolution, corresponding to a particle size = 16.5 m x   , by referring the latter to the panel length L , the ratio / L x  is equal to 4.110 4 . The whole tank depth and width are, respectively, 3 m and 6 m. In order to manage such a small particle size a variable -h technique has been used. Specifically, the particle size gradually changes with a maximum magnification factor of 3 200 between the most refined region and the lowest resolution one (see Fig.  1). The total number of particles is about 310 6 . For small dead-rise angles water compressibility cannot be neglected when calculating the speed of the water jet jet U [11][12][13] . In the experimental conditions the speed of sound in water is 0 =1481m / s 247 c U   , therefore the Mach number Under such a condition an estimate of the jet speed is jet 42 250 m / s U U   . As noted in Ref. [11], this very thin, high-speed jet disintegrates at some distance from the root due to interaction with both the surrounding air and the surface of the body. In this case surface tension plays an important role on the jet evolution. This local and complex physics is not taken into account in the present numerical method but it is not supposed to play are relevant role on the pressure distribution.

Selection of the speed of sound for the SPH model
Before starting the SPH simulations the speed of sound needs to be specified. As mentioned in Section 1, the numerical Mach number usually adopted within the SPH simulations is , which guarantees the weakly-compressible regime (i.e., compressibility plays a negligible role). Nonetheless, for this kind of impact the reference speed for the Mach number cannot be the impact velocity U . Indeed, considering that the intersection point between the horizontal undisturbed free-surface and the wedge surface has a speed equal to inters = /sin( ) 14 U U U   . According to Wagner theory the water jet formed during the impact has a speed higher than inters U . Therefore, if one chooses the speed of sound using the wedge speed U , i.e., 0 = 10 c U , the jet would not format all, its speed being in the supersonic regime. This is a clear example where the weakly-compressible rule 0.1 Ma  needs to be enforced in a proper way, considering the specific problem at hand.
In the present case the reference speed should be the water jet speed jet U which, however, is an unknown of the problem. Using the theory in Ref. [11], the estimate jet = 40 U U can beused. Note that, using the latter constraint, the speed of sound in water would result even higher than the real one, 0 = c 400U versus 0 = 247 c U. This means that for the adopted model scale water compressibility effects are not negligible, at least inside the jet region. However, considering that in the jet zone the pressure is close to the ambient pressure, water compressibility effects should not play a relevant role on the local impact loads.
In order to satisfy the weakly-compressible assumption, at least in the impact region, the reference velocity used is ref is an unknown of the problem and needs to be estimated. Then, an ex-post facto verification of the SPH simulations is required. Using the Wagner theory (which is valid for small deadrise angles as far as the air presence is negligible) the maximum pressure predicted for corresponding to about 9.110 6 Pa in our experiment. The water-hammer pressure for this impact is which is the maximum pressure level that can be physically reached in the experiments. The max P predicted by potential flow theory is higher than the acoustic pressure, and this is a further indication that water compressibility cannot be neglected for this problem. Thus, considering 6 which is 250 times smaller than the sampling rate of the pressure probes used in the experiments (i.e., the SPH sample frequency is 25 MHz versus the 100 kHz of the experimental pressure probes).This aspect is expected to influence the observed pressure peak which, for the considered configuration, needs a high time/space resolution to be captured. Further, in order to verify the appropriateness of such a choice, also a simulation using 0 = 247 c U has been run, the results will be shown at the end of Section 2.2.  Figure 2 shows the pressure field predicted by the SPH for the flat panel impact. In the same plot the free surface deformation evaluated by the potential flow theory by Ref. [10] is shown. Left plot of Fig. 3 shows an enlarged view of the flow velocity predicted by the SPH in the area of the highest pressure levels. A thin water jet is formed with a thickness of about 0.1 mm corresponding to about ten particles, thus justifying the high / L x  ratio needed to properly solve such a flow.

Comparison between SPH results, analytical solution and experimental data
In the right plot of the same figure an enlarged view of the pressure field is shown (in this case the displayed pressure range is enlarged too). From this plot it is seen that the high-pressure region is limited to an area of 1 mm 2 with a pressure peak of about 6.010 6 Pa. In the same figure the size of the probe used in the experiments is depicted. Clearly, the pressure sensor or has a size much larger than the pressure bulb formed below the water-jet root. This aspect will be discussed in the following section.
In Fig. 4 the time histories of the pressure measured at probe P12 are shown. This probe is positioned 0.236 m from the left-hand edge of the panel (see Fig. 1). The SPH solution is compared with the experiments and with the pressure peak predicted by Wagner theory. The SPH prediction is between the two reference data sets, and is characterized by high-frequency components due to the fragmented jet of water that pass over the numerical pressure probe. The latter has a dimension of 0.125 mm (which is the size of the SPH kernel support) and the SPH sampling rate is 25 MHz. Both the SPH and the analytical predictions overestimate the experimental data for which the maximum pressure recorded is 3.410 6 Pa (in the SPH solution the pressure peak reaches 7.010 6 Pa whereas the analytical prediction is 9.110 6 Pa). It has to be noted that, even though the Wagner theory is referred to the case of asymmetric wedge entry, according to Refs. [14] and [15] the value of the pressure peak should be substantially the same of the case of an oblique flat panel impact (it will be shown in Section 2.3 that this approximation can be quite rough for the present case).
Conversely, regarding the entire pressure signal, it is not possible to compare to classical analytical solutions, such as in Ref. [10], since they are all formulated for the symmetric wedge entry case. As mentioned above, air entrainment is expected to play a minor role for deadrise angles greater than o 3 [10,15] . Notwithstanding that, most of the experimental measurements available in the literature for angles close to o 4 exhibit pressure peaks much smaller than the one predicted by potential theory [15][16][17][18] and the values measured in the present study are in fair agreement with previous experiments by Refs. [15,16]. As for possible 3-D effects, these have been checked by comparing the pressure values on gauges aligned at the same distance from the piercing edge. For the probes positioned in the most central region no relevant differences have been observed. However, the maximum pressure impact measured in the experiment for small dead-rise angles can be also. affected by: (1) Changes of the body velocity during the impact stage.
(2) Rotations of the body during the impact stage.
(4) Sampling rate of the pressure signals.
(5) Size of the pressure gauge.
Thanks to the experimental setup adopted the first three points can be neglected, while the last two can play an important role. In order to take into account the experimental sampling rate, the SPH signal has first been filtered using a moving average filter (MAF) reducing the numerical sampling rate from 25 MHz to 100 kHz. The result is illustrated in Fig. 5. The reduction of the SPH peak due to this filtering procedure is not enough to get a good agreement with the experimental data. As a further step the SPH pressure has been measured integrating on a circular area equal to the size of the experimental pressure probes and then filtered at 100 kHz. This result is shown in Fig. 6. In this case the SPH output is much closer to the pressure recorded in the experiment.  In Fig. 7 the experimental pressure time histories recorded on a sequence of pressure probes is reported. Even if the experimental pressure peaks present some fluctuations, very similar pressure evolutions are recorded with an almost constant time shift at each probe. The SPH results show quite good repeatability of the pressure peaks along the wedge surface as well (see Fig. 8). In this regard, it is worth comparing the propagation velocity of the pressure peak along the plate, P U . Indeed, in water entry flows the value of the pressure peak should correlate to 2 P U [19,20] as In Table 1 The average values of P U are reported. For the simulation at = 0.01 Ma the average value of P U is about 117 m/s corresponding to a pressure coefficient = 1.03 P C which is close to the expected value (Eq. (4)). On the other hand, since the value of P U predicted by the SPH is smaller than the analytical one, maximum pressure of the SPH cannot be equal to the one predicted by the Wagner theory as shown above. This difference is essentially due to the fact that Wagner theory is referred to a symmetric wedge entry, while the present case is asymmetric (see Section 2.3). Table 1 Average of the peak propagation velocity P U plotted in Fig. 9. The corresponding pressure peak When considering the experimental pressure probes an average value of P U equal to 104 m/s is obtained (with a standard deviation = 3.1  ). This propagation velocity of the pressure peak corresponds to a pressure coefficient P C equal to 0.56. This confirms that the measurement system adopted is not able to record the real pressure peaks which are therefore underestimated as shown in this section.

Comparison between symmetric and asymmetric water entry
In the last subsection it has been shown that the pressure peak predicted by SPH is consistent with the water impact theory. Indeed, the calculated peak propagation velocity P U and the pressure peak max P give a pressure coefficient P C close to unity (see Eq. (4)). Nonetheless, it still remains a significant discrepancy between SPH results and the analytical solution for this impact angle. In order to investigate the source of this incongruity a further simulation has been performed. The symmetric problem has been set by retaining the initial particle configuration of the asymmetric case and closing the fluid domain at the left edge of the panel with a vertical wall (see Fig. 10). The simulation has been run with = 0.01 Ma . In Fig. 11 the peak propagation velocity for both symmetric and asymmetric problems are shown. It is clear that in the symmetric configuration the SPH solution is now much closer to the potential flow prediction and, because of this increase in the value of P U , a larger pressure peak is expected on the panel surface. The recorded pressure at 0.16 m from the left-hand edge of the panel is shown in Fig. 12 for both symmetric and asymmetric SPH solutions.
Consistently with the observed value of P U , the pressure peak of the symmetric solution is about 8.810 6 Pa. In the same figures the analytical solu-tions from Refs. [21] and [10] are also reported. Evidently, the symmetric SPH solution is now much closer to the analytical prediction and the remaining difference can be mainly attributed to fluid compressibility. This is in agreement with the analytical work in Ref. [22]. In that work it is shown that, generally, asymmetric impacts induce smaller pressure peaks with respect to the symmetric case. This effect is emphasized when the deadrise angle is smaller (in Ref. [22] the smallest angle considered is o 10 ).

Water entry with high horizontal speed
In this section the ditching problem including a large horizontal velocity component is studied, comparing the SPH out come to model test experiments. Given the higher complexity of the problem, in this section also possible 3-D effects are investigated, as their role is expected to be not negligible for this case.

Description of the experimental data
Guided ditching impact experiments [19,23] were performed in the CNR-INSEAN towing tank, which is 470 m long, 13.5 m wide and 6.5 m deep. The dimension of the flat plate were 0.5 m by 1 m (Fig.  13). Pressures at 18 points are measured through Kulite XTL123B pressure transducers. The sampling rate of the latter is 200 kHz while their dimension is 3.8 mm. The horizontal and vertical velocities at the impact are respectively 40 m/s and −1.5 m/s. In the experiments, these velocities are assumed to remain constant during the whole ditching impact.

Numerical methodology
In this second test case series, simulations have been conducted using adaptive particle refinement (APR) technique [24] . This technique allows keeping a high spatial resolution around the flat plate during the simulation as the refinement areas move at the same speed of the plate. In order to avoid the pressure filtering described in Section 2, the numerical adopted pressure sensor size is the same as in the experiments (i.e., 3.8 mm).
Given the high speeds involved in the considered ditching experiments, the simulations were performed with a nominal sound speed water 0 = 1 480 m / s C . Comparisons with experiments are established in terms of pressure coefficient, which is defined as where U and V refer respectively to the horizontal and vertical velocities.  (Fig. 14). At the beginning of the simulation, the total number of particles are about 6.710 4 and 1.110 5 for the test cases involving, respectively, 6 and 8 refinement levels. Figure 15 shows the pressure coefficient field predicted by the 2-D SPH simulation in the area of the highest pressure levels. It is worth noting that the field appears very regular despite the contour lines cross several refinement interfaces. In the same figure also the dimension of the pressure probe is reported.

Impact at o 4 pitch angle
In Fig. 16, SPH pressure time histories for three different probes are compared with the experimental data from Ref. [23]. The origin of the time axis is based on the pressure peak registered at probe P4. It can be observed that a remarkable time shift is present between the SPH and the experimental data, which increase as the plate penetrates in the water. This is clearly visible in the bottom plot of Fig. 16. The time shift is due to the different peak propagation velocities between SPH and experiments. The measured peak propagation velocity in the SPH simulation is SPH = 41.7 m/s P U which is close to the value obtained from the 2-D potential flow solution in Ref. [19] for the same case, which corresponds to POT = P U 47.3 m/s . Conversely, as described in Ref. [19] the experimental data exhibit a much lower value of the propagation velocity which is mainly attributed to 3-D effects. This aspect will be investigated more in details in the next section.
Another important aspect to discuss is the amplitude of the pressure peak. Indeed, looking at top plots of Fig. 16, one could be induced to conclude that the 2-D SPH matches very well the 3-D experimental measurements. This would be a misleading judgement since it would be in contrast with the different peak propagation velocities (according to correlation provided in Eq. (4)). This apparent contradiction can be explained observing that at probe P16 (bottom plot of Fig. 16) the SPH pressure peak is about 50% higher than at probes P4 and P8. The latter presents also more noisy signals and a less clear convergence. The reason is linked to the fact that at the beginning of the impact there are only few particles to resolve the impact region, while going forward in time the peak region grows in amplitude, i.e., it is better resolved. Therefore, at probes P4 and P8 the SPH pressure peak is close to the experimental one because it is under resolved. Note also that the pressure peak predicted by the 2-D potential flow solution in Ref.

P C
. But, even at probe P16, the SPH pressure peak does not reach this value. This is due to the averaging on the experimental probe size which, similarly to vertical impact case in Fig. 6, largely smooth the peak. This is confirmed also by the good agreement at P16 between the solution with 6 and 8 refinement levels. Simulations were performed also for pitch angle of o 10 to investigate also possible 3-D effects. Indeed, the case at o 4 requires a too high resolution to be studied also in 3-D. The study of 3-D effects is, therefore, limited to the case at o 10 which requires a lower computational effort. The 2-D simulation has been set for this case with 6 refinement levels min ( = 1.563 mm) x  . In Fig. 17 the 2D SPH pressure signals registered at probes P4 and P8 are reported. The behavior is similar to the case at o 4 : the pressure peak is close to the experimental one but the peak propagation velocity is different. At probe P16 (Fig. 18) the SPH pressure peak is, again, much higher with respect to probes P4 and P8 and is quite far from experimental data. However, the matching with the potential solution provided in Ref. [19] (represented by a star in Fig. 18) in this case is very good. Therefore, at o 10 pitch angle the averaging over the probe area seems to not affect the agreement between the SPH and the potential flow solution. The reason for that is to be found in the shape of the pressure peak, which at o 10 is less impulsive and much smoother than that at o 4 . angle. The symbol refers to the pressure peak predicted by the potential flow solver in Ref. [19] As for the peak propagation velocity, the potential solution predicts POT = 23.6 m/s P U which is in good agreement with the SPH value SPH = P U 23.9 m/s . Therefore, it is confirmed that, as for the oblique impact at o 4 in Section 3.3, for the 2-D simulation the SPH agrees with the potential solution and remains far from the experimental data.
The 3-D simulation of the case at o 10 pitch angle has been run with 7 refinement levels, the minimum particle spacing being min = 3.123 mm x  . An overall view of the 3-D flow and the different refinement boxes is given in Fig. 19.
In Fig. 20 the contour plot of the pressure coeffi-cient P C for a particle slice at the plate midline is shown. From this plot it can be seen that the values of P C are lower with respect to those measured in the 2-D simulation. This can be explained by the 3-D effect associated with the flow escaping from the lateral sides of the plate as also commented in Ref. [19]. Beside this, it is worth noting how, also in the 3-D case, the adopted APR technique allow for a smooth pressure field across the different refinement boxes. The latter can be also observed in Fig. 21, where a top view of the pressure field is provided. The delay of the spray root close to the lateral edges of the plate is visible in then nearly-parabolic shape of pressure peak front. In Fig. 22 the comparison between 2-D and 3-D SPH pressure signal at P16 is provided. The 3-D solution is much closer to the experimental data with respect to the 2-D simulation. It can be clearly seen that the 3-D effects induce a decrease of the peak amplitude and a delay of the peak occurrence, as consequence of the lower peak propagation velocity. This confirms the conjecture in Ref. [19] regarding the large discrepancies between 2-D potential flow solution and experiments.

Concluding remarks
High speed ditching problems, in both vertical and oblique entry velocity have been discussed. All along the paper the main flow characteristics considered are the pressure signal and the jet propagation velocity (i.e., pressure peak propagation velocity) which are strictly correlated.
For the vertical water entry, the challenging problem of the impact of a flat plate with deadrise angle of o 4 has been considered. A critical discussion about the choice of the speed of sound as well as the key aspects to consider when comparing SPH pressure signals is provided. Comparisons are performed with experimental data and available analytical solutions, showing that the 2-D weakly-compressible SPH is able to match potential flow solutions and to agree with experimental data, after careful analysis of the measure system. For this case the observed 3-D effects are negligible.
Then, analysis has been focused on ditching problems of flat plates with high horizontal velocity, comparing to the experimental data and potential flow solution in Refs. [23] and [19]. For this case, an Adaptive Particle Refinement strategy has been adopted to limit the computational costs.
Test cases with pitch angle of o 4 and o 10 are considered. In both situations it is observed that, differently from the purely vertical case, the 2-D SPH simulations does not correctly reproduce the experimental pressures on the plate (even if, apparently, in some under resolved cases they are quite close) while the agreement between the SPH and the potential solver is fair.
Then, a 3-D simulation for the case at 10° is performed recovering the agreement with the experimental data and showing that, as speculated in Ref. [19], 3-D effects are substantial for this problem for both the pressure peak amplitude and the jet propagation velocity.