Experimental study of coefficients during vertical water entry of axisymmetric rigid shapes at constant speeds

Applied Ocean Research 37 (2012) 183–197

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Review

Experimental study of coefficients during vertical water entry of axisymmetric rigid shapes at constant speeds A. El Malki Alaoui ∗ , A. Nême, A. Tassin, N. Jacques ENSTA-Bretagne, LBMS (Laboratoire Brestois de Mécanique et des Systèmes) – EA 4325, 2 rue Franc¸ois Verny, F-29806 Brest Cedex 9, France

a r t i c l e

i n f o

Article history: Received 28 November 2011 Received in revised form 20 March 2012 Accepted 1 May 2012 Keywords: Dimensionless slamming coefficient Impact water entry Axisymmetric body High and constant velocity

a b s t r a c t This paper presents an original experimental approach of three-dimensional water impact at constant speed. The hydrodynamic force acting on axisymmetric rigid bodies striking a horizontal liquid surface at constant vertical speed is investigated using a hydraulic shock machine. The different aspects of the experimental device allowing to carry out impacts at high-speeds with small deviation of the velocity are initially detailed. Then, results of observations on several axisymmetric shapes are presented and discussed. Good agreements between theoretical model, numerical results and available experimental measurements have been obtained. © 2012 Elsevier Ltd. All rights reserved.

Contents 1. 2.

3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Test machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Measuring device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Calibration of the gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Calibration of the accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Impact moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Dimensioning of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity profile and hydrodynamic load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless slamming coefficient for simple cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless slamming coefficient for hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless slamming coefficient for a cone with knuckles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Ships navigating in different sea conditions suffer from various structural damages that are generally associated to waves repeatedly slamming. The prediction of the hydrodynamic loads acting on the wetted part of an impacting body is very important in the naval field both for structural design and for the prediction of some

∗ Corresponding author. Tel.: +33 298348962; fax: +33 298348730. E-mail address: aboulghit.el malki [email protected] (A. El Malki Alaoui). 0141-1187/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apor.2012.05.007

183 184 185 185 186 186 187 187 187 189 192 193 194 194 195 196 196

sea-keeping properties of the ship. Even if the typical impact duration of a solid entering water is very short (around a few milliseconds), this phenomenon leads to important pressure levels on the hull. In addition, the repetition of slamming loads combined to impact induced vibrations can be responsible for fatigue damage. As a consequence, the possibility of estimating the loads due to water impact would be of great interest to shipbuilders. Several experimental, theoretical and numerical studies with the intention of achieving a better understanding of the slamming phenomenon are carried out. The first studies on slamming were carried out by Von Karman and after by Wagner [1,2]. Von Karman derived a theoretical

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Nomenclature f(ˇ) and Cs dimensionless slamming coefficients  normalized depth b(t) instantaneous penetration [m] ˇ deadrise angle of cone [◦ ] radius of sphere [m] R F vertical hydrodynamic force [N] V impact velocity [m s−1 ] mass density of water [kg m−3 ]  t time [s] t period [s] ıt temporal difference [s] de displacement measured with an internal sensor of the test machine [m] dopt displacement optimized [m] da displacement obtained from double integration of acceleration [m] ˛, parameters of optimization test frequency [Hz] f Em mean deviation between dopt and de

expression for the impact coefficient of a falling sphere in 1929. He does not take into account the deformation of the free surface around the outside of the sphere, Fig. 1. The resulting Von Karman estimate for dimensionless slamming coefficient is then: 1/2 Cs = 3.30(Vt/R1 ) where V is the impact velocity and t is the time. In 1932, Wagner used the same method as Von Karman to derive another impact coefficient which would take into account the deformation of the free surface by simply moving the free surface higher. This resulted in a higher estimate for two-dimensional impact coefficient than Von Karman. The resulting Wagner esti1/2 mate for the impact-slamming coefficient is: Cs = 6.33(Vt/R2 ) . The main difference between the two models is that Wagner used the distance between the center of the sphere and the top of the surface deformation for the instantaneous half-length of the flat plate. This can be seen in Fig. 1. Shiffman and Spencer [3] are among the first to notice that the liquid may separate from the sphere, leading to cavity formation. However the stage of the impact under consideration in this study is before separation has commenced; the depth b is less than half of radius R. Shiffman and Spencer give an explicit relationship for 1/2 Cs , i.e., Cs = a1 (Vt/R) − a2 Vt/R where a1 = 5.4. There are also many others expressions concerning sphere penetrating a free surface with a constant velocity as those proposed by Touvia Miloh [4,5], Moghisi and Squire [6], Laverty [7], and others. Chuang and Milne [8] showed after comparison of test results for the wedges and cones that the maximum impact pressures were

generally lowers for the three-dimensional impact. In this experimental investigation, the maximum impact pressures are generally about 30% lower for the cone than for the wedge (for the same deadrise angle superior than 1◦ ). Based on some experiments of various wedges impacting water calm surface, Chuang studied the effects of deadrise angle. He concludes that air was trapped between the impact wedge surface and the water surface for deadrise angles less than 3◦ Such trapped air provided cushioning during the impact, resulting in lower measured impact pressures [9]. More recently, Huera-Huarte et al. studied slamming with trapped air between the plate and the water free surface, at high impact speeds and small deadrise angles. They show that the trapped air phenomenon significantly cushions flat plate impacts with angles less than 5◦ . Impact with larger angles adhere to Von Karman’s equations [10]. Battistin and Iafrati [11] studied the water impact of two-dimensional and axisymmetric bodies. They focused their investigation on the resulting hydrodynamic loads. In their numerical approach, they developed a fully nonlinear boundary-element model taking into account the jet formation at the intersection of the body contour and the free surface due to the local flow singularity. Battistin and Iafrati [11] completed their study by analysing and comparing the numerical approach with existing theoretical and experimental results for the water impact at constant velocity of a cone, a circular cylinder and a sphere. Based on an analysis of the experimental data and comparison with an analytical model, Yettou et al. [12] proposed a new method to predict the local and maximum pressure. This method takes into account the velocity variation of the wedge during the entry phase. In this experimental study, Yettou et al. show that the main parameter influencing the pressure coefficient is the wedge angle. Constantinescu et al. [13,14] studied the numerical modelling of the impact of 2D body with water and the subsequent appearance of microscopic fractures. They developed and validated a numerical tool, named Impact++ ABAQUS, simulating the fluid–structure interaction during the impact of a body on a free water surface. Impact++ ABAQUS is based on the Wagner theory and the commercial finite element code ABAQUS associated with PYTHON and FORTRAN routines. The fluid-heat transfer analogy is employed to solve the fluid dynamics problem with finite element code ABAQUS. They take into account the effect of the deformation of the structure upon the flow spectrum and vice versa. This numerical tool allows us to calculate the stress and deformations, energies and the global forces for a 2D solid impacting a fluid. A common problem in the body’s design which enters the water at high speeds is the determination of the forces during the water impact. This paper presents some results for this quantity. Rigid-body slamming was experimentally investigated at the ENSTA-Brittany by impacting several aluminium models from various velocity entries in a calm water surface. This paper presents tests results and compares them with numerical calculations. Those experimental investigations give an adequate approximation of the slamming coefficients compared to numerical simulations and others studies. This article is organized as follows. In Section 2, the description of experimental process is presented. The experimental results are given in Sections 3–6. Conclusions are presented in Section 7.

2. Experimental device

Fig. 1. Geometry of the sphere entry.

Many experiments were conducted in LBMS laboratory of ENSTA-Brittany to measure the structural response of shapes entering water at constant speed. In this paper, we provide

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

185

Fig. 3. Cone with sensors (accelerometer and gauges) fixed at low extremity of the piston.

Fig. 2. Picture of the experimental test setup and tank.

the description of an elaborate experimental procedure designed to conduct a slamming test. The effort time history, the position, the first contact between models and water and acceleration–deceleration of the body were recorded during impact. 2.1. Test machine A high-speed shock machine (Fig. 2) makes it possible to carry out impact tests on water (slamming). The main advantages of this machine are its high capabilities in terms of speed and force. The admitted force is 100 KN (or 200 KN) with velocity up to 20 m s−1 (or 10 m s−1 ) for a total stroke 150 mm (or 300 mm). Moreover, the impacting velocities mentioned above are stabilized levels for which the maximum deviation is 5% during the total duration of the impact (total stroke). The experimental set-up is illustrated in Fig. 2. The tank is made of steel. The inner dimensions of the tank are 2 m wide and 3 m long with a maximum depth capacity of 2 m. The water level was set at 1.2 m to allow impacts for all tests. Four circular windows (0.7 m in diameter) made of PMMA (polymethyl methacrylate) embedded in middle of each tank faces. The simulation of the flow in semi-infinite environment is guaranteed by the ratio between the size of the tank and the size of the model. The tank full of water is supported by pneumatic cylinders.

basis of the impacting body to measure its acceleration (Fig. 3). It has a measuring range ±250 g, a sensitivity 0.32 mV/g, an eigen frequency 2500 Hz, a damping coefficient 0.64 and an excitation voltage of 12 V. The accelerometer measurements make it possible to determine the velocity and the inertial effects during the tests; • strain gauges located 50 mm from the lower extremity of the piston (Fig. 3) measuring the axial load applied to the piston. All four gauges of mark KYOWA (model KFG-2-1K-C1-11) constitute a Wheatstone bridge. The sensitivity of the bridge was identified to 1.19 × 109 N (see Fig. 4) by static calibration; • an electric contact sensor located on the impacting body (see Section 2.5).

Measurement is carried out using “Nicolet Odyssey” on which are connected the accelerometer, the strain gauges, the machine displacement sensor and the electric contact sensor detecting the moment of impact. The conversion of these data is achieved by a program developed on Matlab software. This program makes it possible to convert the data expressed into Volt in their respective units. To measure the acceleration of the rigid body and the force of the contact body-water respectively, the accelerometer and gauges were used. A precise calibration of the gauges and accelerometer are necessary to obtain a good quality of shock measurements.

200

2.2. Measuring device An internal sensor of the machine provides a measure of the piston displacement. However, this data alone is insufficient to conduct our study. In order to complete the measures (hydrodynamic load, inertial load and the moment of impact fluid-structure), additional external sensors satisfying specific requirements (fast dynamic, fluid-structure interaction) are used. The experimental assembly is composed of the following instrumentation: • a displacement sensor located on the upper extremity of the piston; • an accelerometer of technology piezoresistive, model EGAS-FS-*250-/V12/L8M/X company MEAS France – Entran, located on the

Load F [kN]

150

100

50

0

0

0.5

1

Variation of relative tension u/ U

1.5

x 10

-4

Fig. 4. Load versus variation of relative tension at gauges terminal.

186

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197 Table 1 Values of ˛ and Em according to temporal difference (tests for cone with deadrise angle ˇ = 15◦ ).

optimized displacement experimental displacement

0.6

V = 10 m s−1

dispalcement [m]

0.5 0.4

˛

0.3 Em

0.2

ıt (ms)

0

0.8

1.2

1.4

1.6

test 1 test 2 test 3 test 4 test 1 test 2 test 3 test 4

1.0463 1.0493 1.0283 1.0316 0.0056 0.0062 0.0051 0.0058

1.0346 1.0362 1.0203 1.017 0.0026 0.0031 0.0022 0.0025

1.0277 1.0289 1.0155 1.0096 0.0014 0.0016 9.11E−04 9.81E−04

1.0239 1.0251 1.0127 1.0059 0.0012 0.0013 7.92E−04 7.29E−04

1.02 1.0212 1.0099 1.0021 0.0012 0.0012 0.0011 0.0011

10 10 10 10 10 10 10 10

0.1

Where, t (1/f with f is the acquisition frequency) is the period and ıt is the measure of temporal difference. In order to determine the best optimized displacement dopt , the following procedure is used:

0

0.02

0.04

0.06

0.08

0.1

time [s] Fig. 5. Displacement measured and displacement obtained after optimization – existence of small temporal difference.

2.3. Calibration of the gauges A calibration stage is necessary (static tests) to establish the relation linking the load F to the variation of relative tension u/U at the gauges terminal (Fig. 4). Eq. (1) establishes the link between load and variation of relative tension. F = 1.19 × 106

u [kN] U

(1)

where u (in [V]) is the tension variation of the gauges, U (in [V]) is the excitation voltage of the gauges and F (in [KN]) is the load measured by a calibrated load cell. The sensibility value determined experimentally (1.19 × 106 KN) is very close to the theoretical value (1.2 × 106 KN). 2.4. Calibration of the accelerometer The present section details the methodology used to determine the velocity, the displacement of the piston and the model’s equivalent mass. Tests were carried out with the model assembled with the piston at its extremity and without impact on the water (i.e., test in air) for four different speeds: 5, 10, 15 and 20 m s−1 . During these tests, the piston displacement, the acceleration of the rigid body and the effort at the extremity of the piston were recorded. The procedure of integration of the acceleration signal takes into account initial conditions (the velocity in particular) and an optimization method is used. It consists to determine the displacement dopt which minimizes the deviation with the measured displacement de (displacement measured with an internal sensor of the machine) in the sense of ordinary least squares (see Appendix A). Fig. 5 presents an example of result for measured displacement and optimized displacement. In this figure both curves are not identical. Indeed, a very small temporal difference between the experimental signal and the optimized signal lower than 2 ms remains. This temporal difference is of the same order as the studied phenomenon (slamming). A temporal retiming is used to correct this difference. The principle of retiming consists to translate the elements of vector de compared to the elements of vector t before applying optimization procedure. In other words, to take into account this temporal difference it is necessary to create a translation of index in de vector. Such as the couple {t(i), da (i)} corresponds, after temporal difference, to couple {t(i), de (i + k)} where the integer k is such as: k · t = ıt

• To fix a temporal difference (0–2 ms); • To carry out the optimization; • To calculate the mean deviation Em between vectors dopt and de . With Em =

1 Nmax



Nmax

|dopt (i) − de (i + k)|

i=1

The results for various ıt are presented in Table 1 (tests for cone with deadrise angle ˇ = 15◦ ). It can be seen that the difference ıt = 1.4 ms leads to the best optimization. Taking into account the temporal difference ıt = 1.4 ms, Fig. 6 corrects Fig. 5. These results are very satisfactory because both curves are mixed. Finally, the signal of load is compared with the signal of acceleration using Newton’s second law. For example, after optimization, an equivalent mass equal of 14.14 kg has been obtained for cone with deadrise angle 15◦ . It corresponds to the total mass of “objects” put up-stream of the strain gauges (until the apex of the cone). This optimized mass is effectively close to the sum of the physical masses of the cone (8.36 kg), of the spider (2.4 kg), of the screws (0.14 kg) and the extremity of the piston (3.06 kg) that is approximately 13.96 kg. Results obtained for velocities V = 5 and 10 m s−1 are depicted respectively in Fig. 7.

optimized displacement experimental displacement

0.6 0.5

dispalcement [m]

0

0.4 0.3 0.2 0.1 0 0

0.02

0.04

0.06

0.08

0.1

time [s] Fig. 6. Displacement measured and displacement obtained after optimization – to allow for temporal difference ıt = 1.4 ms.

5000

5000

0

0

Load [N]

Load [N]

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

-5000

187

-5000 -10000

-10000 -15000

-15000 gauges signal accelerometer signal

0

0.05

0.1

gauges signal accelerometer signal

-20000 0

0.15

0.05

0.1

Time [s]

Time [s]

Fig. 7. Evolution of inertial load for tests in air with impact velocities 5 and 10 m s−1 .

Fig. 8. System used to identify the moment of contact fluid–solid (left), typical signal of the tension R1 at moment of impact (right).

2.5. Impact moment The moment of impact refers to the instant of time of the first contact between the model and the water. In this case, the water has only reached the top of the model and the hydrodynamic load is close to zero. The moment of impact detection is achieved by measuring the instantaneous raise of tension occurring in the resistor circuit depicted in Fig. 8 when the model touches the liquid surface. This provides a simple but efficient indication of the first instant of contact between the solid and the water. In the next sections, the instant of time t = 0 corresponds to the “moment of impact”. 2.6. Dimensioning of the models In order to avoid the edge effects, the model’s diameter must be lower or equal to 1/6th of the smallest characteristic length of the tank (width). Various geometries have been tested numerically (finite elements ABAQUS code) to minimize the mass while respecting the criterion of indeformability. In this paragraph, a result concerning a cone with deadrise angle 30◦ is presented. For this model, the rigid-body condition is converted into a 50 ␮m maximum deflection condition of the extremity under bending (note A in Fig. 9), corresponding to the manufacturing quotation. Dimensions and quotation of the model tests in accordance with the mechanical and geometrical constraints are depicted in Appendixes B and C.

Fig. 9. Cone under bending – field of displacement (m) of the point A.

performed with prescribed constant entry velocities which have been determined using numerical simulations and respecting machine capacities. The objective of those tests is mainly focused on measuring the displacement, the acceleration and the evolution of hydrodynamic load acting on the impacting body. In order to

Table 2 Velocities employed for each model. Velocities tests (m s−1 )

Model ◦

3. Velocity profile and hydrodynamic load Table 2 shows the different bodies that have been tested. The models are made from aluminium. The tests have been

Cone deadrise angle 7 Cone deadrise angle 15◦ Cone deadrise angle 30◦ Cone with knuckles Hemisphere

5, 6.5 and 8 8, 12 and 15 15, 18 and 20 8, 10 and 12 18 and 20

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21

25

Impact velocity V = 20m/s

Impact velocity [m/s]

Impact velocity [m/s]

20 15 10 5

20.5

20

19.5 0 -5 -0.03

-0.02

-0.01

0

19

0.005

0

0.5

1

1.5

Time [s]

2

2.5

3

x 10

Time [s]

-3

Fig. 10. Velocity of the test calculated from acceleration integration.

100

100

80

60

60

Load [kN]

Load [kN]

maximal load

80

40 20

40 20

t=0 Top of impact

0 -10

maximal hydrodinamic load

hydrodynamic load

Impact load Inertial load

t=0 Top of impact

0 0

0.5

1 1.5 Time [s]

2

2.5 x 10

0

0.5

-3

1 1.5 Time [s]

2

2.5 x 10

-3

Fig. 11. Impact load and inertial load versus time [left], hydrodynamic load versus time [right] (ˇ = 15◦ with V = 15 m s−1 ).

curves of Figs. 12, 13 and 14 are nearly identical indicating a good repeatability of the measurements. Moreover, as predicted by the Wagner theory, the evolution of the hydrodynamic load with time is parabolic; neither the deadrise angle nor the velocity presents an influence on parabolic form (Figs. 15, 16 and 17). These figures also show that the increase of the force level depends on impact velocity and the deadrise angle of the cone. The impact duration range from 2 ms for the fastest to 4.95 ms for the slowest. After the peak load, we observe a sudden decrease of the force. At this instant, the jet zone is on the cylinder support and the cone is totally immersed.

x 10 6

cone

4

= 7°

Test1 Test2 Test3 Test4

5 4

Load [N]

ensure the reproducibility of the tests, for each speed, the test is repeated at least three times. Fig. 10 illustrates the velocity during the impact stage determined by the integration of the accelerometer signal as a function of time. For each shape at least three velocities are considered. The value t = 0 refers to the moment of impact. It can be seen that, during the impact, the velocity is controlled with a maximum error of 3%. The velocity can therefore be considered almost constant during the impact duration. Although the velocity deviation is small during the impact on the almost rigid bodies, induced dynamic effects (hydrodynamics and structural dynamics) cannot be avoided during such experiments at high speeds. Structural dynamic effects are expected to be of greater order than the hydrodynamic induced effects. Therefore the method proposed in this section to suppress the induced dynamic effects will only consider the structural effects. As explained in Section 2.4, an equivalent mass is determined during the calibration of the accelerometer sensor. As a consequence, the accelerometer measurement makes it possible to determine part of the measured impact load due to the “rigid body” inertia. Fig. 11(left) displays the total impact load together with the inertial load during the impact of a 15◦ cone at 15 m s−1 . One can see that the inertial load is not significant but however can’t be neglected. Thus, the maximum load occurring at the total immersion of the model is affected by the dynamic effects. The drop of hydrodynamic load occurring after the total immersion of the model leads to significant dynamic effects which can be seen on the inertial load. The total immersion occurs at 2.4 ms, which is in good agreement with the Wagner theory predicting the total immersion at tmax = 2.31 ms (tmax = R tan ˇ/4V ). In the case of the simple cones; three or four tests for each speed are realized and showed very good repeatability of the experiments. The

3 2 1 0 0

0. 5

1

Time [s]

1. 5

2

-3

x 10

Fig. 12. Hydrodynamic loads versus time for four slamming tests with 8 m s−1 velocity for cone with deadrise angle 7◦ .

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197 4

10

cone

x 10

= 15°

9

Test 2

= 15°

V=15m/s V=12m/s V=8m/s

6

Test 4

Load [N]

Load [N]

cone

7

Test 3

6

4

8

Test 1

8

x 10

189

4

5 4 3

2

t=0 Top of impact

2

0

t=0 Top of impact

1 0

-2

0

0. 5

1

1.5

Time [s]

2

2. 5

Fig. 13. Repeatability of tests for cone 15◦ under 15 m s−1 for impact velocity. 4

5

cone

x 10

Load [N]

1

1. 5

2

2. 5

Time [s]

3

3. 5

4

x 10

-3

Fig. 16. Evolution of hydrodynamic loads for cone 15◦ versus time at several velocities.

the curves obtained being almost superposed. In Figs. 20 and 21 the hydrodynamic load versus time is presented for various velocities respectively for cone with knuckles and hemisphere. Also in the hemisphere case, the form of hydrodynamic load is the same found by numerical calculation [6,11,15,16].

3

4. Dimensionless slamming coefficient for simple cones 2

The numerical simulations using Impact++ ABAQUS, ABAQUS/Explicit and FLUENT codes supplied the values of dimensionless slamming coefficient f(ˇ). Its expression for a conical rigid shape is the following one [11,18]:

1

0 0

0. 5

1

1. 5

2

2. 5

Time [s]

3

3. 5

4

x 10

Figs. 18 and 19 show the comparison of the hydrodynamic load respectively for the cone with knuckles and for hemisphere. The impact velocity is 12 m s−1 for cone with knuckles and 18 m s−1 for hemisphere. For those models, the repeatability is still very good; cone

4

x 10 6

f (ˇ) =

Ftg 3 (ˇ) V 4 t 2

where ˇ is the deadrise angle, F the vertical hydrodynamic force, t the time, V the impact velocity and  is the water density. Under the assumptions of inviscid and incompressible fluid without tension surface, parameter f(ˇ) should remain constant during the impact of a cone. Table 3 presents the values of this non-dimensional parameter calculated for three cones (ˇ = 7◦ , 15◦ and 30◦ ) with the various simulation codes. Globally, Table 3 shows that the Impact++

= 7° 5

V=5m/s V=6.5m/s V=8m/s

5

(2)

-3

Fig. 14. Repeatability of tests for cone 30◦ under 20 m s−1 for impact velocity.

x 10

4

cone

= 30°

V=15m/s V=20m/s V=18m/s

4

3

Load [N]

4

Load [N]

0.5

= 30°

Test1 Test2 Test3 Test4

4

0

-3

x 10

3

2

2 1 1 0 0 0

0.5

1

1. 5

Time [s]

2

2. 5

3

x 10

-3

Fig. 15. Evolution of hydrodynamic loads for cone 7◦ versus time at several velocities.

0

1

2

3

Time [s ]

4

5

x 10

-3

Fig. 17. Evolution of hydrodynamic loads for cone 30◦ versus time at several velocities.

190

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197 ests w t V

/s

30

80

Test 1 Test 2 Test 3

Hydrodynamic load [kN]

60

V=18m/s V=20m/s

25

Hydrodynamic load (kN)

70

50 40 30 20

20 15 10

10 0

5 0

-1 0

0

2

4

6

8

10

12

14

Time [s]

0

16

x 10

1

Fig. 18. Hydrodynamic loads versus time for three slamming tests with 12 m s−1 velocity for cone with knuckles.

3

Time (s)

4

5

-3

x 10

Fig. 21. Evolution of hydrodynamic loads for hemisphere versus time at two velocities.

Table 3 Values of the non-dimentional slamming coefficient f(ˇ) for various deadrise angles of cones.

tests V=18m/s

20

Hydrodynamic load (kN)

2

-4

Test 1 Test 2 Test 3

15

Value of f(ˇ)

ˇ (◦ )

7

15

30

Impact++ ABAQUS ABAQUS/Explicit FLUENT

7.27 7.18 7.2

6.11 6.25 6.26

4.44 4.65 4.68

10

5

0 0

1

2

3

4

Time (s)

5

6

x 10

-3

Fig. 19. Hydrodynamic loads versus time for three slamming tests with 18 m s−1 velocity for hemisphere.

ABAQUS results are in good agreement with ABAQUS/Explicit and FLUENT. The value of f(ˇ) depends only on the deadrise angle. Total hydrodynamic loads have been recovered for the impact with constant entry velocity of cones with ˇ = 7◦ , 15◦ and 30◦ . In Fig. 22 the slamming coefficient f(ˇ) of the cone entry has been plotted versus the time t. The slamming coefficient is given by Eq. (2). This slamming coefficient is poorly measured with the time close to 0 because the effort F is almost null (F ≈ 0). On the other hand f(ˇ) seems to tend well towards a constant value until F is at maximum (total wetting of the cone). The error made to calculate f(ˇ) is evaluated with Eq. (3). err f (ˇ) =

|F| |V | |t| +4 +2 F V t

(3)

80

V=8m/s V=10m/s V=12m/s

60

20 Slamming coefficient f( )

15 50

Slamming coefficient is constant

10

40

f( )

Hydrodynamic load [kN]

70

30 20

5 0

10 -5

0 0

0. 5

1

Time [s]

1. 5

2

2. 5 -3

x 10

Fig. 20. Evolution of hydrodynamic loads for knuckles versus time at several velocities.

-1 0

t=0 Top of impact

0.5

1

1.5

Time [s]

2

2.5

x 10

-3

Fig. 22. Value of non-dimensional slamming coefficient versus time (cone with ˇ = 15◦ and V = 15 m s−1 ).

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

191

Table 6 Results for cone with deadrise angle of 30◦ .

7 f( )

Slamming coefficient f( )

mean-f( )

V (m s−1 )

Testing time (ms)

Impact load max (KN)

Hydro. load max (KN)

f(ˇ) men

15 15 15 18 18 18 20 20 20 20

4.75 4.9 4.75 4 3.9 3.95 3.54 3.5 3.54 3.5

25.1 26.1 26.25 37.76 37.4 38.21 47.7 47.93 49.65 48.99

26.6 25.9 26.27 37.17 37.6 38.14 47.72 47.69 48.16 47.75

4.9 4.78 4.88 4.73 4.64 4.27 4.79 4.94 4.73 4.9

6. 5

6

5. 5

5

0

5

10

15

20

25

30

1.4

120

Point index Fig. 23. Calculation of f(ˇ) with err f (ˇ) ≤ 10% (cone 15◦ with V = 15 m s−1 ).

1.2

100

1

80

Table 4 Results for cone with deadrise angle of 7◦ .

0.8

V (m s−1 )

Testing time (ms)

Impact load max (KN)

Hydro. load max (KN)

f(ˇ) men

5 5 5 5 6.5 6.5 6.5 8 8 8 8

3.1 3.2 3.2 3.2 2.55 2.45 2.5 2 2.05 2.05 2.05

22 22.8 22.7 22.9 40 39.5 38.7 59.2 60 63.9 62

23.6 23.6 23 23.2 38.7 39.8 38.1 59.6 60.5 59.4 59.5

6.92 6.67 6.69 6.95 6.72 6.43 6.4 7.06 6.87 6.97 6.89

60 0.6

To determine exactly the “asymptotic” value of f(ˇ), we chose to preserve the points such as err f (ˇ) ≤ 10%, then we made the mean of the values of f(ˇ) which check this condition (Fig. 23). The cones are impacted into calm water. At least three tests were performed for each velocity to ensure repeatability of the measurements. The repeatability of tests is illustrated by the experimental values presented in Tables 4–6 which correspond respectively to impact tests of rigid cones with deadrise angles 7◦ , 15◦ , and 30◦ . In order to propose a value of f(ˇ) experimental fexp (ˇ), a statistical study of all values of f(ˇ) checking the two following conditions was achieved:

40

0.4

20

0.2

0

3.75

4

4.5

4.75

5

5.5

5.75

0

Fig. 24. Column diagram and associated normal law distribution for the cone with ˇ = 30◦ .

The values of f(ˇ) are also assumed to be uncorrelated. The histogram of Fig. 24 presents the frequency distribution as well as the probability following a normal law for the cone with deadrise angles of 30◦ . In order to conduct the better predictions, a Chisquared test is achieved to accept or reject the hypothesis on the form of probability law which fits in best the experimental values. This test assures us that the distribution is not significantly different from a normal law with 95% of confidence level as shown in Fig. 25. We assume that the dispersion of f(ˇ) is essentially due to the measurement errors. The following experimental values are proposed with a 95% of probability, respectively for the cone of 7◦ ,

• err f (ˇ) ≤ 10% • For each test, the condition err f (ˇ) ≤ 10% must be verified for a duration representing at least 20% of the total duration of the impact experiment.

Normal Probability Plot 0.999 0.997 0.99 0.98

Table 5 Results for cone with deadrise angle of 15◦ . V (m s−1 ) 15 15 15 15 15 15 12 12 8

Testing time (ms)

Impact load max (KN)

Hydro. load max (KN)

f(ˇ) men

2.4 2.4 2.45 2.4 2.4 2.4 2.98 3 4.44

88.1 79.5 88.1 87.5 87.5 87.6 55.8 54.3 24.1

88.6 83.1 87.6 88.2 87.3 89.4 57 55.1 24.2

5.87 6.2 5.97 6.27 6.12 5.8 6.48 6.26 6.17

Probability

0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 4.4

4.5

4.6

4.7

4.8

4.9

5

Data Fig. 25. Fitting with Henry line for cone with ˇ = 30.

5.1

192

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

1.2

ABAQUS Battistin

1

0.8

0.8

0.6

Cs

Ftan3( )/( (VR)2)

1

0.6

0.4 0.4 0.2 0.2 0

0

0.1

0. 2

0. 3

0. 4

0.5

0 0

Vt/R

0.05

0.1

0.15

0.2

0.25

0.3

2

Fig. 26. Value of [F tan3 (ˇ)/(VR) ] for cone (ˇ = 30◦ ) versus the dimensionless time (Vt/R).

15◦ and 30◦ : fexp (ˇ = 7◦ ) = 6.79 ± 0.56, fexp (ˇ = 15◦ ) = 6.18 ± 0.5 and fexp (ˇ = 30◦ ) = 4.75 ± 0.35. The slamming coefficient fexp (ˇ) has been compared with the theoretical prediction by Schiffman and Spencer [18]: F = 3k(ˇ) tan3 

 2



− ˇ V 4t2

Here f(ˇ) = 3k(ˇ), and t = 0 is the moment when the cone first touches the free surface. Battistin and Iafrati [11] and Kleefsman et al. [19] considered the dependence of dimensionless factor k(ˇ) on the deadrise angle. They found k(ˇ) = 1.6 for the deadrise angle 30◦ . For the same deadrise angle, we find the experimentally determined equivalent as k(ˇ) = 1.58. Comparison of test results for the cones indicated that the variation (error) on the dimensionless slamming coefficient f(ˇ) was lower for the 30◦ deadrise angle cone than for the lower deadrise angle cones. This can be explained by increase of the wetted area at small deadrise angle for higher speed. Indeed, for example, for the same speed (15 m s−1 ) the cone with deadrise angle of 15◦ is completely submerged in 2.4 ms whereas only half of the cone with deadrise angle 30◦ is wetted. The results of this study are very satisfactory. They showed a good repetitiveness as in Figs. 26 and 27 where we plotted respectively for cone with 30◦ and 7◦ the value of expression [F tan3 (ˇ)/(VR)2 ] according to [Vt/R] for all test points. These last ones were superimposed in the parabola y = f (ˇ) · x2 .

0.35

Ftan3( )/( (VR)2)

0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.05

0. 1

0.15

0.2

0.25

Vt/R 2

Fig. 27. Value of [F tan3 (ˇ)/(VR) ] for cone (ˇ = 7◦ ) versus the dimensionless time (Vt/R).

Fig. 28. Comparison of Cs versus  for hemisphere between ABAQUS and Battistin’s results.

The numerical prediction of the non-dimensional slamming coefficient for cones with different deadrise angles is presented in Table 3. The numerical results are in good agreement with experimental results presented above. To summarize, in this section, it has been shown experimentally that the dimensionless coefficient f(ˇ) is principally governed by the deadrise angle of the cone. Its value is almost constant over time and neither shape nor velocity show any influence. 5. Dimensionless slamming coefficient for hemisphere Dimensionless slamming coefficient for hemisphere defined in Appendix C is also studied. The numerical simulation is conveniently presented (Fig. 28) in terms of a dimensionless impact coefficient Cs , defined as: Cs =

2F AV 2

(4)

where F is the impact force, A is the projected area of the object,  is the fluid density and V is the velocity of the body. The principal condition is that Cs depends on the depth of immersion b with  = b(t)/R = Vt/R [5–7,15,19]. The obtained calculation was then compared with Battistin results as shown in Fig. 28. The results provided by ABAQUS are in good agreement with Battistin’s results [11]. For the hemisphere model, it was decided to perform two series of tests ranging in impact speed from 18 m s−1 to 20 m s−1 . Each test was repeated three times for test repeatability. The final results are shown graphically in Figs. 29 and 30. These show that the impact slamming coefficient, and hence the impact force (Fig. 21), rise rapidly to a maximum when the depth of penetration is inferior a twelfth of the radius ( < 1/12). Thereafter it declines more gradually towards a value of  between 0.2 and 0.4, when the cavity is formed. In fact, the cavity begins to form, only after the sphere, has submerged a distance somewhere between a quarter of a radius and a radius [3]. As can be seen in the above graph (Fig. 30) the experimental data match closely with the ABAQUS simulation and consequently match closely with Battistin’s results (Fig. 28). The results of this work permit us to test the relationship between dimensionless slamming coefficient Cs and normalized 1/2 depth of immersion  [3,5,6,15,18]: Cs = a1 () − a2 . After divi1/2 it my be written sion by  Cs ()

1/2

1/2

= a1 − a2 ()

(5)

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

1.2

16

Test 1 Test 2 Test 3

1

Test 1 Test 2 Test 3 ABAQUS

14 12

0.8

10

0.6

Cs

Cs

193

8 6

0.4 4 0.2

0 0

2 0 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

Fig. 29. Value of Cs for hemisphere versus  at V = 18 m s−1 .

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 32. Experimental and numerical result of Cs versus  for cone with knuckles at V = 12 m s−1 .

1.2

6. Dimensionless slamming coefficient for a cone with knuckles

1

Cs

0.8

0.6

0.4

ABAQUS Test V=18m/s Test V=20m/s

0.2

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 30. Experimental and numerical result of Cs versus  for hemisphere.

1/2

Thus a graph of Cs /() against  1/2 should be linear, with slope (−a2 ) and intercept a1 . Fig. 31 shows the resulting plot over 1/2 1/2 the range 0.01 ≤  ≤ 0.38 then Cs /() = 5.99 − 8.55() . The obtained experimental results are very close to those found by Moghisi and Squire (Fig. 31).

Eq. (4) was used to determine the experimental impact coefficient for each of the experimental tests concerning cone with knuckles. The particularity of such geometry compared to a classical cone is that there are sharp evolutions of the local deadrise angle as the radius increases. As a consequence, the average slamming coefficient (Eq. (4)) depicted in Fig. 32 is no more parabolic after the first sharp change of local deadrise angle ( ≈ 0.046). Three different impacts were carried out for each of the three different speeds: 8, 10 and 12 m s−1 . It can be seen in Fig. 32 that there are only small discrepancies between the three experiments at the same speed (12 m s−1 ). However, much larger differences can be observed in Fig. 33 between experiments at different speeds. These important differences are due to the hydroelastic effects occurring after the first change of local deadrise angle. Indeed, as the local deadrise angle increases sharply (from 6◦ to 14◦ ), the hydrodynamic load also decreases sharply, leading to vibrations of the model. Although the period of the oscillations, that can be observed in Fig. 33, do not be the same because of the abscissa scale, the period can be related to the period of the first dry vibration-mode of the model computed numerically (T = 0.35 ms). Furthermore, Tassin et al. [21] showed that the slope of the load drop after the first knuckle was related to the time necessary to the release pressure wave to propagate from the knuckle to the centre region. This

6 18

Test with V=8m/s Test with V=10m/s Test with V=12m/s

16

5

14 12 10 3

Cs

C s /√ξ

4

8 6

2

experiemtal data linear interpolation Moghisi and Squire data

1

4 2 0

0

-2 0

0.1

0.2

0.3

√ξ

0.4

0.5

0.6

Fig. 31. Determination of a1 and a2 values for Eq. (5).

0.7

0

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 33. Value of Cs for cone with knuckles versus  at various velocities.

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A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

process. To avoid problems with slamming, attention should be paid to the ship’s geometry: the local deadrise angles that are too small should be avoided, since small deadrise angles imply large impact pressures and forces.

14 ABAQUS V=8m/s ABAQUS V=12m/s

12 10

New experimental tests are planed to measure pressure distributions during impact.

Cs

8 6

Appendix A.

4 2 0 0

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 34. Cs versus  for cone with knuckles. Table 7 Numerical and experimental values of f(ˇ). ˇ

7◦ 15◦ 30◦

The displacement dopt is calculated from the displacement da obtained numerically after double integration of the acceleration signal. A sensitivity of displacement da to initial conditions suggests that dopt takes into account of initial speed. The sensitivity of the accelerometer determined in factory (0.32 mV/g) may change slightly once integrated in the experimental assembly (light slope during the gluing process, effect of humidity, etc.). The displacement to be optimized is under the following form:

f(ˇ) ABAQUS/ Explicit

Fluent

Impact++ ABAQUS

Experiment

7.18 6.25 4.65

7.2 6.26 4.68

7.27 6.11 4.44

6.79 ± 0.56 6.18 ± 0.5 4.75 ± 0.35

phenomenon can be observed in Fig. 34 depicting the slamming coefficient computed with ABAQUS, assuming a rigid body and a compressible fluid. These two phenomena are responsible for the observed discrepancies, the structural vibrations effects being of greater order than the fluid compressibility effects. 7. Conclusions In this paper, slamming phenomena on axisymmetric bodies have been experimentally studied by means of high-speed shock machine. Several rigid shapes have been impacted an initially calm water with constant velocities. Experimental equipment and results were described in detail. The main conclusions of this study are: • The hydrodynamic forces are proportional to the square of velocity. • The slamming coefficients for the simple cones have been calculated and compared with available experiments and theoretical estimates. All are in a satisfactory agreement [15,18]. At the end, numerical results were well confirmed by the experimental ones as shown in Table 7. In conclusion, this study confirms the parabolic form of the effort according to time (Fig. 11). Obviously the deadrise angle of the cone has a huge influence on the magnitude of the impact force. At constant velocity, the value of dimensionless slamming coefficient f(ˇ) depends only of deadrise angle ˇ. Thus neither the shape nor the velocity presents an influence on f(ˇ) value (Figs. 26 and 27). • It is important to remember that Cs for hemisphere, on the contrary to the cones, depends on the depth of immersions. In that case, a Cs new approach according to  between 0.01 and 0.38 1/2 1/2 leads to: Cs /() = 5.99 − 8.55() • According to all results, the difference in peak loads for the same impact velocity between the 30◦ cone and the hemisphere is a factor of 1.9. A ratio of approximately 3.3 is found between the peak loads of the 15◦ and 30◦ cones. This illustrates the importance of considering slamming phenomena during the hull shape design

dopt (i) = ˛da (i) + v0 t(i) Whose: t is the time, ˛ and  are variables of optimization and v0 the impact velocity. The optimization consists in minimizing the deviation E between dopt and de in the sense of ordinary least squares. That is to resolve the following problem:

 E=

min

 dopt (i) − de (i) 2



c i

where c is the piston length (c = 0.6 m). The solution of the problem verifies the following conditions: ∂E ∂E =0 = ∂ ∂˛ We must resolve the system:



M11 M12

M12 M22



˛ 

=

F1 F2



with M11 =

1 2 da (i) , c2

M12 =

1 de (i) · da (i), F1 = 2 c i

v0  c2

F2 =

i

da (i) · t(i),

v0 

c2

M22 =

i

de (i) · t(i)

i

Finally, the pair {˛,}, solution of problem, is given by: ˛=

1 [F1 · M22 − F2 · M12 ] 

=

1 [F2 · M11 − F1 · M12 ] 

2 with  = M11 · M22 − M12

v20  c2

i

2

t(i) ,

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

Appendix B. Detailed views of the rigid cone with different deadrise angles (7◦ , 15◦ and 30◦ ).

195

196

A. El Malki Alaoui et al. / Applied Ocean Research 37 (2012) 183–197

Appendix C. Design of hemisphere and cone with knuckles (6◦ , 14◦ and 10◦ ).

References [1] Von Karman T. The impact on seaplane during landing. NACA TN 321; October 1929. [2] Wagner H. Uber Stoss – und Gleitvorgange an der oberflacke flussigkiten. ZAMM 1932;4(4):193–235. [3] Shiffman M, Spencer DC. The force of impact on a sphere striking a water surface. AMP Report 42. 2R. AMG-NYU Nos. 133; 1945. [4] Miloh T. Hamilton’s principle, Lagrange’s method and ship motion theory. Journal of Ship Research 1984;28(4):229–37. [5] Miloh T. On the initial-stage slamming of a rigid sphere in vertical water entry. Applied Ocean Research 1991;13(1):43–8. [6] Moghisi M, Squire PT. An experimental investigation of the initial force of impact on a sphere striking a liquid surface. Journal of Fluid Mechanics 1981;108:133–46. [7] Laverty Jr SM. Experimental hydrodynamics of spherical projectiles impacting on a free surface using high speed imaging techniques. Thesis, Massachusetts Institute of Technology; August 8, 2004. [8] Chuang SL, Milne DT. Drop tests of cones to investigate the three-dimensional effects of slamming. Washington: Report No. 3543, Naval Ship Research and Development Center; April 1971.

[9] Chuang SL. Experiments on slamming of wedge-shaped bodies. Journal of Ship Research 1967;11:190–8. [10] Huera-Huarte FJ, Jeon D, Gharib M. Experimental investigation of water slamming loads on panels. Ocean Engineering 2011;38:1347– 55. [11] Battistin D, Iafrati A. Hydrodynamic loads during water entry of twodimensional and axisymmetric bodies. Journal of Fluids and Structures 2003;17:643–64. [12] Yettou EM, Desrochers A, Champoux Y. Experimental study on the water impact of a symmetrical wedge. Fluid Dynamics Research 2006;38: 47–66. [13] Constantinescu A, Neme A, Jacques N, Rigo P. Finite element simulation and experimental investigations of simple 2D geometries in slamming. In: ASME, 27th international conference on offshore mechanics and arctic engineering. 2008. [14] Constantinescu A. 2D modelling of the impact of a body on water. Damage initialization. Ph.D. Thesis, Ensieta, Brest, France; 2004. [15] De Backer G, Vantorre M, Frigaard P, Beels C, De Rouck J. Bottom slamming on heaving point absorber wave energy device. Journal of Marine Science and Technology 2010;15(2):119–30.

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[19] Kleefsman KMT, Fekken G, Veldman AEP, Iwanowski B, Buchner B. A volumeof-fluid based simulation method for wave impact problems. Journal of Computational Physics 2005;206:363–93. [21] Tassin A, Jacques N, El Malki Alaoui A, Nême A, Leblé N. Assessment and comparison of several analytical models of water impact. International Journal of Multiphysics 2010;4(2):125–40.