Sensitivity Analysis of the Effect of Speed and Inclination Angle on Water-Entry Slamming Pressure of the Bow

In this paper, a method about the water-entry slamming of a two-dimensional (2D) bow structure has been proposed based on the experimental and simulation results. According to this method, the sensitivity analysis has been carried out about the effect of speed and inclination angle on the slamming pressure of the bow. Firstly, a 2D ship bow experimental model was performed to obtain the slamming pressure distribution at different measuring points under different speeds. Then, numerical simulation for the water-entry slamming of this experimental model was conducted to obtain the pressure distribution on the experimental model under different working conditions. Finally, the experimental results were compared with the numerical simulation results to evaluate the effect of speed and inclination angle on the slamming pressure of the bow. The results show that the slamming pressure is more sensitive to speed variation within the low-speed range. The effect of inclination angle on the slamming pressure is more obvious in the small angle condition. When the inclination angle is larger than 45°, the effect is limited.


Introduction
Vessel slamming is a complex process with fluid-solid coupling interactions involving the relationship among three factors: air, water, and structure, which have a certain nonlinear and special characteristic. The retroaction of structural deformation on the flow field has not been considered in the traditional theoretical methods, and it cannot truly reflect the essential properties and physical mechanism of the slamming phenomenon. Although many numerical simulation methods are known, they are not mature enough, and some artificial errors exist in the results. Hence, the experimental method is still one of the most reliable ways to analyze the slamming problem. Chuang (1966Chuang ( , 1967 tested the water-entry problem of a simple rigid body and two-dimensional (2D) wedgeshaped structure, and the pressure at different positions of the wedge-shaped structure was measured for water angles of 1°-15°. The existence of air cushion was verified by measuring the water-entry impact for a rigid flat-bottomed object. Zhao et al. (1996) conducted a free-fall slamming test of a V-shaped section with an inclination angle of 30°. The experimental results show that the maximum slamming pressure occurs near the root section of fluidic splash initially and transfers to the keel part of the wedge as the flow separates. Luo et al. (2012) and Wang (2011) simulated the water-entry slamming loads of a free-falling wedge using the LS-DYNA software, and the results were close to those obtained from experiments. Veen and Gourlay (2012) combined the SPH method with the slicing method to predict the slamming loads of a flaring bow under the working condition of head seas. Register (2011) and American Bureau of Shipping (2011) provided calculation ideas for designing the slamming loads of partial naval ships. Tian et al. (2014) used the three-dimensional (3D) Rankine source method to solve the motion of hull in irregular waves and then combined the slamming pressure coefficient to generate the maximum value of slamming pressure. Moreover, the design value of the slamming load was obtained by a shortterm analysis. Wang et al. (2010) conducted a water-entry slamming test on a ship frame structure. Unlike the traditional wedge drop tests, this model consists of two sections, namely 0° and 25°. The reduction coefficient of the slamming pressure was obtained by experiments, and at the same time, the linear relationship between the reduction coeffi-cient and water-entry velocity was established.
In this study, a 2D ship bow experimental model with two curvatures was designed based on two typical frame structures of a real ship according to similar rigidity and a certain scale ratio. First, the slamming test of this structure was performed to obtain the slamming pressure distribution at different measuring points of the experimental model under different speeds. Meanwhile, the feasibility of the experimental model was verified by numerical simulation, i.e., the results of this experimental model were compared with the numerical simulation results of the slamming pressure of a symmetric structure. Then, numerical simulation for the water-entry slamming of this experimental model was conducted to obtain the pressure distribution on the experimental model under different working conditions. Finally, the test results were compared with the numerical simulation results to verify the accuracy of the numerical simulation method, thus laying a foundation for the numerical simulation of the water-entry slamming of a 3D structure.

Experimental model design
According to the shape line of the ship bow, to reduce the test errors under different working conditions and increase the comparability of the effect of different inclination angles on the slamming pressure and save the experimental cost, a model composed of two semi-sections with different inclination angles (different frames of the ship bow) was developed. The advantage is that it is possible to obtain the slamming pressures of the bow frame at two different inclination angles when conducting the water-entry test only once, and they have the same loading condition, making the test results more comparable.
The model was designed as a 2D form, whose shape and structure remain unchanged in length. To make the fluid model show a 2D flow pattern during free falling as far as possible, stop plates were installed at the front and back of the model to limit the longitudinal flow of the flow field along the model. The main dimensions of the entire model are 0.738 m (length)×0.6 m (width)×0.27 m (height). Its scale ratio with the original vessel is 1:20, and the total mass of the model is 24 kg. Fig. 1 shows a typical cross-section of the model; Fig. 2 shows the internal structure of the experiment model.

Measuring method
The experiment was carried out in the experimental pool of Jiangsu University of Science and Technology. In the experiment, the subsection model of ship body was made to fall from different heights to a still water surface to simulate the slamming phenomenon. The water-entry speed and the slamming pressure during the slamming were tested. An optoelectronic switch was used to test the water-entry speed, and a pressure sensor was used to measure the slamming pressure. The specific testing process is shown in Fig. 3. Fig. 4 shows a schematic diagram of the water-entry device for the model. The slamming test device consists of a slamming test pool, test tower, and polyester lifting rope. The size of the experimental pool is 100 m×4 m×3 m. The height of the crane for the slamming test is 5 m. It was fixed over the experimental pool. The lifting hook was suspended at the midpoint of the beam. At the beginning of the test, the model was lifted to a predetermined height using the crane. Then, the lifting rope was cut, thus allowing the model to fall freely. To reduce the friction between the lifting rope   ZHANG Jian et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 432-440 433 and beam, the beam was coated with lubricating oil.

Measuring point arrangement
The testing contents mainly include the following aspects: (1) Slamming pressures of points P1 and P4 at the center of the right bottom plate of the model under different heights; (2) Slamming pressures of points P2, P3, and P5 at the center of the right hypotenuse of the model under different heights; (3) Slamming pressures of points P6, P7, and P10 at the center of the left bottom plate of the model under different heights; (4) Slamming pressures of points P8 and P9 at the center of the left hypotenuse of the model under different heights.
Ten pressure measuring points (P1−P10) were placed in the middle, front, and back of the grillage on the bottom. Note that the pressure measuring points P1, P2, P6, P7, and P10 correspond to the previous simulation values obtained from five stress elements Elm1-5, and the rest of the measuring points were observed as needed.
According to the above mentioned testing contents, the position descriptions and labels of the measuring points of the model are shown in Fig. 5. The dimensions marked horizontally in the figure are the positions of the longitudinal in the model.

Testing procedures
In this test, free-fall water entry was used to measure the slamming pressure on the structure of the model. During the measurement, the structure was first hung on the crane beam, and then the structural model was made to fall at a distance from the water surface, 0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, 0.6 m, 0.7 m, 0.8 m, 0.9 m, and 1 m high. Pressure sensors were used to measure the slamming pressure on the structure in experiments with different falling heights. The working conditions of the model were designed according to the principle of hydrodynamic similarity, and the free-falling height of the model was determined according to the possible water-entry slamming speed of a real ship. To eliminate the discrete type of slamming and reduce the minimization of human factors, tests were repeated for more than four times for each height.
Before each test, all the testing instruments return to zero. The lifting device was activated to increase the structural model to the height of current working condition, and then the lifting rope was cut with scissors for the free falling of the structure. During the water-entry process of the structure, the bottom first collides with the water surface, and the splashing water spray extends along the middle part to both sides. At this moment, the data acquisition instrument can automatically record the voltage signal history of the pressure sensor. After each test, the structure was lifted again to the next target height. The next test was conducted after the water surface became calm.
As shown in Fig. 6, a drop height of 1 m was set as an example to show the time-varying process of the waterentry slamming of the model. When t = 1/30 s, the bottom of the experimental model started to contact with the water surface. When t = 2/30 s, the model entered the water, and slamming was triggered. Water sprays start splashing on both sides of the model; when t = 3/30 s, the waves around the model splashed along the hypotenuse of the model with the maximum ramp angle. When t = 4/30 s, the splashing waves of the model continued to rise along the transverse direction, and the disorder occurred along the longitudinal direction of the model at the same time. When t = 5/30 s to t = 18/30 s, the splashing water sprays of the experimental model gradually splashed to the maximum position. When t = 19/30 s, the model reached its lowest point. At this time, the model completely entered the water, and the slamming

Analysis of experimental results
4.1 Test results for slamming pressure on the right side of the model Fig. 7 shows the curves for the maximum pressure of measuring points on the right side of the model with drop heights. With the increase in drop height, i.e., the increase in slamming speed, the maximum slamming pressure shows a trend of gradual increase. Fig. 8 shows a time-history diagram of the water-entry slamming pressure of a typical measuring point P1 on the right of the experimental model with a height of 0.2 m. When the slamming occurred, i.e., the bottom touched the water surface, the slamming pressure of the model rapidly increased to the maximum value. The slamming pressure gradually decreased with the increase in the water-entry depth until the water surface was calm.
4.2 Test results for slamming pressure on the left side of the model Fig. 9 shows the curves for the maximum pressure of measuring points on the left side of the model with drop heights. With the increase in drop height and water-entry slamming speed, the maximum value of the slamming pres-   sure gradually increased. As the vertical curvature increased, i.e., the constant increase of inclination angle, the maximum slamming pressure of various measuring points gradually decreased. Fig. 10 shows a time-history diagram of the slamming pressure of a typical measuring point P10 on the left of the experimental model with a height of 0.2 m. The distribution law of water-entry slamming pressure of measuring points on the left of the model is similar to that on the right of the model. When the slamming occurs, that is, the bottom touches the water surface, the slamming pressure of the model rapidly increases to the maximum value. The slamming pressure gradually decreases with the increase of water entry depth until the water surface is calm. Fig. 11 shows the free fall of the typical measuring Point P1 on the right of the model with a height of 0.1-1 m, namely, the trend of the maximum slamming pressure with a water-entry speed from 1.4 m/s to 4.47 m/s based on the free fall. With the increase in drop height, the water-entry slamming speed increased, and the maximum value of slamming pressure gradually increased. At a low speed, i.e., when the height was relatively small, the change was obvious. When the height increased, the speed increased, and the slamming pressure changed slowly, indicating that the speed change is more sensitive to the slamming pressure at a low speed.

Test results for slamming pressure at different meas-
uring points Fig. 12 shows a curve for the change in the slamming pressure of typical measuring points P6, P7, P8, and P9 on the left of the model with a height of 0.1 m. In the vertical direction, the maximum value of the slamming pressure for various measuring points of the model gradually decreased with the increase in the vertical height of the measuring point from the center of the bottom plate, i.e., the increase in inclination angle. For various measuring points on the left side of the model, when the height is 0.1 m, the maximum slamming pressure appears at P6, which is at the bottom of the model, reaching 182.37 kPa; and the maximum slamming pressure at P9, where the hypotenuse is the closest to the sheer strake, is only 3.9 kPa. The results show that the effect of inclination angle on the slamming pressure is limited to a small angle, generally within the range of 45°; when the inclination angle was very large, the effect of slamming pressure gradually became smaller and even negligible.

Simulation study of asymmetric wedge slamming
The feasibility of simplifying two curvature models into one model was considered. In this section, V-shaped wedge structures with three different inclination angles (the angle between the outside surface of the structure and water surface) were designed. Figs. 13 and 14 show two symmetrical models with 20° and 30° inclination angles, respectively.   An asymmetric model with 30° and 20° inclination angles is shown in Fig. 15. First, the maximum slamming pressure of the symmetrical model was calculated at a constant waterentry velocity and compared with the maximum value of the slamming pressure for an asymmetric model with a corresponding angle at the same water-entry velocity. By observing the error of maximum pressure values at different measuring points, the feasibility and accuracy of using an asymmetric model in place of two corresponding symmetrical models to conduct the slamming test were verified to enhance the interference of other factors on the test results during the experimental process. Table 1 shows a comparison of the maximum values of water-entry slamming pressure at several typical measuring points of a structure with 30° inclination angle on the left side and 20° inclination angle on the right side and the corresponding symmetric structures. Clearly, the slamming pressures of the asymmetric and symmetric structures under the same conditions are similar. Fig. 16 shows a comparison of the maximum values of water-entry slamming pressure at several typical measuring points of a structure with 30° inclination angle on the left side and 20° inclination angle on the right side and its corresponding symmetric structure with 30° inclination angle on both sides. As shown in the following figure, for the asymmetric and symmetric structures, the slamming pressures are almost the same under the same conditions when the vertical curvature is small. As the vertical curvature increases, the error of the hypotenuse measuring point increases, but the errors are all within a reasonable range of 20%. The wedge-shaped structure with two different curvatures can be approximately simplified into a 2D wedge model with two curvatures. Thus, the initial conditions are identical for both the left and right sides in a water-entry process, improving the impact comparability of different curvatures on the maximum value of slamming pressure.

A 2D ship bow simulation model
In this paper, the finite element software MSC.Dytran is used to carry out the numerical simulation. In the previous study (Zhang et al., 2016), the software was used to simulate the process of water slamming in two dimensional wedges and the results were compared with the experimental results of Chuang (1966Chuang ( , 1967. The error is very small, which verifies the accuracy of the numerical simulation software used in this paper to deal with the problem of water slamming. In order to make the model test and numerical simulation calculation comparable, the constructed numerical simulation model is consistent with the experimental model as much as possible.The dimensions of the mathematical model established in this paper and the experimental model are identical in Y (the direction of ship width) and Z (the direc-    ZHANG Jian et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 432-440 tion of ship height). In X (the direction of ship length), effective units are used instead of contact length, while the influence of factors that cannot be changed, such as water weight and air field, are ignored. Although such a generalization model could not completely construct equivalent simulation test scenes in accordance with the similarity criteria, which led to a certain deviation of the test results in longitudinal pressure, it did not affect our comparison of the results of the ship model entering water and thudding under different working conditions. The simulation results shown in the previous section indicate that to improve the computational efficiency, it is feasible for an asymmetric simulation model to calculate the 2D water-entry slamming of a real ship. The cross-sections of bow frames #156 and #167 of a bulk cargo ship were selected as the research objects. To improve the computational efficiency, two finite-element models were established for the frames in the same plane, as shown in Fig. 17. The finite-element model for the water-entry simulation computation of a 2D model for a part of ship bow is shown in Fig. 18. For the structure of ship bow, the length in the x direction is 0.738 m, the height is 0.27 m, and the distance from the water surface is 0.2 m. A rigid structure was used. The outside surface of the structure is defined as a fluid-solid coupling surface, and the general coupling algorithm was used. Euler region is divided into two parts: The upper region is an air medium, and the lower region is an aqueous medium. Nonviscous and compressible materials with linear fluid-constitutive relations were used to fill these elements, and the pressure within a water area can be described by a polynomial state equation. Free and nonreflective condition was used for other boundaries.

Slamming simulation computation of a 2D ship bow model
In this section, the simulation was used to calculate the water-entry slamming of the model under five working con-ditions, the water-entry height is 0.2 m, 0.4 m, 0.6 m, 0.8 m, and 1.0 m. Table 2 shows the slamming pressures of five typical Elements 1-5 existed at the bottom of the structural model during the slamming. With the height of 0.2 m as an example, Figs. 19 and 20 show the time-varying maximum value of water-entry slamming pressure at two typical positions (Elm 1 and Elm 2).
Compared with the above data, the slamming pressure increased with water-entry height (velocity). In the vertical direction, the pressure decreased with the increase in the inclination angle, and the stress of the element measuring point on the left side was greater than that of the corresponding point on the right side.     China Ocean Eng., 2020, Vol. 34, No. 3, P. 432-440 6 Comparative analysis between test and simulation results

Comparison between test and simulation results
In order to validate the numerical simulation results with test results, several typical points of the experimental model were selected, such as P1, P2, P6, P7, and P10. The slamming pressures measured from the tests were compared with the calculation results of the simulation model. Tables 3 and  4 show comparisons between the test and simulation results of typical measuring points on the left and right sides of the model. A comparison shows that the maximum slamming pressures obtained from the numerical simulation are identical with those obtained from the experiment, and the variation trends with the height (speed) of the two results are consistent. Table 4 shows that the values could not be measured for several heights during the testing, and therefore they are not listed. A comparison shows that the errors are relatively small, indicating that it is feasible and accurate to use numerical simulation to study the water-entry slamming of 2D bow structures.

Error analysis
The errors between the simulation and experimental results were analyzed, and the main reasons for these errors are as follows.
(1) The initial conditions of simulation and experiment have a certain difference. The initial velocity of the model was provided in the simulation, and a constant speed was maintained to simulate the velocity at the time of water entry in the test. The influence of natural factors such as air was neglected. Moreover, the water and air areas were described by ideal equations, where the water area of such materials are nonlinear. Therefore, it affects the results to some extent.
(2) The release device was lifted in the test, and a polyester rope was used. Due to the tightness of the rope and the high friction, the natural stretch under compression affected the load balance of the structure. However, these factors  ZHANG Jian et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 432-440 439 were not considered in numerical simulation.

Conclusions
In this study, the water-entry slamming test of a 2D partial ship body with different curvatures was designed. The effect of velocity and curvature on the slamming problem was studied. Test devices and an experimental model were designed. The slamming test of a 2D structure with different curvatures was conducted in a pool to test the slamming pressure of the model at different heights during the waterentry slamming. Based on the test results, the following conclusions can be drawn.
(1) Both the numerical simulation and experimental results show that it is feasible to design a model with an asymmetric structure and use this model to simulate water-entry slamming at two different inclination angles. This method is more advantageous than that with two models of different inclination angles. In addition to increasing the calculation efficiency, more importantly it ensured the same initial condition for water-entry slamming and increased the comparability of the calculation results for the left and right sides, making the evaluation of the effect of different inclination angles on the slamming pressure more accurate.
(2) A comparison shows that the maximum slamming pressures of the numerical simulation results are identical with those of experimental results, and the variation trends with the height (speed) of two results are consistent. Therefore, it is feasible to simulate the water-entry slamming of a 2D bow structure using a numerical method.
(3) The test results show that the maximum value of the slamming pressure of a 2D bow structure increased with the increase in the slamming speed. The change in slamming pressure is obvious at a low speed; when the speed gradually increased, the slamming pressure changed slowly, indicating that the slamming pressure is more sensitive to the speed change within a low-speed range.
(4) The larger the inclination angle of the structure, the smaller the maximum slamming pressure. The effect of inclination angle on the slamming pressure is only limited to a small angle, generally below 45°. When the inclination angle was larger than 45°, the effect of inclination on the slamming pressure gradually decreased.