We present a study of the forces, velocities, and trajectories of slender (length/diameter = 10) axisymmetric projectiles using an embedded inertial measurement unit (IMU). Three nose shapes (cone, ogive, and flat) were used. Projectiles were tested at vertical and oblique impact angles with different surface treatments. The trajectory of a half-hydrophobic and half-hydrophilc case impacting vertically was compared to the trajectory of symmetrically coated projectiles impacting the free surface at oblique angles. The oblique impact cases showed significantly more final lateral displacement than the half-and-half case over the same depth. The amount of lateral displacement was also affected by the nose shape, with the cone nose shape achieving the largest lateral displacement for the oblique entry case. Instantaneous lift and drag coefficients were calculated using data from the IMU for the vertical, half-and-half, and oblique entry cases. Impact forces were calculated for each nose shape and the flat nose shape experienced the largest impulsive forces up to 37 N when impacting vertically. The impact force of the flat nose decreased for the oblique entry case. The location of the center of pressure was determined at discrete time steps using a theoretical torque model and values from the IMU. Acoustic spectrograms showed that the sound produced during the water entry event predominately arises from the pinch-off for the cone and ogive nose shapes, with additional sound production from impact for the flat nose shape. Each test run was imaged using two Photron SA3 cameras.

## I. INTRODUCTION AND BACKGROUND

The splash behavior and cavity formation of many previous studies have focused on canonical shapes and their associated cavity dynamics, yet few authors have been able to accurately measure the accelerations and forces associated with these impacts and none has utilized an untethered accelerometer or simultaneously measured the acoustic signature. This paper presents findings for the water entry dynamics of slender axisymmetric bodies including the effects of nose shape, wetting angle, and impact angle on the object dynamics (e.g., forces, trajectory, and velocity) and cavity acoustics. The experiments were performed at relatively high Reynold's number (Re > 68 000), high Weber number (We > 2470), and high Bond number (Bo ≈ 22), indicating that surface tension does not significantly affect the experiments.^{1} Fig. 1 shows the major events accompanying water entry of a slender axisymmetric projectile in time, including radial sheet formation, cavity pinch-off, and jet formation.

Studies of objects entering into a liquid have been published for decades, beginning with water droplets falling into a water-milk mixture utilizing short-duration flash and hand drawing reconstruction by Worthington and Cole.^{2} Worthington then used short-duration flash photography to capture splashes^{2} and later extended his studies to include cavities.^{3} We present images using two Photron Fastcam SA3 cameras to illustrate both symmetric and asymmetric cavities as well as to provide visualization of subsurface trajectories of slender axisymmetric bodies.

A study of the comparison between the cavity dynamics of spheres and other shapes was performed by Duclaux *et al.*^{4} They derived an approximate analytic solution showing that there is a shape difference between the cavity formation of spheres and cylinders. The cavities formed by spheres were larger than the impacting sphere while the cavities formed by cylinders were roughly the same radial size as the cylinder. They showed that the cavity pinch-off time was independent of the impact velocity, but increased with an increase in diameter. They also showed that an increased diameter resulted in a decrease in pinch-off depth. Duclaux assumed a constant velocity model, but Aristoff *et al.*^{5} expanded on the model to include acceleration changes due to hydrodynamic forces and showed that the density of a sphere significantly changed the pinch-off depth of the cavity, but the pinch-off time remained constant when compared to normalized sphere densities. Gilbarg and Anderson^{6} examined the effects of atmospheric pressure on the cavity formation and collapse, and showed that the cavity would remain open at the free surface permanently for pressures of 1/16 atmosphere or less. They also calculated non-dimensional pinch-off times for spheres under varying atmospheric pressures. Non-dimensional pinch-off times for slender axisymmetric bodies are examined in the present study and compared with their findings.

Rather than using a symmetrical surface condition for their test objects, Techet and Truscott^{7,8} coated one hemisphere of spheres with a hydrophobic spray, which resulted in the formation of an asymmetric cavity and trajectory deflection upon entering the water. These findings were similar to the results from dropping spinning spheres into water^{7,8} and indicated that the wetting angle could alter the trajectory of objects. Recently, Bodily *et al.*^{9} showed the effects of a half-and-half coating on an axisymmetric body versus an oblique entry and that even a 5° deviation from vertical has a larger effect on trajectory than coatings. The present study expands on ideas presented in these two studies to show the effects of asymmetric cavities on slender axisymmetric bodies.

Uber and Fegan^{10} studied the acoustic signatures of solid metal spheres, missile-like projectiles, and bowling balls being dropped into the water. They showed that an object entering the water could be potentially identified by its unique sound frequency signature. Honghui and Makoto^{11} measured the acoustic pressure at several distances below the free surface in the wake of a water entry projectile. They observed a somewhat uniform pressure field, with increased uniformity as distance increased from the impact location. They also showed that the initial pressure rise occurred microseconds after the impact. Grumstrup *et al.*^{12} noticed that cavity rippling occurs after pinch-off for spheres, hemispheres, and cones. They reported that the rippling effect is a reflected acoustic wave induced by the oscillations of the impacting body, while Mansoor *et al.*^{13} studied it further and found multiple sources for the rippling events.

The study of water entry of slender axisymmetric bodies is not limited to cavity dynamics.^{29} The abrupt deceleration of high-speed bodies can cause damage, sometimes significant, to both the body and onboard instrumentation.^{14} Baldwin^{15} obtained simple acceleration data for cones to be used for engineering purposes. He used a single-axis crystal accelerometer to determine true velocity at impact and at subsequent times during the water entry. These data were used to determine the coefficients of drag for cone shapes at various deadrise angles as they decelerated through the water. Moghisi and Squire^{16} studied the impact forces on a steel sphere at depths of up to one quarter of the sphere radius and velocities from 1 to 3 m/s. Theoretically, the water impact phenomenon was reviewed by Korobkin and Pukhnachov^{17} and progressively improved upon from Wagner^{18} to Faltinsen and Zhao.^{19}

Currently, a widely used method for obtaining and analyzing the data for water entry is high-speed imaging.^{5,22,4,21,8} This method works well for observing the cavity formation and collapse, trajectory changes, and even velocities. However, using images to estimate the accelerations of an object at impact and subsequent times will not be as precise as using a device specifically designed for the purpose of measuring acceleration due to image resolution and loss of visualization due to cavity turbulence. Experimental studies of forces have used tethered accelerometers, but only to study the impact event.^{16,23,24} Advances in technology have decreased accelerometer size and have enabled the construction of small embedded inertial measurement units (IMU), such as the one used in this study. Herein, tests were performed with an embedded IMU and the data from the impact event were extracted after the experimental tests and analyzed to show the unhindered accelerations of the test object. Simultaneously, a hydrophone and two high speed cameras were utilized to record the sounds and motion of the impact, respectively. Comparisons of the impact forces, trajectories and acoustic signatures of several different nose shapes, material densities, and surface conditions are presented. The data show that major trajectory changes can occur when the projectiles impact the free surface at just 2° or are coated half in hydrophobic, half in hydrophilic coatings.

## II. EXPERIMENTAL METHODS

Experimental tests were conducted for three nose shapes, three surface conditions, and three impact angles. Each test was performed by releasing a projectile into a glass tank measuring 91 × 91 × 122 cm (width × depth × height) with water filled to 91 cm. Two different release heights (36 cm and 50 cm) were used to vary the impact velocity from 2.66 m/s to 3.13 m/s. The tests were recorded at 2000 frames per second (fps) by two Photron Fastcam SA3 high-speed cameras that were positioned orthogonal to each other and normal to the tank as shown in Fig. 2(a). The resolution for each image was 512 pixels × 768 pixels. A bank of 20 fluorescent bulbs with a diffuser sheet provided backlighting for the camera images and was placed in line with each camera behind the tank. For the vertical water entry tests, the projectile was suspended from an electromagnet by a screw that was attached to the tail of the projectile (Fig. 2(A)). The projectile was released from the electromagnet after the projectile became stationary and aligned vertically. The angled water entry cases were performed by securing the projectile to a mount using a vacuum pump (Fig. 2(B)). The mount was set to the appropriate angle using a three-axis positioning mechanism. The projectile dropped when the vacuum pump was turned off.

Material properties of acrylic, aluminum, and Delrin^{®} acetal resin used in this study are outlined in Table I. The effective density ρ/ρ_{w} (hereafter referred to as density) is the mass of each projectile divided by its respective volume. Calculating the density in this manner takes into consideration the different amount of material removed from the nose and the tail (space for embedded IMU). The nose shapes used were a cone, an elliptical ogive, and a flat nose. The machining process used to fabricate the nose shapes caused differences in the wetting angles, even among like materials, likely due to the differences in machined surface roughnesses. Kubiak *et al.*^{20} in 2011 showed that surface roughness can alter wetting angles. Table I presents the measured wetting angle for each material-nose shape combination. A profile of each nose shape is shown in Fig. 3 along with geometric parameters. The profile of the elliptical ogive nose shape follows the function

*D*is the diameter of the projectile in the

*y*direction and

*L*is the length of the nose section in the

*x*direction. The 1.02-cm section on the nose of the flat corresponds to a flat region with two 0.76-cm-radius contours on either side.

Material . | Nose shape . | ρ/ρ_{w}
. | d
. | γ, Hydrophilic (deg) . | γ, Hydrophobic (deg) . |
---|---|---|---|---|---|

Cone | 1.10 | 13.7 | 65 | 120 | |

Acrylic | Ogive | 1.24 | 12.8 | 73 | 119 |

Flat | 1.10 | 12.4 | 79 | 112 | |

Cone | 2.26 | 12.7 | 64 | 125 | |

Aluminum | Ogive | 2.54 | 12.0 | 62 | 127 |

Flat | 2.28 | 11.4 | 67 | 112 | |

Cone | 1.27 | 13.5 | 61 | 133 | |

Delrin | Ogive | 1.42 | 12.5 | 84 | 125 |

Flat | 1.31 | 12.1 | 77 | 108 | |

Ogive | 1.31 | 8.6 | 84 | 125 | |

Delrin CM | Ogive | 1.31 | 10.6 | 84 | 125 |

Ogive | 1.31 | 12.6 | 84 | 125 |

Material . | Nose shape . | ρ/ρ_{w}
. | d
. | γ, Hydrophilic (deg) . | γ, Hydrophobic (deg) . |
---|---|---|---|---|---|

Cone | 1.10 | 13.7 | 65 | 120 | |

Acrylic | Ogive | 1.24 | 12.8 | 73 | 119 |

Flat | 1.10 | 12.4 | 79 | 112 | |

Cone | 2.26 | 12.7 | 64 | 125 | |

Aluminum | Ogive | 2.54 | 12.0 | 62 | 127 |

Flat | 2.28 | 11.4 | 67 | 112 | |

Cone | 1.27 | 13.5 | 61 | 133 | |

Delrin | Ogive | 1.42 | 12.5 | 84 | 125 |

Flat | 1.31 | 12.1 | 77 | 108 | |

Ogive | 1.31 | 8.6 | 84 | 125 | |

Delrin CM | Ogive | 1.31 | 10.6 | 84 | 125 |

Ogive | 1.31 | 12.6 | 84 | 125 |

The surface of each projectile was prepared prior to the tests based on which case was to be performed. Immediately before each test of the hydrophilic case, the surface of the projectile was first cleansed with isopropanol, scrubbed with Kimtech Kimwipes^{®}, rinsed with isopropanol, dried using canned air, and finally ethanol was used to remove any residual film remaining from the isopropanol. For the hydrophobic cases, the projectiles were cleaned using the same procedure as the hydrophilic case and then coated with Cytonix WX2100^{™}, a hydrophobic aerosol spray. The half-and-half condition was achieved by coating only one-half of the surface along the centerline with WX2100^{™}. The coatings were allowed to dry for at least 2 h before being used in experiments.

Experiments to evaluate the effect of changing the location of the center of mass of the projectile were also performed. The projectile was a modified Delrin ogive shown in Fig. 4. A 1.91-cm-diameter hole was drilled from the tail, leaving 2.54 cm of material in the nose section. A lead slug (Fig. 4(c)) weighing 102.1 g and measuring 1.91 cm in diameter and 3.81 cm in length was used to vary the location of the center of mass of the projectile. Two spacers (Figs. 4(b) and 4(d)) measuring 1.85 cm in diameter and 3.33 cm in length were machined out of Delrin to adjust the location of the lead mass for a total of three locations. The lead slug and spacers could be arranged in different orders to change the location of the center of mass.

The range of the Reynolds, Weber, and Froude numbers are presented in Table II. The transition from laminar to turbulent in a flow over a sphere is around Re = 5 × 10^{5}, thus the boundary layer is likely laminar for both the 36-cm and 50-cm drop heights.^{25} A thorough understanding of the importance of the transition between laminar and turbulent boundary layers upon water entry has not been well studied.

Parameter . | Equation . | 36 cm (U_{0} = 2.66 m/s)
. | 50 cm (U_{0} = 3.13 m/s)
. |
---|---|---|---|

Reynolds | $\frac{\rho _w U_0 D}{\mu }$ $\rho wU0D\mu $ | 68 193 | 80 241 |

Weber | $\frac{\rho _w U_0^2 D}{\sigma }$ $\rho wU02D\sigma $ | 2470 | 3420 |

Froude | $\frac{U_0}{\sqrt{g D}}$ $U0gD$ | 5.3 | 6.3 |

Parameter . | Equation . | 36 cm (U_{0} = 2.66 m/s)
. | 50 cm (U_{0} = 3.13 m/s)
. |
---|---|---|---|

Reynolds | $\frac{\rho _w U_0 D}{\mu }$ $\rho wU0D\mu $ | 68 193 | 80 241 |

Weber | $\frac{\rho _w U_0^2 D}{\sigma }$ $\rho wU02D\sigma $ | 2470 | 3420 |

Froude | $\frac{U_0}{\sqrt{g D}}$ $U0gD$ | 5.3 | 6.3 |

The IMU shown in Fig. 5 was embedded in the tail of the projectile during each test to record the instantaneous accelerations and rotation rates that the projectile experienced during the water entry event. The IMU consists of an ADXL345 accelerometer and ITG-3200 gyroscope. The accelerometer sampled at a rate of 3200 Hz, with a measurement range of ±16 g, ±10% scale factor tolerance, 0.5% nonlinearity, and a 1% cross-axis sensitivity. The gyroscope sampled at a rate of 384 Hz, with a ±2000°/s measurable rate, a ±6% scale factor tolerance, 0.2% nonlinearity, and 2% cross-axis sensitivity, and ±10% scale factor tolerance over temperature. The temperature scale factor tolerance was important for tests where the IMU must remain embedded for long periods of time, such as before performing a 0° test where the projectile must become stationary and aligned perfectly vertical before proceeding with the test. The total error of the IMU system was calculated from the manufacturer specifications to be 11.9% if temperature can be neglected, or 15.5% for worst case when the IMU is embedded for long periods of time. The total errors were calculated using the root-sum-square method. The Appendix describes the processing of the IMU data and the experimental error analysis.

The hydrophone was positioned 26 cm from the impact location, which was as far from the walls of the tank as possible while still being out of the field of view of both cameras. The horizontally omnidirectional hydrophone has a sensitivity of −180 dB and a useful range of 10 Hz to 100 kHz.^{28} It was connected to a computer using one channel of standard 3.5-mm microphone stereo input and the signal coming from the hydrophone was recorded using Audacity software at a rate of 96 kHz. The hydrophone data and the images were synchronized by utilizing the record state output from the Photron Fastcam SA3 cameras which was connected to the unused stereo channel of the microphone input. The synchronized data were processed over the portion of the hydrophone data containing the impact and pinch-off events using the MATLAB spectrogram function. The spectrogram was computed using a Hanning window of size 512 with 250 samples of overlap.

## III. RESULTS AND DISCUSSION

This section presents a discussion of the trajectory, velocity, and force results for three key water entry cases. Fig. 6(a) presents an image sequence of a hydrophobic Delrin ogive impacting the free surface at 5° with a velocity *U*_{o} of 3.13 m/s. Fig. 6(b) shows the axial acceleration measurements. The free-fall region demonstrates the IMU signal in the presence of only gravity (see the Appendix). The impact of the projectile caused an acceleration spike (*A*), followed by a gradual increase in acceleration until pinch-off of the main cavity occurred on the side of the projectile (*B*), and at pinch-off of the trailing cavity (*C*). The oscillations between B and C are of the order of 500 Hz, which happen to be the natural frequency of the IMU in water as discussed at the end of the Appendix. Further, the 500 Hz signal persists, though much lower amplitude, between B and the end of the run. The oscillations immediately following *C* will be discussed in Sec. III C 1 and are directly related the pinch-off phenomenon. There were only very small disturbances in the radial accelerations at impact, shown in Figs. 6(c) and 6(d), indicating that most of the impact acceleration occurred in the axial direction due to the completely vertical velocity and projectile orientation. Both Figs. 6(c) and 6(d) show pronounced accelerations during the on-body pinch-off (*B*), but do not show any measured accelerations for the pinch-off of the trailing cavity (*C*), whereas the axial accelerations do, Fig. 6(b).

### A. Trajectory and velocity

The trajectory of each test was computed by numerically integrating the acceleration results twice. A comparison of the findings from the tests for the hydrophilic, hydrophobic, and half-and-half cases will be presented and discussed. The trajectory results will be presented in the form of *x* and *z* positions that were obtained by integrating the acceleration data from the IMU twice. The *x* displacement values are the magnitude from the two radial IMU axes which represent the total lateral displacement of the projectile and are always shown as positive in the trajectory plots even though negative *x* displacement occurred for some cases (see Fig. 2 for coordinate system). Effectively, all rotation and lateral displacement occurred in the *x*-*z* plane with no contribution in the *y* direction (confirmed by camera two). The mean trajectories for multiple cases (e.g., Fig. 9(a)) were computed by averaging the *x* displacement for each test at 5-cm intervals along the *z* axis. The values presented in the velocity plots and tables are ratios of the absolute value of the instantaneous vertical velocity *w* to the initial vertical impact velocity *U*_{o}. All of the positions and velocities are relative to the location of the IMU, which was embedded in the tail of the projectile.

The image sequences in Fig. 7 demonstrate the five main water entry cases used in this study performed for a Delrin ogive. Fig. 8 shows the trajectories of one drop from each case plotted with drops from similar cases. The cases shown in Fig. 8(a) correspond to image sequences given in Figs. 7(a) and 7(b) depicting vertical (0°) water entry for hydrophilic and hydrophobic cases. Fig. 8(b) relates to image sequence given in Fig. 7(c) for half-and-half cases. The angled cases in Figs. 8(c) and 8(d) show trajectory plots for image sequences given in Figs. 7(d) and 7(e), respectively. The trajectory plots of the Delrin ogive hydrophilic and hydrophobic 0° cases in Fig. 8(a) show that each projectile descends straight down through the water after impact with a peak lateral displacement of 0.55 diameters. The 0° case provides the baseline for lateral translation to compare with the half-and-half and oblique entry cases. Fig. 8(b) shows single drops for the half-and-half cases. For each half-and-half drop, the projectile was aligned such that the hydrophobic half was on the left side relative to the camera. The result was a controlled lateral displacement always to the hydrophilic side with 2 diameters of displacement and approximately 0.85 diameters of variation. Figs. 8(c) and 8(d) show the lateral displacement of 2° and 5° cases with maxima of 7.7 and 13.2 diameters, respectively.

In general, increasing the density of the projectile results in greater resistance to lateral displacement, regardless of nose shape. The cone nose shapes generally have the most lateral displacement and the flat nose shape has the least for similar material densities.

#### 1. Symmetric surface condition, 0° impact angle

The trajectory curves from Figs. 9(a)–9(c) show that minimal (<0.5 diameter) lateral displacement occurs over 10 tests for both the hydrophobic and hydrophilic Delrin cases. The 0° impact angles are similar for all other materials. The trajectory of the cone nose shape showed the most variation in the final position and all three nose shapes showed similar variation over the first 20 diameters of descent following impact with the free surface.

The velocities in Figs. 9(d)–9(f) were normalized by the impact velocity *U*_{o} (3.13 m/s). The *w* velocity of the projectile continued to increase for 0.0625 s after impact, at which point the deceleration caused by hydrodynamic forces reversed the direction of *w*, which then decreased at a relatively constant rate, as illustrated in Fig. 10. The peak and final *w* velocities for each case are presented in Table III (taken at *z*/*D* = 34). The hydrophobic cases for both the ogive and flat nose shapes demonstrate a slower final *w* velocity than for the hydrophilic case. This slower velocity was caused by the momentum transfer from the projectile to the water which was manifest by the splash curtain and cavity. The hydrophobic case for the cone shows a slight increase in velocity compared to the hydrophilic case. Fig. 11 depicts the differences in splash curtain formation for each nose shape in both the hydrophilic and hydrophobic cases. As seen in Figs. 11(c) and 11(f), the nose geometry was responsible for the splash curtain that forms for the cone hydrophilic case and is due to the sharp corner transition at the base of the nose. The minimal difference between the hydrophilic and hydrophobic cone velocity values was due to the formation of a cavity in both cases.

Nose shape . | Case . | Final x/D
. | Peak w/U_{o}
. | Final w/U_{o}
. |
---|---|---|---|---|

Cone | Hydrophilic | 0.24 | 1.11 | 0.97 |

Hydrophobic | 0.19 | 1.11 | 0.98 | |

Ogive | Hydrophilic | 0.13 | 1.10 | 0.99 |

Hydrophobic | 0.37 | 1.10 | 0.98 | |

Flat | Hydrophilic | 0.17 | 1.07 | 0.91 |

Hydrophobic | 0.21 | 1.05 | 0.86 |

Nose shape . | Case . | Final x/D
. | Peak w/U_{o}
. | Final w/U_{o}
. |
---|---|---|---|---|

Cone | Hydrophilic | 0.24 | 1.11 | 0.97 |

Hydrophobic | 0.19 | 1.11 | 0.98 | |

Ogive | Hydrophilic | 0.13 | 1.10 | 0.99 |

Hydrophobic | 0.37 | 1.10 | 0.98 | |

Flat | Hydrophilic | 0.17 | 1.07 | 0.91 |

Hydrophobic | 0.21 | 1.05 | 0.86 |

The normalized theoretical terminal velocity *U*_{t}/*U*_{o} in water for each case is shown in Figs. 9(d)–9(f) as a value corresponding to the reference velocity plot. These were calculated using the method described in Sec. III B 3. The ogive demonstrates a faster *U*_{t}/*U*_{o} due to the decreased measurement drag coefficient that was used to compute *U*_{t}.

Truscott *et al.*^{30} showed that from impact to pinch-off hydrophobic spheres had increased velocity compared to hydrophilic ones due to the diminished formation of vortices behind the cavity-forming hydrophobic spheres. Here it appears that the slender bodies have relatively the same speeds up to pinch-off whether hydrophilic or hydrophobic. However, the flat nose hydrophobic case does seem to have a significantly slower speed at depth. This difference between spheres and slender axisymmetric bodies could be due to the long body inhibiting vortex shedding in both hydrophilic and hydrophobic cases near the surface coupled with the increased hydrodynamic drag for the hydrophobic case. Further study of the slender axisymmetric bodies using PIV would be beneficial to understand the exact causes of the velocity differences.

#### 2. Symmetric surface condition, oblique 5° impact angle

The initial impact angle significantly affects the rotation and lateral translation of the projectile. The plots in Fig. 12 present the mean trajectories and velocities for 10 repeated cases of hydrophilic and hydrophobic Delrin cones, ogive, and flat noses. Fig. 12(a) shows that both the hydrophilic and hydrophobic cone cases experiences 8.72 and 8.55 diameters of displacement, respectively. The hydrophilic ogive case presented in Fig. 12(b) shows 7.25 diameters of displacement and the hydrophobic case shows a displacement of 4.92 diameters. The flat nose in Fig. 12(c) experiences 6.92 diameters for the hydrophilic case and 3.66 diameters of displacement for the hydrophobic case. The cone also has much more variation in the final lateral position than either the flat or the ogive noses, which is partially a result of the increased IMU error over longer distances traveled. The cone nose shape is responsible for the greater lateral displacement. Comparing the trajectories from Fig. 9 with Fig. 12 reveals the large effect both a slight angle and different nose shape have on the movement of slender projectiles during water entry.

The peak and final normalized velocities for the oblique 5° case are shown in Table IV. The final values are taken at *z*/*D* = 34. It is interesting to note that the final *z*-direction velocity of the cone was less than that of the flat. This was due to rotation of the projectile which increased vertical drag and the horizontal side force of the cone more than the flat nose (see Secs. III B 3 and III B 4). Comparing the velocities of the oblique entry to the vertical entry shows a significant (>0.1) decrease in final velocity for all cases, but only a slight (≈0.01) decrease in peak velocity.

Nose Shape . | Case . | Final x/D
. | Peak w/U_{o}
. | Final w/U_{o}
. |
---|---|---|---|---|

Cone | Hydrophilic 5° | 8.72 | 1.09 | 0.66 |

Hydrophobic 5° | 8.55 | 1.09 | 0.66 | |

Ogive | Hydrophilic 5° | 7.25 | 1.09 | 0.76 |

Hydrophobic 5° | 4.92 | 1.09 | 0.85 | |

Flat | Hydrophilic 5° | 6.92 | 1.08 | 0.74 |

Hydrophobic 5° | 3.66 | 1.04 | 0.77 |

Nose Shape . | Case . | Final x/D
. | Peak w/U_{o}
. | Final w/U_{o}
. |
---|---|---|---|---|

Cone | Hydrophilic 5° | 8.72 | 1.09 | 0.66 |

Hydrophobic 5° | 8.55 | 1.09 | 0.66 | |

Ogive | Hydrophilic 5° | 7.25 | 1.09 | 0.76 |

Hydrophobic 5° | 4.92 | 1.09 | 0.85 | |

Flat | Hydrophilic 5° | 6.92 | 1.08 | 0.74 |

Hydrophobic 5° | 3.66 | 1.04 | 0.77 |

#### 3. Asymmetric surface condition (half-and-half), 0° impact angle

The initial impact of the projectiles with a hydrophobic coating on one half produces an asymmetric splash curtain (see Fig. 13). The momentum imbalance resulting from the asymmetry causes the nose of the projectile to move toward the hydrophilic side of the projectile. Once the projectile is no longer aligned vertically, a moment couple between gravity, acting through the center of mass, and the hydrodynamic forces acting at the center of pressure, applied toward the nose of the projectile, causes the projectile to rotate in a counter-clockwise fashion. The projectile experiences both continued rotation and a lateral translation toward the hydrophilic side. Fig. 14 presents the trajectories and velocities for 10 averaged half-and-half tests performed for Delrin cone, ogive, and flat nose shapes to compare how the water entry of different nose shapes is affected by half a cavity. Comparing Fig. 14(c) to Figs. 14(a) and 14(b) shows that the projectiles with the flat nose shape undergo a larger lateral translation than the projectiles with the cone or ogive nose shapes due to the larger momentum transfer on the hydrophobic side as is shown in Fig. 13 by a larger splash curtain. The protruding region *A* in Fig. 14(c) demonstrates the displacement of the IMU that is located in the tail of the projectile instead of the nose. As the nose is moved to the right (positive *x*) and the projectile rotates counter-clockwise, the tail moves to the left (negative *x*). As the projectile continues to descend, the position of the IMU crosses the centerline into the positive *x* region. Since the magnitude of the lateral displacement is plotted, all of the values appear positive. A similar protruding region occurred for the oblique angle cases, but was much smaller and not noticeable on the plots.

The peak and final normalized *w* velocities are given in Table V. The final values are taken at *z*/*D* = 34. The final normalized *w* velocities of the cone and ogive are higher for the half-and-half case than for the oblique 5° case from Sec. III A 2 because the projectiles were initially aligned vertically and did not experience as much rotation as they did in the oblique 5° case. The final normalized velocities for the cone half-and-half case show no change compared to the hydrophilic 0° case and a decrease of 0.01 compared to the hydrophobic 0° case. The final normalized velocities for the flat nose do not change for the hydrophobic and half-and-half cases, but the final normalized vertical velocity deceases from 0.91 to 0.87 when comparing the hydrophilic and half-and-half cases. The ogive nose shape results in a slight decrease of 0.01 from hydrophobic to half-and-half, and a decrease from 0.99 to 0.97 going from hydrophilic to half-and-half. The decrease in the normalized vertical velocity between the 0° entry cases and the half-and-half case is due to the rotation of the projectile in the half-and-half case which results in more vertical drag. Additionally, the conversion of momentum from the *z* to *x* direction contributes to the decrease in vertical velocity. The peak normalized *w* velocities for the cone and ogive half-and-half case are the same as the hydrophobic 0° case, and the flat half-and-half case *w* velocity increases from 1.06 to 1.05 when compared to the hydrophobic 0° case.

Nose shape . | Case . | Final x
. | Peak w
. | Final w
. |
---|---|---|---|---|

Cone | Half-and-half | 0.51 | 1.11 | 0.97 |

Ogive | Half-and-half | 0.76 | 1.10 | 0.97 |

Flat | Half-and-half | 1.21 | 1.06 | 0.87 |

Nose shape . | Case . | Final x
. | Peak w
. | Final w
. |
---|---|---|---|---|

Cone | Half-and-half | 0.51 | 1.11 | 0.97 |

Ogive | Half-and-half | 0.76 | 1.10 | 0.97 |

Flat | Half-and-half | 1.21 | 1.06 | 0.87 |

#### 4. Center of mass effects

A series of tests were performed to validate the theory of a moment couple between the center of gravity and the center of pressure being responsible for the rotation and lateral displacement discussed in Secs. III A 1–III A 3. The location of the center of mass was varied and the effect was observed both for an angled hydrophobic case as well as for a half-and-half case for a modified Delrin ogive dropped from 50 cm (*U*_{o} = 3.13 m/s). Fig. 15 qualitatively demonstrates the effect of moving the center of mass a distance *d* (see Fig. 3) from *close* to the nose (*d* = 8.63 cm), to approximately the *middle* of the projectile (*d* = 12.63 cm), and to approximately *halfway* between the nose and the middle (*d* = 10.63 cm) of the projectile. The center of mass for the *middle* case was similar to the center of mass for the original Delrin ogive of 12.5 cm. A similar location was chosen due to the change in relative density with the inclusion of the lead slug. It is clear from the images in the final column in Fig. 15 that the amount of rotation increases as the location of the center of mass is moved farther from the nose and the reason for this will be discussed in Sec. III B 2.

Figs. 16 and 17 present the trajectories and velocities for three locations of the center of mass tests. The effect of moving the center of mass along the axis of the projectile is clear from the differences in lateral displacement for the hydrophobic case (*U*_{o} = 3.13 m/s) shown in Fig. 16 and the half-and-half case (*U*_{o} = 3.13 m/s) shown in Fig. 17. Fig. 16(c) presents the case with the center of mass closest to the nose which shows the least lateral displacement at 2.35 diameters. A lateral displacement of 3.77 diameters is observed when the center of mass is farther from the nose in Fig. 16(a). The velocities for all three cases are shown in Table VI. All three cases reach the same normalized peak velocity at 1.13. The *middle* case has the largest decrease in vertical velocity, ending at 0.97 due to the greater rotation and lateral displacement of this case. The velocity profiles for these center of mass cases have a different form than the velocity plots seen in Sec. III A 1–III A 3 due to the additional mass from the lead slug (Table I). The lead slug is extremely close to the nose for the tests presented in Fig. 16(f) and between the minimal relative rotation and the additional mass there is very little deceleration after the projectile reaches peak velocity. Similar effects are seen in Figs. 16(d) and 16(e) with the final velocity remaining above impact velocity.

Case . | Final x/D
. | d (cm)
. | Peak w/U_{0}
. | Final w/U_{0}
. |
---|---|---|---|---|

Hydrophobic 5° | 3.77 | 12.63 | 1.13 | 0.97 |

Hydrophobic 5° | 3.17 | 10.63 | 1.13 | 1.08 |

Hydrophobic 5° | 2.35 | 8.63 | 1.13 | 1.13 |

Half-and-half | 1.65 | 12.63 | 1.14 | 1.08 |

Half-and-half | 1.17 | 10.63 | 1.13 | 1.13 |

Half-and-half | 0.65 | 8.63 | 1.15 | 1.15 |

Case . | Final x/D
. | d (cm)
. | Peak w/U_{0}
. | Final w/U_{0}
. |
---|---|---|---|---|

Hydrophobic 5° | 3.77 | 12.63 | 1.13 | 0.97 |

Hydrophobic 5° | 3.17 | 10.63 | 1.13 | 1.08 |

Hydrophobic 5° | 2.35 | 8.63 | 1.13 | 1.13 |

Half-and-half | 1.65 | 12.63 | 1.14 | 1.08 |

Half-and-half | 1.17 | 10.63 | 1.13 | 1.13 |

Half-and-half | 0.65 | 8.63 | 1.15 | 1.15 |

Changing the location of the center of mass also affects the half-and-half case as shown in Fig. 17. The regular half-and-half ogive case from Sec. III A 3 had a peak lateral displacement of 0.7 diameters. Moving the center of mass to the *half* and *middle* locations increases the amount of lateral displacement to 1.19 and 1.69 diameters, respectively. The 8.63-cm case results in a slightly decreased displacement of 0.65 diameters.

Finally, the maximum difference between the standard nose shapes (no variable CG) is 1.4 cm between the flat and cone nose shapes and the others. This will obviously affect the rotation somewhat. However, the nose geometry will have a greater effect on the rotation than the CG differences. This can be seen when comparing Figs. 16 and 12. A change of 4 cm CG causes a 1.5 diameter difference in trajectory (Figs. 16(a)–16(c))), whereas the nose shape change alters the trajectory by 4 diameters (Figs. 12(a)–12(c)).

#### 5. Rotation

Fig. 18 compares the rotation angle and rotation rate of half-and-half cases to 5° angled cases versus projectile depth. Figs. 18(a)–18(c) provide additional evidence that the nose shape has a significant effect on the rotation of the projectile. The cone nose shape in Fig. 18(a) achieves a maximum rotation angle of 42.8° for the hydrophobic 5° angled case, but only 4.7° for the half-and-half case, a difference of 38.1°. It is interesting that for the ogive nose shape, the disparity between the rotation angle achieved by the time the projectile reaches the bottom of the tank is 14.3° (24.4° for the hydrophobic 5° case and 10.1° for the half-and-half case). Likewise, the flat nose hydrophobic 5° case and flat nose half-and-half case shows an achieved rotation angle difference of 0.4° (14.3° for the hydrophobic 5° case and 13.9° for the half-and-half case).

When comparing angled entry versus coating variation the final measured rotation amount decreases as the nose shape flattens (i.e., cone > ogive > flat) for the hydrophobic 5° cases. For the half-and-half case, the final measured amount increases as the nose shape flattens (i.e., flat > ogive > cone). As discussed in Sec. III A 3, the rotation for the half-and-half case is caused by an initial momentum imbalance due to the asymmetry of the splash curtain, which forces the nose of the projectile to one side. The splash curtain of the flat nose shape was qualitatively shown in Fig. 13 to have a larger asymmetry, thus we would expect a larger rotation for the flat nose than the cone or ogive noses. For the hydrophobic 5° angled case, the nose shape affects the amount of rotation that occurs immediately following impact with the surface of the water.

Figs. 18(d)–18(f) show the rotation rate of the projectiles for both cases versus depth. Most notable is the rapid increase in rotation rate until the first pinch off point (represented by the vertical line in regions A). The rate of increase is directly related to the shape of the nose, with the more angled cone nose inducing the largest rotation rate (Fig. 18(d) 20.8°/s at A) and the flat nose inducing the smaller rotation rate (Fig. 18(f) 6.5°/s at A).

#### 6. Non-dimensional pinch-off time

The occurrence of two pinch-off events was mentioned in Sec. I and a discussion of the non-dimensional time to these pinch-off events follows. Fig. 19 shows the approximate pinch-off location that occurs on the sides of the projectile and pinch-off of the trailing cavity. Table VII presents the non-dimensional pinch-off times (

*U*

_{o}= 3.13 m/s) used in the present study as well as results for spheres and disks presented in previous studies. The values for the cone and ogive nose shapes for pinch-off on the sides of the projectile are of the same order as the findings for spheres by Truscott and Techet

^{31}and Gilbarg and Anderson.

^{6}The lower values are a result of pinch-off being measured at the time when the cavity touches the sides of the projectile, instead of touching the opposite cavity wall, as in the two sphere studies. This occurs across the 2.54-cm diameter and results in a lower τ

_{A}. Comparing the value for the flat nose with the disk results in a similar decreased value. The τ

_{B}values are in a similar range for all three nose shapes. More variation was observed in the actual pinch-off time due to the more chaotic nature of the trailing cavity pinch-off after the first pinch-off on the sides of the projectile.

Nose shape . | τ_{A}
. | τ_{B}
. |
---|---|---|

Cone | 1.530 ± 0.155 | 3.371 ± 0.460 |

Ogive | 1.437 ± 0.048 | 3.014 ± 0.266 |

Flat | 1.814 ± 0.042 | 3.133 ± 0.229 |

Sphere (Truscott and Techet^{31}) | 1.726 ± 0.0688 | ... |

Sphere (Gilbarg and Anderson^{6}) | 1.74 | ... |

Disk (Glasheen and McMahon^{32}) | 2.285 ± 0.0653 | ... |

### B. Water entry forces

This section discusses the forces that the projectile experiences during the water entry event. The impact forces will be discussed first, followed by a discussion of the drag coefficients and lift coefficients.

#### 1. Impact forces

Understanding how nose shape and wetting angle affect the impact force will enable engineers to design the nose geometry to mitigate potential destructive impact forces. Table VIII presents the peak impulsive force experienced by each nose shape for four impact cases. Each force measurement represents the magnitude of all three accelerometer axes calculated by

where *m*_{p} is the mass of the projectile and *a*_{i} acceleration of the center of gravity of the projectile in the direction of the accelerometer axis see Fig. 2.

Nose shape . | Hydrophilic 0° (N) . | Hydrophobic 0° (N) . | Hydrophobic 5° (N) . | Half-and-half (N) . |
---|---|---|---|---|

Cone | 0.45±0.30 | 0.54±0.31 | 0.58±0.29 | 0.62±0.30 |

Ogive | 1.56±0.41 | 1.61±0.43 | 1.54±0.36 | 1.68±0.39 |

Flat | 30.63±3.09 | 29.79±4.32 | 16.75±7.40 | 33.19±3.84 |

Nose shape . | Hydrophilic 0° (N) . | Hydrophobic 0° (N) . | Hydrophobic 5° (N) . | Half-and-half (N) . |
---|---|---|---|---|

Cone | 0.45±0.30 | 0.54±0.31 | 0.58±0.29 | 0.62±0.30 |

Ogive | 1.56±0.41 | 1.61±0.43 | 1.54±0.36 | 1.68±0.39 |

Flat | 30.63±3.09 | 29.79±4.32 | 16.75±7.40 | 33.19±3.84 |

The average maximum force (10 cases) experienced by the projectiles at impact is presented in Table VIII. The projectiles with the flat nose shape experienced approximately 20 times more impact force than projectiles with either the cone or the ogive nose shapes due to the blunt flat geometry! The flat nose impact force values for the hydrophilic, hydrophobic, and half-and-half cases include axial forces that exceed the measurable limit of the accelerometer (±16 g), so the measured values are the maximum measurable value. An accelerometer capable of measuring accelerations greater than ±16 g was not available in a size for use in the embedded design used in this study. The impact force experienced by the flat nose shape can be nearly halved by angling the projectile, as shown in the Hydrophobic 5° column of Table VIII. Changing the impact angle for the flat from 0° to 5° eliminates the impact directly on the blunt portion of the nose and decreases the overall force experienced by the flat projectile. Changing the angle for the cone and ogive does not result in a significant change in the impact force, as shown in Table VIII. These findings demonstrate that varying the wetting angle has a slight effect on the total mean impact force, but angling the flat nose shape will significantly decrease the mean impulsive force of impact.

A comparison of these impact values to the data from Moghisi and Squire^{16} for spheres show impact forces of the same order of magnitude. The ogive nose shape (most similar to a sphere) impact values of 1.56±0.41 N (hydrophilic) and 1.61± 0.43 N (hydrophobic) with *U*_{o} = 3.13 m/s are similar to the extrapolated impact force value of 1 N with *U*_{o} = 3 m/s for a 2.5 cm steel sphere from Moghisi and Squire.^{16}

#### 2. Center of pressure

A projectile entering the water at an oblique angle will experience a moment couple from gravity acting through the center of mass and from the hydrodynamics forces acting at the center of pressure acting toward the nose of the projectile. If the center of gravity is farther from the nose than the center of pressure, the projectile will be rotationally unstable. The concept of rotational instability of a floating or submerged body is well known and will be discussed briefly inasmuch as it relates to the rotation of the projectiles herein.

The center of pressure is the point at which the resultant force from the fluid acts on a submerged portion of a body. The resultant force acts normal to the body at the center of pressure and is the product of the surface area of the body and pressure at the centroid of the body.^{25} In this study, the resultant force is the hydrodynamic force *F*_{T}. Furthermore, *F*_{T} can be resolved into drag and lift forces, *F*_{D} and *F*_{L}, and their effects will be discussed in Secs. III B 3 and III B 4.

Fig. 20 illustrates the placement of key forces relevant to the lateral displacement of slender projectiles in the water. The point *T* represents the total center of pressure of the body at an instant in time. Point *B* is the center of pressure of just the body and point *N* is the center of pressure for the nose section of the projectile. The center of pressure *N*, at the nose, plays an important role in the rotation and lateral displacement of the projectiles and changes based on nose geometry. The angle of the resultant force *F*_{N} on the long, angled surface of the cone produces a larger force component in the +*z* direction, which increases the magnitude of the moment couple and results in more rotation and lateral displacement. Likewise, the ogive would be expected to have more lateral displacement than the flat and comparing Figs. 12(b) and 12(c) shows that this is indeed true.

An approximate location of the center of pressure at discrete depths was derived from a theoretical torque model using measured force and angle values from the IMU (Fig. 21(a) is a diagram of the variables used in the analysis). The locations of the IMU sensor (IMU), projectile center of gravity (CG), and center of pressure (CP) along the projectile longitudinal axis were used as reference points. The instant center of rotation (IC) of the projectile was calculated by finding the intersection of perpendicular lines (r_{IMU}, r_{CG}, r_{CP}) passing through corresponding velocity vectors (v_{IMU}, v_{CG}, v_{CP}), with the assumption that all rotation is in a single plane. A moment balance about IC was calculated using the well known equation for torque

where *I* is the moment of inertia of the projectile, α is the angular acceleration of the projectile, and τ_{IC} is the net torque on the projectile. Equation (3) presents the net torque τ_{IC} separated into the component of the projectile weight acting in the direction of the velocity vector at the center of gravity, and the component of the hydrodynamic force acting in the direction of the velocity vector at the center of pressure, each multiplied by their respective distances to the instant center of rotation.

where *r*_{CG} and *r*_{CP} are the distances from the center of gravity and center of pressure to the instant center of rotation, respectively, θ_{VCG} and θ_{VCP} are the angles of the velocity vectors, *W* is the weight of the projectile, and *F* is the hydrodynamic force acting normal to the projectile at the center of pressure.

Angle θ_{F} was easily obtained by inspection, for convenience angle β was found in terms of *r*_{CP} using the law of sines with the known distance *r*_{CG} and known angle γ (from IMU data), and angle θ_{VCP} was found in terms of β as follows:

_{P}is the angle of the projectile longitudinal axis. Substituting Eqs. (4a)–(4c) into Eq. (3), solving for

*r*

_{CP}, and simplifying yields

Finally, the distance *b* (Eq. (6)) between the center of pressure and the center of gravity was calculated using the distance formula between the known coordinates of the instant center and the coordinates of the center of pressure, where *x*_{CP} = *b*sin θ_{P}, *y*_{CP} = *b*cos θ_{P}, and the *r*_{CP} is the result from Eq. (5) and is simplified as

Figs. 21(b) and 21(c) present the results for the distance from the center of gravity to the center of pressure (*b*/*D*) for all three nose shapes of the hydrophilic Delrin case. Fig. 21(b) shows many spikes in *b*/*D* that are unrealistically large and obviously not located on the projectile. In regions up to approximately *z*/*D* = 17 the projectile is subjected to disturbances from impact and the cavity collapsing which causes the angular velocity (and subsequent derived angular acceleration) to change values drastically. This observation reveals that the theoretical model for determining the location of the center of pressure is very sensitive to changes in the measured data and would require an estimate for the cavity forces at shallow depths. However, Fig. 21(c) shows the region of the plotted data from approximately *z*/*D* = 17 until the end of the run at *z*/*D* = 34 and reasonable values for the distance b/D. It is important to recognize that in this region (post pinch-off) the projectile has already undergone the majority of its rotation and therefore one would expect the location of the center of pressure to change very little.

#### 3. Instantaneous drag coefficients

Instantaneous drag coefficients for each projectile case were calculated by

*F*

_{D}is the drag force,

*m*

_{p}is the mass of the projectile,

*g*is the acceleration of gravity,

*a*

_{D}is the component of acceleration of the center of gravity in the direction parallel to the instantaneous velocity of the center of gravity,

*V*is the volume of the projectile, ρ

_{w}is the density of the water,

*U*is the velocity of the projectile, and

*A*is the frontal area of the projectile (5.07 × 10

^{−4}m

^{2}, for 2.54-cm diameter). The

*m*

_{p}

*a*term includes added mass and other unsteady forces caused by rotation through time that are measured by the IMU. Fig. 22(a) presents the instantaneous drag coefficient of a hydrophobic flat nose projectile versus time overlaid with the same data filtered with a 50 Hz low-pass Butterworth filter. Similar filtered data were averaged over 10 data sets for each of the cases in Figs. 22(b) and 22(c). The drag coefficient increases at a constant rate until

*t*= 0.0947 s, at which time the cavity pinch-off occurs and a ripple phenomenon is observed on the sides of the bubble cavity (see Sec. III C 1).

Fig. 22(b) shows that the drag coefficients for the ogive and flat nose shapes are larger for the hydrophobic case, but the drag coefficients for the cone shape for both hydrophobic and hydrophilic cases are almost identical. The increased drag for the hydrophobic cases indicates that the formation of the cavity subjects the projectile to higher forces than the hydrophilic case. In Fig. 22, the drag coefficients do not show a spike at impact due to the Butterworth low-pass filter used to smooth the acceleration response. For the cone, a cavity was seen to open even for the hydrophilic case in Fig. 11 and this is the cause for the nearly identical drag coefficients. Fig. 22(c) shows drag coefficient results for the half-and-half case for the cone, ogive, and flat nose shapes that are very similar to the hydrophilic drag coefficients in Fig. 22(b). The similarity is due to the half-and-half cases undergoing only a little more rotation than the hydrophilic and hydrophobic 0° case. Fig. 22(d) shows a decrease in drag coefficient because more of the resultant force is contributing to lift, as will be shown in Sec. III B 4. For all cases, the flat nose produced the largest drag coefficient due to the blunt nose geometry and the cone and ogive are similar in the amount of drag produced by the nose geometry.

Terminal velocities for each case were computed to determine the final theoretical velocity of the projectile in water using

where *F*_{b} is the buoyant force acting on the projectile, *C*_{D} is the drag coefficient, and *A* is the frontal area of the projectile. The final drag coefficient value for each nose and surface coating are presented in Table IX and were used to calculate the terminal velocities in Eq. (8). The terminal velocity values were shown in Figs. 9(d)–9(i), 12(d)–12(i), and 14(d)–14(i) are reproduced in Table X for convenience and represent a fraction of the impact velocity. The effect of projectile rotation was included in *C*_{D} to facilitate comparison between cases.

Nose . | Hydrophilic 0° . | Hydrophobic 0° . | Half-and-half . | Hydrophilic 5° . | Hydrophobic 5° . |
---|---|---|---|---|---|

Cone | 0.253 | 0.251 | 0.253 | 0.431 | 0.541 |

Ogive | 0.288 | 0.310 | 0.317 | 0.371 | 0.70 |

Flat | 0.320 | 0.385 | 0.417 | 0.380 | 0.386 |

Nose . | Hydrophilic 0° . | Hydrophobic 0° . | Half-and-half . | Hydrophilic 5° . | Hydrophobic 5° . |
---|---|---|---|---|---|

Cone | 0.253 | 0.251 | 0.253 | 0.431 | 0.541 |

Ogive | 0.288 | 0.310 | 0.317 | 0.371 | 0.70 |

Flat | 0.320 | 0.385 | 0.417 | 0.380 | 0.386 |

Nose . | Hydrophilic 0° . | Hydrophobic 0° . | Half-and-half . | Hydrophilic 5° . | Hydrophobic 5° . |
---|---|---|---|---|---|

Cone | 0.692 | 0.695 | 0.692 | 0.530 | 0.473 |

Ogive | 0.803 | 0.774 | 0.765 | 0.707 | 0.707 |

Flat | 0.692 | 0.631 | 0.607 | 0.635 | 0.630 |

Nose . | Hydrophilic 0° . | Hydrophobic 0° . | Half-and-half . | Hydrophilic 5° . | Hydrophobic 5° . |
---|---|---|---|---|---|

Cone | 0.692 | 0.695 | 0.692 | 0.530 | 0.473 |

Ogive | 0.803 | 0.774 | 0.765 | 0.707 | 0.707 |

Flat | 0.692 | 0.631 | 0.607 | 0.635 | 0.630 |

#### 4. Instantaneous lift coefficients

Lift is the component of the force that acts perpendicular to the direction of the fluid flow^{25} and the lift coefficient is calculated by

*F*

_{L}is the lift component of the force,

*m*

_{p}is the mass of the projectile, and

*a*

_{L}is the component of acceleration of the center of gravity in the direction perpendicular to the instantaneous velocity of the center of gravity. The lift force causes the projectile to translate in the

*x*direction and also contributes to the rotation seen in Figs. 7(c)–7(e).

Fig. 23 presents the calculated instantaneous lift coefficients for the hydrophilic and hydrophobic 0° and 5° cases as well as the half-and-half case. Fig. 23(a) shows that there is no significant lift force when the projectile descends straight down, as expected. Fig. 23(b) shows a maximum lift coefficient of 0.21 for the flat nose shape in the half-and half case, with 0.2 and 0.05 for the ogive and cone, respectively. This finding corroborates the trajectory plots from Fig. 14, showing that the flat and ogive have more lateral displacement than the cone. The lift coefficients for the hydrophilic and hydrophobic 5° cases are shown in Fig. 23(c). The lift coefficients for both the hydrophilic and hydrophobic cone approach 1 and the lift coefficients for the hydrophilic cases for the ogive and flat are significantly larger than their respective hydrophobic lift coefficients. Comparing these values to trajectory plots in Fig. 9 shows that the hydrophilic and hydrophobic cones have essentially the same lateral displacement. The trajectories for the hydrophilic and hydrophobic cases for the ogive and flat in Figs. 9(b) and 9(c) achieve a displacement of approximately 6 diameters. Figs. 23(b) and 23(c) show a dip around *z*/*D* = 8 which is due to the squared radial force components in Eq. (9b), similar to the discussion presented in Sec. III A 3.

The final lift coefficient values for each nose and case are presented in Table XI. The hydrophilic and hydrophobic 0° cases for all noses show almost no lift, and the half-and-half cases show only a small amount. However, the 5° case for all noses shows a significant increase in the lift. These lift coefficient values are similar to what would be expected from various National Advisory Committee for Aeronautics (NACA) airfoils^{26} with an angle of attack between approximately 6° and 10°, which supports the lateral displacement observed in each case.

Nose . | Hydrophilic 0° . | Hydrophobic 0° . | Half-and-half . | Hydrophilic 5° . | Hydrophobic 5° . |
---|---|---|---|---|---|

Cone | 0.029 | 0.021 | 0.131 | 0.566 | 0.398 |

Ogive | 0.022 | 0.036 | 0.175 | 0.624 | 0.419 |

Flat | 0.030 | 0.055 | 0.188 | 0.601 | 0.414 |

Nose . | Hydrophilic 0° . | Hydrophobic 0° . | Half-and-half . | Hydrophilic 5° . | Hydrophobic 5° . |
---|---|---|---|---|---|

Cone | 0.029 | 0.021 | 0.131 | 0.566 | 0.398 |

Ogive | 0.022 | 0.036 | 0.175 | 0.624 | 0.419 |

Flat | 0.030 | 0.055 | 0.188 | 0.601 | 0.414 |

Finally, we present a quick comparison between the theoretical values one would predict given C_{D} and C_{L} for a fully immersed projectile in Fig. 24 using Eqs. (7b) and (9b). Here, we put a projectile under the surface and then released it after letting the water came to rest. Fig. 24(a) shows the actual trajectory of the projectile versus the theoretical assumption that there was no deviation from the vertical. Figs. 24(b)–24(d) show how the velocity, acceleration, and drag force differ from the theory, respectively. Overall, the match is what one would expect with such a rudimentary comparison, and confirms that a model would need to account for the unsteady effects^{30} we have ignored in this simplified model.

### C. Acoustics

May^{14,27} showed that the nose shape can affect the formation of the splash curtain and cavity and Uber and Fegan^{10} showed some acoustic differences between various shapes. This section describes the acoustic differences between three Delrin nose shapes. Using a hydrophone positioned 26 cm from the area of impact, the sounds produced by the three different nose shapes were recorded and processed and are represented in Fig. 25 as a spectrogram showing the intensity of frequencies up to 48 kHz through time. Spectrograms are well suited for observing sudden, dramatic changes in the signal, such as the flat impact or the pinch-off event for all three nose shapes. Line *A* in Fig. 25 indicates the moment of impact. Line *B* represents the moment of pinch-off. The transition of the projectile from free-fall to impact with the water is obvious for the flat nose shape (Fig. 25(c)). However, the transition for the cone (Fig. 25(a)) and ogive (Fig. 25(b)) is very subtle and synchronization is necessary to determine the instant of impact.

#### 1. Cavity ripples

Fig. 26 depicts an interesting phenomenon that is seen after pinch-off for all cases. Ripples form on the sides of the bubble of entrained air behind the projectile, a phenomenon that has been previously studied by Grumstrup *et al.*^{12} and Mansoor *et al.*^{13} From the images, the time between the formation of successive ripples is found to be 5 ms, which corresponds to a frequency of 200 Hz. Acceleration data from the IMU show that at the corresponding time there are oscillations along the axial direction. The frequency of these oscillations is approximately 210 Hz, as shown in Fig. 27 and can also be seen in the spectrograms of Fig. 25 (after pinch-off B). Using this frequency, a new ripple should form every 4.76 ms, which matches well with the findings from the images of 5 ms. The 0.24-ms difference is due to the 0.5-ms time step between images. The synchronized image-acceleration data (Figs. 26(a) and 26(b)) show that the ripples begin at pinch-off and decay through the remainder of the descent of the projectile. Continued oscillations are seen from the acceleration data, but obvious ripples are absent from the images after t = 0.155 s. However, the IMU data confirm that the sides of the trailing cavity continue to oscillate with a decayed rippling pattern until the projectile impacts with the bottom of the tank (*t* = 0.266 s on Fig. 26(b), 35/D), a finding that cannot be observed from image data alone.

Fig. 27 presents the frequencies of the 10 runs of each nose shape for hydrophilic cases from Sec. III A 1 to the normalized amplitudes of each run. The mean frequencies for the cone, ogive, and flat nose shapes are 208.5 Hz, 210.9 Hz, and 212.7 Hz, respectively. Grumstrup *et al.*^{12} showed oscillation frequencies of 190 Hz for spheres, and both the acoustic frequency and acceleration frequency are in the same range even for slender axisymmetric projectiles.

## IV. CONCLUSION

This paper presented trajectory, force, and acoustic findings for slender axisymmetric projectiles with three nose shapes and three surface coating conditions, entering the water vertically and at oblique angles. An embedded inertial measurement unit (IMU) was used to measure accelerations from which the forces, velocities, and positions were calculated and compared.

The acceleration data from the IMU were integrated to obtain velocities and positions for the entire water entry event. The calculated positions showed a larger lateral displacement for the oblique entry cases than for the half-and-half coated ones. The hydrophobic coating caused a cavity to form at impact and resulted in increased drag and decreased velocity compared to the hydrophilic case. The effect of adjusting the center of mass of the test projectile was shown, resulting in greater lateral displacement as the center of mass was moved farther from the nose of the projectile, as predicted. Finally, non-dimensional pinch-off times were calculated and compared to a previous study with good agreement.

Impact forces were calculated for the cone, ogive, and flat nose shapes for four cases. The flat showed the largest impact force in the vertical entry cases, which was lessened during oblique water entry cases. The wetting angle does not significantly affect the impact force, but the flat nose shape experiences 20 times more impact force than the ogive or cone nose shape. Drag and lift coefficients were computed for all four cases and showed that the lift force was responsible for the lateral displacement seen in the trajectory plots.

A spectrogram was calculated from synchronized acoustic data that showed that the majority of the sound produced by the cone and ogive nose shapes was a result of pinch-off and not from the actual impact with the free surface, except for the flat nose for both hydrophilic and hydrophobic cases which produced an intense sound at impact and less sound at pinch-off. In addition, the impact sound for the flat nose can be reduced by entering the water at an oblique angle. Frequencies calculated from the acoustic and acceleration data agreed well with ripple frequency findings previously studied by other researchers.^{12}

Overall, the data show that even small angles of entry can cause extreme trajectory changes greater than those caused by surface treatments alone. In fact, it is apparent that the trajectory is affected from most by angle, less by nose shape, and least by the center of gravity. Further, cones tend to be more sensitive to angular impact and the largest forces of impact are experienced by flat nosed projectiles. The IMU proved to be a valuable instrument and performed well when compared to data from known positions and image processing techniques. The greatest utility of the IMU is perhaps the accuracy, relatively small size for use in conditions where a camera is not a viable option, and the ability to calculate the center of pressure.

## ACKNOWLEDGMENTS

We gratefully acknowledge the support of Maria Medeiros at the Office of Naval Research University Laboratory Initiative Grant No. N000141110872 for funding this research. We thank J. Ellis, M. Hansen, D. Richardson, B. Doolin, and Z. Smith for their help in performing the experiments.

### APPENDIX: GYROSCOPE CALCULATIONS

Each measurement of the acceleration data was converted to SI units by multiplying by 0.0383 m s^{−2} bit^{−1}. The conversion factor is based on the maximum measurable acceleration value of 16*g*, divided by 4096 bits (the digital value corresponding to this maximum acceleration value), and then converting the 16*g* to m/s^{2}. A rotation matrix (described below) was then applied to the three-axis acceleration values at each instant in time in order to perform the integration with respect to known fixed directions. The gravity vector was applied in the vertical direction and the result was integrated numerically. The acceleration data were integrated numerically using the trapezoidal method in MATLAB to compute velocity and then integrated again to compute the position of the projectile.

A three-axis gyroscope was used to relate the orientation of the accelerometer axes at each instant to the initial orientation in order to calculate the components of acceleration in the vertical and horizontal directions. The instantaneous angular velocities measured by each axis of the gyroscope were represented by ω_{1}, ω_{2}, and ω_{3}. Fig. 28 shows the Euler angle transformation performed using a body-fixed 3-2-1 rotation sequence.^{33} The Euler angle rates can be separated into individual components for convenience as follows (where *s* and *c* represent the sine and cosine functions, respectively):

in the correct 3-2-1 sequence yields a total transformation matrix to express the current orientation in terms of the previous orientation,

The Euler angle rates in Eq. (A1) were integrated at each time step and the computed Euler angles ϕ, θ, and ψ were substituted into the combined three-stage rotation matrix in Eq. (A3) to obtain the rotation for the current time step. The current orientation can be expressed in terms of the initial orientation by post-multiplying successive rotation matrices.

An additional step was necessary to correctly orient the coordinate frames for the oblique cases relative to the gravity vector. Using another 3-2-1 body-fixed rotation sequence, the **a**_{3} axis of the coordinate frame was transformed from the vertical direction to the initial angle of 2° or 5°:

_{1}, α

_{2}, and α

_{3}were equated to their respective elements from the third column of Eq. (A3) to compute the rotation about each axis from the 3-2-1 transform of the oblique cases. Solving for ϕ, θ, and ψ yields the angles

obtained by translating the three-axis accelerations along the longitudinal axis of the projectile ensured that the accelerations were expressed at the center of gravity (where *l* is the distance from the IMU to the center of gravity, ω_{i} is the angular velocity about axis *i*, and **b**_{i} is the direction axis *i*). The inverse of the transformation matrix was applied to the rotation matrix at each time step in order to correctly apply the gravity vector before integrating and calculating the velocity and trajectory of the projectile.

The validity of the IMU measurements and the code used in the analysis procedure were also obtained. A pendulum validation mechanism was created because of the similarity between the circular arc and the curving that was expected that the projectile would undergo in the actual experiments. The IMU was embedded in a projectile and the two were secured to the pendulum arm such that the actual accelerometer chip was located along the centerline of the pendulum arm at a distance of 91.4 cm from the pivot point. Fig. 29(a) shows the validation setup in the initial (A) and final (B) test locations as well as an intermediate position. The results of 20 validation tests are compared to the prescribed trajectory of a quarter circle with a radius of 91.4 cm, shown in Fig. 29(b). The total IMU position error was calculated from the mean of 20 tests to be 3.97%, with a standard deviation of 0.88% using

where *p* is the prescribed coordinate, *m* is the measured coordinate, and *r*_{p} is the prescribed radius from the pivot point to the center of mass (92.2 cm). The error value is within the total error calculated from the manufacturer specifications discussed previously (see Sec. II).

In a second test, the trajectory of the projectile measured by the IMU was compared to the trajectory obtained by tracking the projectile using the images from the high-speed camera (28 mm lens). The projectile was attached to an apparatus that made a 47-cm arc and placed completely underwater as shown in Fig. 29(c). The general curve of the IMU and image trajectory plots are similar, but both deviate somewhat in the middle and the IMU does better in the final position. This discrepancy is partially due to index of refraction error in the images especially as the projectile moves toward the top and bottom of the frame, noticeable in the image trajectory plot as a shortening of the *z*/*D* depth. This comparison of the trajectory from the images and from the IMU is evidence of the benefit of using an IMU in water entry position tracking, especially when refraction error is likely be significant.

Fig. 30 presents trajectory, velocity, and acceleration plots for an in-air free-fall release from 150 cm (*U*_{0} = 5.42 m/s). A small amount of drift from the IMU measurements can be seen in Fig. 30(a) with a lateral deviation of 0.05 diameters over 60 diameters. While there is some drift in the IMU system, it is not enough to be of concern for the data presented herein. Additionally, Figs. 30(d) and 30(e) demonstrate a small amount of noise inherent in the raw accelerometer data. As discussed in Sec. III B 3, a Butterworth low-pass filter with a cut-off frequency of 50 Hz was employed to reduce these noise spikes.

The IMU signal was analyzed using a Fast Fourier Transform (FFT) to determine the natural frequency of the system. Fig. 31(a) presents a 750 Hz spike related to a ping test performed in air of a Delrin ogive projectile. Figs. 31(b) and 31(c) show the frequency response of the IMU during the in-air free-fall of Fig. 31(a). Small frequency spikes are shown throughout the frequency domain, but no dominant frequencies are noted that would cause concern for system instability and the 500 Hz signal is not excited because there has not been any external forcing. Underwater, analysis of the IMU signal immediately following impact of the projectile with the bottom of the tank reveals a 500 Hz signal for both radial axes in Fig. 31(d). After significant testing we believe this 500 Hz signal is the natural frequency of the IMU in water and is strongly evident between B, C and through to the end of the run (0.32 s) of Fig. 6 as well as all cases where the IMU is impacted by a cavity or other resonant forcing.