Experimental study on the trajectory of projectile water entry with asymmetric nose shape

During the water entry process, the projectile may have an unstable trajectory due to instabilities such as bounce and sinking. Some of these instabilities are related to the unsteady flow field and cavitation phenomenon.¹ Therefore, ensuring the stability of the water entry trajectory has important significance in the field of naval engineering. When the projectile impacts the free surface, it transfers momentum to the surrounding liquid, which leads to cavity formation. Meanwhile, the development and collapse of the cavity will have a significant impact on the trajectory attitude of the projectile. Therefore, the investigation of the trajectory and that of the cavitation in the water entry problem are often carried out at the same time.

The factors for the change in the trajectory of the projectile after entering the water can be attributed to the hydrodynamic forces exerted on the projectile itself. Von Karman² first established a theoretical model based on the momentum theorem to predict the maximum pressure of the two-dimensional wedge during landing. Armand and Cointe³ obtained a new formula for the impact force by using the method of matched asymptotic expansions. This formula considered the wetting and drag corrections, which differs from the classical Von Karman formula. Wagner⁴ built a basic theory of water impact to predict the loads on an entering body, and it is found that the results are overestimated compared with the measured ones. Based on Wagner’s wedge theory, many researchers have carried out further studies on the small deadrise angles effect,⁵ two-dimensional water impact,⁶ three-dimensional water impact,^7–9 and asymmetric water entry.^10–14 Recently, Wang et al.¹⁵ investigated the unsteady hydrodynamic forces of solid objects vertically entering the water with an air cavity behind the falling body within the framework of potential flow. They decomposed the unsteady hydrodynamic forces into three components, namely the acceleration term, the gravity term, and the velocity term. For different geometries, the physical models proposed by Wang gave a good approximation of hydrodynamic forces. The impact load of high-speed vertical water entry of a cylinder with a flat bottom was investigated numerically by Hong et al.,¹⁶ and the influence of the impact velocity and the aeration effect on the slamming loads were considered. They noted that both the impact loads and the affected area will decrease as the aeration level grows.

The characteristics of the flow field around the projectile will significantly affect its surface pressure distribution and drag coefficient,¹⁷ and the development of cavity will also affect the motion of the projectile.¹⁸ In our previous study,³³ we investigated the cavity dynamics of projectiles with a series of different symmetric nose shapes and obtained detailed hydrodynamics characteristics of the symmetric nose shape during the water entry process. Hou et al.¹⁹ used experimental and numerical methods to investigate the cavity dynamics and vortex evolution for oblique water entry of a cylinder. The results showed that the vortex structures are closed relative to the evolution of the cavity, and the trajectory characteristics are sensitive to the impact angle.

Many researchers investigated water entry of a steel sphere to simplify the theoretical model of the water entry trajectory. Richardson¹ used a high-speed camera to capture the water entry process of different small diameter spheres and measured the drag coefficient, and he found that under certain angle conditions, the sphere will ricochet near the liquid surface. Johnson and Reid²¹ made a more in-depth study of the ricochet phenomenon and given the critical impact angle above which ricochet does not occur. For a spinning sphere, Tan²⁰ performed the experiments of the inclined oil entry of rotating steel spheres to analyze the jet formation and deep seal phenomena. The results indicated that the deep seal time was independent of both linear and angular sphere velocities, while the vertical deep seal displacement increased with the Froude number. Truscott and Techet^22,23 investigated the trajectories, forces, and cavity formation of water entry by a spinning sphere, and results revealed that the sphere’s trajectory exhibits greater curvature as the spin rate of the sphere is increased. By assuming that the drag and added-mass coefficient is constant, Aristoff et al.²⁴ established a theoretical model to describe the vertical water entry trajectory of low-density spheres.

The water entry of slender axisymmetric projectiles is more common in the field of naval engineering. During World War II, it was discovered that trajectory sinking and instability would occur in a large number of underwater weapons, such as torpedoes, after the water entry, which has aroused the attention of many researchers.^25,26 Waugh²⁷ introduced the basic theory of water entry ballistics and systematically studied the behavior of projectiles with nose shapes such as disks, cones, ogives, spheres, cusps, and disk ogives during the cavity-running phase mainly to provide the necessary data for the design of water-entry weapons.

In the field of high-speed water entry, researchers are not only concerned about the impact load of the projectile but also about trajectory stability. Chen et al.²⁸ discussed the effects of the nose shape, impact velocity, and attitude angle on the trajectory stability of high-speed water-entry projectiles. The results show that the flat projectile generated the largest peak pressure, and it had a perfect trajectory stability, while the ogival projectile showed significant instability of trajectory and attitude deflection. Truscott et al.²⁹ investigated projectile dynamics for various bullet geometries with shallow angles and found that bullets with lower length-to-diameter ratios will tumble inside the vapor cavity, while higher length-to-diameter ratios can mitigate the tumbling behavior.

In addition, some studies focus on the projectile motion within a relatively low range of impact velocity. Xia et al.³⁰ investigated the hydrodynamic characteristics of a cylinder with different initial horizontal velocities and inclined angles. The results showed that the average change rate of the lift coefficient increases almost linearly with an increase in the initial angle of attack. Wu et al.³¹ established a dynamical model to study the trajectory optimization problem of the unmanned aerial–aquatic vehicle (UAAV) by simplifying the shape as a wingless configuration, and the hydrodynamic force acting on the vehicle is calculated by the forces induced by the ideal fluid and viscous fluid, respectively. Mirzaei et al.³² developed a transient model to predict the shape of the water-entry cavity for circular cylinders so that it can predict the impacts of the cavity on the cylinder body and apply the related forces and moments to the cylinder during the motion. The proposed model took all of the forces and moments, which act on the cylinder, and the results calculated by the proposed model gave a good accuracy relative to the experimental data.

In summary, most of the research studies on water entry trajectory are focused on the water entry process of the axisymmetric slender body. In other words, there is lack of research on the water entry trajectory of a slender body with an asymmetric nose shape. Therefore, in this paper, the water entry trajectory of the projectile with an asymmetric nose shape and the effects of nose shape, impact velocity, and impact angle on those trajectories are investigated experimentally. High speed photography was used to capture the development of cavity and splash when the projectile with the asymmetric nose shape enters the water and detected the ballistic trajectory, displacement curve, and attitude angle curve of the projectile under each experiment condition. In order to predict the trajectory characteristics of a projectile with an asymmetric head in the initial stage of water entry, we constructed a theoretical model of the water entry trajectory and compared the calculate results with the experimental results; it is found that the theoretical calculation showed good agreement with the experimental results.

In this study, we designed four different nose shape projectiles, and the detail parameter definition of each nose shape is shown in Fig. 1(a) in which R₀ is the radius of the cylinder. The projectile with a flat nose shape is obtained by the reduction ratio of the MK46 torpedo, and its scale ratio is λ = 0.123 457. The key parameter of the asymmetric nose shape is the cutting angle α and the cutting distance δ, which determined the angle and the area of the cutting plane. Thus, each nose shape can be uniquely defined by the cutting angle α and the cutting distance δ. The models of four projectiles with different nose shapes are shown in Fig. 1(b), including the 35° cut angle nose shape projectile, 40° cut angle nose shape projectile, 45° cut angle nose shape projectile, and flat nose shape projectile. The material of the projectile is aluminum alloy, and the contact angle of the projectile surface is 61.48°, which is measured by the static drop method. In order to connect the projectile to the launching tube, a ring armature with a thickness of 4 mm was embedded in the end of the projectile so that it could be adsorbed by the magnet at the end of the launching tube.

Figure 2 shows the schematic of the experimental setup. The entire water entry process was carried out in a transparent water tank with a size of 1.840 × 1.200 × 1.240 m³, which was composed of acrylic sheets with a thickness of 20 mm, and the water depth of the water tank was 0.85 m. In order to capture high quality water entry pictures, two 500 W lamps are arranged on the back of the water tank for the experimental background light source, and a large sheet of white paper with a 50 * 50 mm² grid is attached to the back of the water tank for light diffuse and size calibration. In order to enter the projectile into the water at a specific impact velocity and impact angle, the projectile needs to be accelerated in the launch tube. The launch system is mainly composed of a launching frame, a launching tube, and an air compressor. The angle between the launching tube and the horizontal plane can be adjusted by a screw, and the adjustment range is 0°–90°. Through the air compressor, compressed air can be introduced into the air vessel, and the highly compressed air can be released to the launch tube and can accelerate the projectile along the direction of the launch tube by using an electronic trigger. This experimental setup can adjust the pressure of the air in the air vessel so that the projectile is accelerated to different impact velocities. Due to the instability of the compressed air-based system, there is a certain fluctuation in the impact velocity under the same pressure of compressed air in the air vessel. Therefore, for each case presented below, we give the specific impact velocity value.

During the experiment, a high-speed camera (Phantom-RS320) was used to record the entire water entry process. The resolution of the high-speed camera was 1216 × 1024, the frame rate was 2000 fps, and the exposure time was 200 µs. In order to reveal the trajectory behavior of projectile and the cavity morphology behavior, we processed the experimental images taken by the high-speed camera. The ratio of the actual distance to the pixel distance in the image is determined based on the gridded background paper, and the value is 1.515 mm/pixel in the air and 1.395 mm/pixel in the water. From the positions of projectiles and the contour of cavities, the trajectory, velocity, and dimensions of the cavities can be obtained.

Experiments were performed with four different nose shapes (35° cut angle nose shape projectile, 40° cut angle nose shape projectile, 45° cut angle nose shape projectile, flat nose shape projectile), three impact velocities (U₀ = 8.47 m/s, 9.93 m/s, 11.07 m/s), and four different impact angles (θ = 60°, 70°, 80°, 90°). Before each impact experiment, the surface of the projectile was cleaned with a soft absorbent cloth, heated at 110 °C for 10 min, and then cooled to room temperature so that the surface of the projectile was fully dried.

To make this paper more readable, all symbols presented in this paper are listed in Table I.

When a projectile travels across the air–water surface at a certain initial impact velocity, an obvious water-entry cavity will be generated below the water surface. According to Waugh’s description,²⁷ the existence and development of the water-entry cavity will have a significant effect on the water-entry resistance and trajectory of the projectile. Therefore, in order to accurately establish the water-entry trajectory model of the projectile with the asymmetric nose shape, it is necessary to study the cavity characteristics of the asymmetric nose shape after entering the water. Referring to the investigation of May,²⁶ the evolution of water entry cavity can be divided into four stages: the impact stage, open cavity stage, closed cavity stage, and collapsing cavity stage. It is foreseeable that, compared with the axisymmetric nose shape, the asymmetric nose shape has a unique geometric shape, so it will inevitably have unique effects on the surrounding flow field during the flow formation process.

Figure 3 (Multimedia view) shows the comparison images of the cavity evolution at the same time after vertical water entry under the conditions of U₀ = 9.93 m/s for the experimental projectiles with a flat nose shape and three cutting-angle nose shapes. The time zero t = 0 ms in the figure is the moment when the projectiles just touch the free liquid surface.

When the projectile with a flat nose shape impacts the water surface, the projectile transfers momentum to the surrounding fluid so that the surrounding fluid gains radially outward momentum, thus forming the initial cavity below the liquid surface, and a uniform and symmetrical splashing water film is formed near the axis of the projectile [Fig. 3(a) t = 8 ms], which calls the open cavity stage. Due to the continuous development of the cavity, the pressure in the cavity is lower than the external atmospheric pressure. Therefore, under the action of surface tension and internal and external pressure difference, the top of the water film moves to the axis until it contacts the surface of the projectile, thus forming the “surface seal” phenomenon [Fig. 3(a) t = 16 ms]. Then, the closed water film moves downwards so that the top of the cavity is below the free liquid surface, forming a pull-away phenomenon [Fig. 3(a) t = 40 ms]. With the continuous downward movement of the projectile, the cavity is continuously stretched. However, because the cavity has been closed, the external air no longer flows into the cavity, causing the pressure of the gas inside the upper half of the cavity to drop rapidly, and the wall of the cavity shrinks and becomes rough [Fig. 3(a) t = 48 ms]. Then, the cavity near the tail of the projectile began to shrink and collapse under the action of static pressure. The cavity separates into two parts: the upper part formed flocculent cavities due to the collapse, while the lower part remained relatively complete [Fig. 3(a) t = 64 ms]. At the same time, the tail of the lower half of the cavity began to cloud rapidly and repeat the previous process [Fig. 3(a) t = 80 ms].

Different from the flat nose shape, the projectile with a cutting-angle nose shape generates an asymmetric water film after impacting the liquid surface [Figs. 3(b)–3(d), t = 8 ms]. The water film on the side with cutting-angle is obviously farther away from the projectile axis, which leads to a delay in surface seal [Figs. 3(b)–3(d), t = 16 ms]. Thus, more air flows into the cavity and causes the dimensions of the cavity to increase. Due to the asymmetry of the nose shape, the direction of the resultant hydrodynamic force on the cutting-angle nose shape does not coincide with the axis of the projectile, which causes the projectile to deflect counterclockwise [Figs. 3(b)–3(d), t = 40 ms]. The deflection of the projectile’s trajectory caused the cavity to significantly bend. After the projectile’s attitude was deflected to a certain angle, the tail-slap phenomenon occurred, and an obvious strip-shaped cavity was formed at the tip of the tail, which destroyed the original cavity shape [Fig. 3(b), t = 80 ms, Fig. 3(c), t = 64 ms, and Fig. 3(d), t = 56 ms]. Since the cavity is completely closed, the tail slap causes the original cavity to lose part of the gas, thus making the internal pressure of the original cavity drops, which accelerates the collapse of the cavity. It caused the internal pressure of the original cavity to drop rapidly. Under the action of the surrounding static pressure, dense small cavities are peeled off from the tail of the original cavity and quickly spread to the entire cavity wall. It can be found that as the cut angle of nose shape decreases, the timing of the tail-slap phenomenon also advances.

It can be found that the trajectory of the projectile with asymmetric nose shapes has been significantly deflected after impacting the water vertically, and the attitude angle of the projectile has also changed (Fig. 3). In order to obtain the influence characteristics of different asymmetric head shapes on the trajectory and attitude angle of the projectile under different water entry conditions, we extracted the trajectory and attitude angle data of the projectile under different experiment conditions and plotted them into curves. We define the intersection of the projectile axis and the nose plane as the nose midpoint, and the trajectory given in this paper depicts the spatial position of the nose midpoint at different times. In this section, we present the influence of the nose shape, impact angle, and impact velocity on the trajectory.

Figure 4 shows the horizontal and vertical displacements and attitude angle changes with the time of projectiles with different nose shapes after water enters vertically at an impact velocity of 9.82 m/s. As expected, the projectile with a flat nose shape is not affected by lateral force, its trajectory is a vertical straight line, the horizontal displacement is always zero, and the attitude angle is maintained at 90° (Fig. 4). Using the trajectory of the flat nose shape as a benchmark, the trajectory of the projectile with asymmetrical nose shapes shows obvious counterclockwise deflection after water entry. The trajectory deflection of the asymmetrical nose shape with a smaller cut angle α is greater than that with a larger cut angle α after water entry [Fig. 4(a)]. Compared with the flat nose shape, the attitude angle of the projectile with asymmetric nose shapes continuously decreases with time after water entry, and the speed at which the attitude angle decreases is also increasing. Moreover, the attitude angle of the projectile with a smaller cut angle α decreases more rapidly, which causes earlier tail-slap phenomenon. The projectile with a 35° cut angle nose shape reduces its attitude angle almost by 30° at t = 80 ms [Fig. 4(b)]. It can be inferred that as the water penetration depth of the projectile continues to increase, the attitude angle of the projectile will reach 0° at a certain moment, but due to the limited size of the water tank, this phenomenon is not observed under the condition of vertical water entry.

The projectile with an asymmetrical nose is subjected to lateral force after water entry so that the projectile obtains an angular acceleration that deflects to one side. When the velocity of the center of mass of the projectile u and the angular velocity of the projectile ω are determined, according to the rigid body kinematics, the velocity of the projectile nose u_n can be expressed as

where the x_n is a vector that coincides with the axis of the projectile and points to the nose midpoint and its modulus is equal to the distance between the center of mass of the projectile and the nose midpoint. Due to the velocity u_n of the projectile nose, the nose midpoint produces a horizontal displacement [Fig. 4(c)], and the horizontal displacement of the projectile with the 35° cut angle nose shape increases more rapidly than those nose shapes with a larger cut angle. The main reason can be attributed to the fact that the cut area is larger when the cut angle is smaller, which results in a larger hydrodynamic force. As shown in Fig. 5, where C₁, C₂, and C₃ are the cut surface areas of 35°, 40°, and 45° cut angle nose shapes, respectively, it can be obviously found that the length comparison of different cut surface areas has the following relationship: C₁ > C₂ > C₃. The hydrodynamic force acting on the cut surface is equal to the integral of the surface pressure on the area. With the same impact velocity, it can be assumed that the pressure distribution on the cut surface is uniform, and the surface pressure for each nose shape is approximately equal. This means that the hydrodynamic force depends only on the area of the cut surface, thus the hydrodynamic force will be larger for a larger area of the cut surface. Therefore, the hydrodynamic forces comparison of different cut angles also follows this relationship, F₁ > F₂ > F₃, and the horizontal force will be F_H = F cos α; that is why the nose shape with a smaller cut angle will have greater lateral deflection during the water entry process. Nevertheless, the trend of the vertical displacements of projectiles with different nose shapes is almost similar, while the vertical displacement of the projectile with a flat nose shape increases more rapidly [Fig. 4(d)].

In order to investigate the influence of impact angle on the trajectory of projectile with a 35° cut angle nose shape, four impact angles that are equal to 60°, 70°, 80°, and 90° are adopted in the experiment. Figure 6 presents the evolution of the horizontal and vertical displacements and attitude angle of projectile with the 35° cut angle nose shape at an impact velocity of 9.82 m/s and impact angles of 60°, 70°, 80°, and 90°. As shown in Fig. 6(a), under the four impact angle conditions, the projectile with a 35° cut angle nose shape deflected to the right after entering the water, and the deflection amplitude increased with the decrease in the impact angle. At the same time, the four curves of attitude angle change in Fig. 6(b) are nearly parallel, and it reveals that the change trend of the projectile’s attitude angle is the same under different impact angle conditions. This means that the deflection moments on the nose of the projectile are likely to be approximately equal under different impact angle conditions. Therefore, the projectile with the 35° cut angle nose shape at a small impact angle can make the attitude angle reach 0° in a shorter time, which means that the projectile attitude reaches the horizontal state. Due to the same impact velocity, the initial horizontal and initial vertical velocity of the projectile will also change when the impact angle changes. It indicates that by decreasing the impact angle, the projectile will obtain a higher horizontal velocity so that the horizontal displacement increases more rapidly [Fig. 6(c)]. Under the effect of nose deflection force, the growth rate of the vertical displacement gradually slows down, which is more obvious at a small angle of water entry [Fig. 6(d)].

Although we have obtained some laws of different cut angle nose shapes under the condition of vertical water entry, the influence of different cut angles on the deflection of the trajectory under the condition of oblique water entry still needs to be studied. Figure 7 shows the horizontal and vertical displacements and attitude angle changes with time of projectiles with different nose shapes after water entry at an impact velocity of 9.93 m/s and impact angles of 70°. The projectiles with asymmetrical nose shapes all deflected counterclockwise after water entry, and the amplitude of the deflection increased with the decrease in the cut angle [Fig. 7(a)]. At the same time, due to the difference in the area of the cut surfaces, the attitude angle decreases more rapidly for the projectile with a smaller cut angle [Fig. 7(b)]. Different from the result shown in Fig. 4, the projectile with a flat nose shape deflected clockwise after water entry, and the attitude angle increased linearly with time [Figs. 7(a) and 7(b)]. The increase in horizontal displacement is mainly related to the amplitude of deflection, and thus, the smaller cut angle can increase the horizontal displacement of the projectile [Fig. 7(c)]. As the attitude angle of the projectiles with asymmetric nose shapes decrease, the vertical velocities continue to decrease, which are especially obvious in the case of smaller cut angles [Fig. 7(d)].

In order to investigate the influence of impact velocity on the trajectory of the projectile with a 35° cut angle nose shape, we considered three impact velocities that are equal to 8.47 m/s, 9.93 m/s, and 11.07 m/s in the experiment. Figure 8 presents the evolution of the horizontal and vertical displacements and the attitude angle of the projectile with a 35° cut angle nose shape at an impact angle of 60° and impact velocities of 8.47 m/s, 9.93 m/s, and 11.07 m/s. It is clear that the trajectory characteristics of the projectile with a 35° cut angle nose shape are not sensitive to the impact velocity. Under the condition of different impact velocities, the trajectory of the projectile is basically the same in the inertial coordinate system [Fig. 8(a)]. However, the hydrodynamic force on the nose is lager due to the increasing impact velocity, which causes the attitude angle of the projectile to decrease faster [Fig. 8(b)], whereas it seems that the contribution of the increment of impact velocity becomes weaker with the increase in impact velocity. Therefore, increasing the impact velocity can make the attitude angle of the projectile with a cutting-angle nose shape reach the 0° faster, but the effect will also decrease with the increase in the impact velocity. With the increase in impact velocity, the impact velocity in horizontal and vertical direction also increases correspondingly, which leads to the increase in the displacement in horizontal and vertical directions more rapidly. At the same duration, the attenuation rate of the slope of the vertical displacement curve is obviously less than that of the horizontal displacement curve [Figs. 8(c) and 8(d)]. Figure 9 (Multimedia view) presents images of water entry cases by a projectile with a 35° cut angle nose shape at the same impact angle and different impact velocities. It is can find that the shapes of cavities formed at different impact velocities are basically similar, and the larger impact velocity will elongate the cavity. Compared to a smaller impact velocity, the tail-slap phenomenon of the projectile with a 35° cut angle nose shape will occur earlier, which results in a lager wetted area, and the depth of the tail-slap also increases [Figs. 9(a)–9(c)]. Therefore, the hydrodynamic force on the projectile will be lager to slowdown the vertical displacement increasing rate [Fig. 8(c)].

Figure 10 shows the horizontal and vertical displacements and attitude angle changes with the time of projectiles with different nose shapes after water entry at an impact velocity of 11.07 m/s and impact angles of 70°. When the impact velocity increases, the trajectory of the projectiles with different nose shapes still maintains a similar trend, as shown in Fig. 7, after impacting the liquid surface. However, it is worth noting that at this impact velocity, the projectile with a flat nose shape also deflects clockwise when it launches into the water [Fig. 10(b)]. This means that the projectile with a flat nose shape is likely to show a trajectory dive when water enters obliquely.

The force analysis of the projectile water entry process before the tail-slap occurs is shown in Fig. 11. We set two coordinate systems that are the inertial coordinate system (x₀oz₀) and projectile-fixed coordinate system (x_poz_p), the origin of inertial coordinate system is made to coincide with the impact point of projectile water entry, the x-axis of the inertial coordinate system is in the horizontal direction and coincides with the free liquid surface, and the z-axis of inertial coordinate system is in the gravity direction. The origin of the projectile-fixed coordinate system is set at the center of gravity of the projectile, the x-axis of the projectile-fixed coordinate system coincides with the axis of the projectile, the direction points to the nose of the projectile, and the y-axis of the projectile-fixed coordinate system is perpendicular to the x-axis and lies in the x_poz_p plane. In addition, θ is the attitude angle of the projectile relative to the inertial coordinate system.

The forces acting on the projectiles mainly include the gravity force F_G and the hydrodynamic force on the nose plane F_N and cutting plane F_C. The moment of the projectile is mainly the moment of the nose M_C. Therefore, the 3-DOF motion equation of the projectile can be expressed as

where F_Nx, F_Cx, and F_Gx represent the components of these forces in the x direction in the projectile-fixed coordinate system and F_Nz, F_Cz, and F_Gz represent the components of these forces in the z direction in the projectile-fixed coordinate system. u and v are the velocities of the projectile pointing to the x-axis and the z-axis in the projectile-fixed coordinate system, respectively, and ω is the pitch angular velocity. I_yy is the moment of inertia of the projectile.

In order to calculate these forces and moments, it is first necessary to obtain the conversion matrix from the projectile-fixed coordinate system (x_poz_p) to the inertial coordinate system (x₀oz₀). The conversion matrix is expressed as follows:

Also, the conversion matrix from the inertial coordinate system (x₀oz₀) to the projectile-fixed coordinate system (x_poz_p) is expressed as follows:

Thus, the gravity force F_G in the projectile-fixed coordinate system can be represented as

Assuming that the water is incompressible, the viscosity is negligible and the flow is irrotational, and then, the flow field can be described by the potential-flow model. With a high Reynolds-number limit, the motion of the liquid can be described by the Euler equation,

Using the velocity potential, the velocity of the liquid can be expressed as u = ∇ϕ, and we can obtain the generalized Bernoulli equation by integrating Eq. (8),

where A is located near the wetted surface of the projectile head and B is located at the free liquid surface at infinity. z is the vertical distance between point A and point B. Then, the pressure on the wetted surface of the projectile can be represented as

For simplicity, we approximated that the cavity expands axisymmetrically with the projectile axis as the center when water enters obliquely. To determine the ∂ϕ/∂t term, we established an axisymmetric cylindrical coordinate system in which the z axis coincides with the projectile axis and points to the nose of the projectile, r is the radial coordinate, and the origin of the coordinate system is the water entry point. Assuming that the flow field is irrotational, the velocity potential ϕ satisfies the Laplace equation,

According to research work of Yan,³⁴ the total velocity potential can be regarded as the sum of ϕ_c and ϕ_b. Therefore, the total velocity potential can be represented as

where ϕ_b and ϕ_c associated with the line source q(z, t) of cavity contraction and the point source σ(t) of cavity expansion, respectively, whose detailed derivations are presented in the  Appendix. If the n is the interior normal vector to the boundary surface, the hydrodynamic force F_N and F_C on the nose plane S_N and cutting plane S_C can be represented as

where P_N is the pressure on the nose plane, P_C is the pressure on the cutting plane, n_N is the outer normal vector of the nose plane, and n_C is the outer normal vector of the cutting plane. In order to obtain dS_N and dS_C, the nose plane coordinate system x_no_nz_n and cutting plane coordinate system x_co_cz_c are established, respectively, as shown in Fig. 12. Therefore, the dS_N and dS_C can be correspondingly described by the following equations:

where r_n is the radius of the nose plane, Δ_n is the distance from the axis o_ny_n to the side boundary, a_c is the major axis of the cutting plane, b_c is the minor axis of the cutting plane, and Δ_c is the distance from the axis o_cy_c to the side boundary. In this study, the cut surfaces are approximated as incomplete ellipses. For different cut angles, the parameters of the cut surface are also different, and these parameters have been provided in Table I.

Using the classic Runge–Kutta iterative method to numerically solve the 3-DOF equation [Eqs. (2)–(4)] and set the time step t = 0.0001 s, the initial conditions are set as follows: the nose midpoint of the projectile is at the origin of the inertial coordinate system, the initial impact velocity is 9.93 m/s, and the initial impact angles are 60°, 70°, 80°, and 90°, respectively. The initial angular velocity of the projectile is 0 rad/s, and the initial acceleration is 9.8 m/s due to the gravity. Combination of the above formulas can be solved to obtain the position and attitude angle of the projectile at any time water enters before the tail-slap happened.

In order to validate the mathematical model, we compared the experimental data with the calculate results from the mathematical model. Figure 13 shows the evolution of the horizontal and vertical displacements and an attitude angle of a projectile with a 35° cut angle nose shape at an impact velocity of 9.93 m/s and impact angles of 60°, 70°, 80°, and 90° after impacting the water with time and compares the results with the results calculated by the theoretical model. It is observed that the deflection rate of the attitude angle of the projectile calculated theoretically is slightly larger than the experimental results. However, the evolution trend of the change curves of the four attitude angles is consistent with the experimental results and close to parallel. The main reason for the error is that when the projectile crosses the air–water interface, the splash formed hits the surface of the projectile, thereby forming additional forces and moments. At the same time, the water entry cavity will form upwards and downward jet after the surface seal. This downward jet will also act on the tail of the projectile and affect the initial posture of the projectile. Similarly, the trajectory characteristic curves and displacement characteristic curves obtained from the theoretical calculation give a good approximation with the experimental results. In the present study, the maximum error of the horizontal displacements between the theoretical results and the experimental results is only 3.25%, and the maximum error of the vertical displacements is 2.47% at t = 81 ms. From the above, the mathematical trajectory model can predict the trajectory characteristics of a projectile with an asymmetric head in the initial stage of water entry.

In the present study, we conducted an experimental study on the water-entry trajectory of an asymmetric nose shape with 35°, 40°, and 45° cut angles at a low impact velocity in the range of 8.47 m/s–11.07 m/s. Through high-speed camera images, we captured the development of cavity and splash while the asymmetric nose shape enters the water and obtained the ballistic trajectory, displacement curve, and attitude angle curve of the projectile under each experiment condition. The influence of the nose shape, impact velocity, and impact angle on cavity shapes and trajectory characteristics is explored. Based on the theory of potential flow, a mathematical trajectory model was developed in this paper to predict the projectile motion and trajectory of the projectile with an asymmetric nose shape before the tail-slap process.

Compared to the projectile with a flat nose shape, the projectile with an asymmetric nose shape will significantly affect the shape and development process of the cavity and splash. The asymmetric nose shape can generate an asymmetric water film, which can put off the closure of the cavity, and thus, the volume of the cavity increases obviously. At the same time, the cavity of the asymmetric nose shape is also asymmetric, and it will bend with the deflection of the projectile. In addition, the asymmetric nose shape with a smaller cut angle will cause the tail-slap to occur earlier.

The nose shape, impact velocity, and impact angle will influence the trajectory in different ways. Due to different hydrodynamic effects, the nose shape with a smaller cut angle will have greater lateral deflection during the water entry process. The results presented that the change trend of the projectile’s attitude angle is the almost same under different impact angle conditions. In addition, we found that the trajectory characteristics of the projectile with a 35° cut angle nose shape are not sensitive to the impact velocity when other parameters are fixed.

Finally, by calculating the hydrodynamic forces on the nose of the projectile, a theoretical model of the water entry trajectory was established. We compared the experimental data with the calculate results and the theoretical calculation gives a good approximation with the experimental results. However, the downward jet and tail-slap can generate more glaring errors in the later stage water entry process, and thus, the theoretical model needs to be supplemented in further study.

This work was supported by the National Natural Science Foundation of China (Grant No. 51709229) and the Natural Science Foundation of Shaanxi Province (Program No. 2018JQ5092).

The data that support the findings of this study are available within the article.

In the water entry process with an open cavity, the dynamic boundary condition on the free liquid surface S_F(x, t) in the Lagrangian form is

where the P_F is the pressure on S_F(x, t), and in this study, we set the P_F to be equal to the atmospheric pressure. The velocity potential satisfies the far-field condition,

At an initial time, the free surface is still in a static state, and the velocity potential satisfies the initial condition,

We make a slender body hypothesis for water entry cavity, d_r/l_c = ε ≪ O(1), where d_r is the diameter of the cavity and l_c is the length of the cavity, and thus, the flow in the near field of the cavity is two-dimensional,

Then, the Laplace equation (11) reduces to

On the cavity wall r = a(z, t), the cavity radius develop equation can be written as

Taking the leading order term, Eq. (A6) becomes

Similarly, the dynamic boundary condition (A1) can also be written as

Ignoring the splash generated by the projectile, the free surface at z = 0 satisfies the boundary condition,

A line source of strength q(z, t) is arranged along the centerline of the cavity, which plays a major role in the shrinkage of the cavity. The expansion of the cavity is mainly affected by the three-dimensional point source of strength σ(t) located at the center point of the projectile nose. We define the velocity potential ϕ_c and ϕ_b associated with the q(z, t) and σ(t), respectively, and then, the total velocity potential can be written as Eq. (12). To satisfy the zero Dirichlet condition (A9), the velocity potential of this point source and its negative image above z = 0 can be expressed as

The strength value of σ(t) can be obtained by the volume flux across the wetted body surface S_B of the projectile nose. For the sphere, if the cavity separation line is at the maximum radius, then we obtain σ(t) = 2πR²V(t). For circular disk and symmetric projectile, the value of σ(t) is given by σ(t) = 3πR²V(t). For nose shape with cut angles, we placed an additional point source ϕ_ba at the center of cut surface, which makes the cavity develop asymmetrically. The wetted surface S_B of an asymmetric nose shape is mainly composed of the nose plane S_N and cutting plane S_C. For the nose plane S_N, σ(t) = 3S_NV(t), and for the cutting planeS_C, σ(t) = 3S_CV(t).

The line source q(z, t) of cavity contraction always coincides with the axis of the projectile, and for the inner solution, the problem is two-dimensional. The velocity potential of the line source and its negative image can be expressed as

where the function f₁(z, t) is determined by later asymptotic matching. The outer problem is three-dimensional, and the line source ϕ_c and its negative image can be written as

Equation (A12) has an inner expansion for r ≪ l_c, which has the form of (A11) but with f₁(z, t) given by³⁴ 

Then, the potential ϕ_c is uniquely specified in terms of the unknown line source q(z, t) with this matching. To determine q(z, t), we impose the dynamic boundary condition (A8) for total inner solution,

By integrating (A8) with respect to time, the inner solution can be written as

where h is the penetrate depth of the projectile and the t₀(h) is the time when the projectile arrives at the depth h. The integration constant C(z) is given by the velocity potential on the cavity wall at z at t = t₀,

At any time t > t₀, evaluation of (A14) at r = R₀ to satisfy (A15) gives an integral equation for the unknown q(z, t),

The expression −ϕ_b + ϕ_b0 − f₁(z, t) + f₁(z, t₀) = O(ε), and then, the value of q(z, t) is obtained by the following simple formula:

By substituting Eqs. (A18) and (A13) into Eq. (A11), we obtain