# Modeling of the Tail Slap for an Underwater Projectile within Supercavitation

Authors: Xuefeng Zhao; Xujian Lyu; Da Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Detailed process of the tail slap for an underwater projectile in supercavitation is studied in this paper. Firstly, the horizontal equation of motion for the projectile and a dynamic model of the projectile’s tail slap are introduced through a mathematical derivation, including a simple harmonic motion model of the projectile within the supercavitation layer and the tail slap reflection movement model inside the cavity. Subsequently, the MATLAB software package is used to model the projectile’s motion. Characteristics for force from the projectile tail on the water layer during a single tail slap and the behavior of the projectile’s angular motion as well as the characteristics of projectile tail motion through multiple tail slaps are investigated. It is found that the force magnitude of the water on the projectile tail decreases gradually while the relative motion angle of the projectile tail into the water layer and the contact time increase obviously as the projectile repeatedly slaps the cavity interface. When the projectile velocity decreases into a certain range, i.e., less than 50 m/s, the contact time of the tail in the water layer increases dramatically, and the supercavitation can tend to collapse and lead to unstable projectile motion. With increasing lateral deviation and angular velocity of the projectile tail, the number of tail slaps and the angle of reflection both increase significantly. In the case of supercavitation, the parameters of different tail slap points tend to be stable with time.


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## Body

## 1. Introduction

When an underwater vehicle moves within a certain speed high enough, its surface pressure is lower than the saturated vapor pressure of local water, thus forming a cavity that can surround nearly the entire projectile. This phenomenon is called natural supercavitation [1, 2]. When supercavitation occurs, only a small part of the tip of the vehicle is in contact with liquid water, and the rest only contacts the vapor in the cavity. In this case, surface friction is greatly reduced, which can ensure that the underwater vehicle can maintain high speed efficiently, greatly improving its performance [3, 4]. However, since the cavity itself is inside the interface of liquid phase and gas phase, the instability of this two-phase flow is obvious [5]. Meanwhile, the environmental pressure of the vehicle is affected by multiple external factors, and there are problems such as perturbation of hydrodynamic coefficients and unknown interaction from wake flow [6]. These further affect the persistence of the cavity and the force on the vehicle and greatly increase the complexity of the motion of the supercavitation vehicle [7, 8].An underwater high-speed projectile is a typical supercavitation vehicle. Tail swing is very likely to occur when the underwater high-speed projectile moves in the supercavitation cavity, due to external disturbance or its own unstable characteristics [9]. That is, slaps occur between the tail and the upper and lower walls of the cavity, and the projectile body attitude is corrected to maintain stability by these slap forces. Rand [10] made a very valuable pioneering exploration on the tail slap dynamics analysis of supercavitating projectile. Subsequently, Zhaoet al. [11] researched tail slap phenomenon during projectile horizontal movement. Their work included the influences of initial disturbance on tail slap motion, the cavity shape, and ballistic characteristics as well as the tail slap motion characteristics of supercavitating projectiles with different center of mass positions. Liet al. [12] studied the tail slap of supercavitation vehicles for different motion conditions and compared the effects of gravitational acceleration on tail slaps at different speeds. Zhang [13]et al.developed an improved method to obtain the length of a projectile that was immersed in a cavity boundary. Heet al. [14] established the equations of motion for tail slap coupling in supercavitation vehicles. They studied the fluid-solid coupling response of the supercavitating projectile during the tail slap process and analyzed the influence of fluid-solid coupling on the attitude and dynamic characteristics of the projectile. Zouet al. [15] established an analysis model of supercavitating projectile motion through the gas mass conservation equation and studied factors such as gravity and angle of attack on the tail slap dynamics of a supercavitating vehicle. Mirzaeiet al. [16, 17] established a nonlinear dynamic model to simulate the process of two-dimensional planar force acting on the tail of a supercavitation vehicle. They provided an equation to model the case in which the tail of the high-speed supercavitating projectile slaps on the cavity wall surface and established an empirical model for predicting the tail slap of the projectile. Kulkarniet al. [18] directly simulated the slaps of the projectile tail and cavity walls and predicted the behavior of slaps between the projectile’s tail and the cavity walls. Zhaoet al. [19] established a computational fluid dynamics model under different initial disturbances and studied the ballistic characteristics when tail slaps occur. Zhanget al. [20] investigated the tail-slap loads of supercavitating projectiles based on Logvinovich’s Principle. They concluded that the impact duration increases while frequency of impacts decreases as the slenderness ratio of projectiles increases.Tail slap has been extensively studied as a main stable or quasistable motion model for supercavitating bodies in water. The motion characteristics of tail slap remain unclear due to the instability of the two-phase flow of the supercavitation and the complexity of projectile motion in the cavity. Although plenty of works have been carried out to study the motion characteristic of tail slap as stated above, only minor of them are focused on the detailed process of the motion of the tail slaps, which is always significant for the application of a supercavitating projectile. In view of this, the characteristics of projectile forward motion and tail slap motion are studied through a simplified supercavitating projectile with both single tail slap in the water layer and multiple tail slaps in the cavitation being taken into account. A mathematical model is introduced to demonstrate the tail slap process, and influence of parameters including reflection angle, impact time, and cavity radius on tail slaps is discussed. Diagrams of tail slaps on various conditions are proposed to provide a different perspective to sketch the motion of projectiles in supercavitation, which may provide a reference for a better understanding of the interesting tail slap phenomenon.


## 2. Dynamic Model of Supercavitating Tail Slap

### 2.1. The Model of Forward Motion

A supercavitating projectile is affected by hydrodynamic force, vapor force in the cavitation, and external disturbances during movement. The movement is also closely related to the shape of the projectile and characteristics of the fluid medium. The actual dynamic characteristic of the projectile is very complicated. Considering that the supercavitating projectile is in horizontal motion, the following assumptions are made according to Rand [10]:(1) the effect of gravity on the trajectory of the projectile's center of mass is negligible;(2) the projectile is treated as a rigid body and its elastic deformation is not taken into account;(3) in the process of projectile motion, the projectile body can move forward steadily during the contact between the tail and the cavitation liquid-vapor interface, regardless of the influence of vapor in the supercavitation;(4) the projectile nose is always located at the central line of the cavity. The projectile body swings around the nose and does not rotate around its own axis of symmetry during its motion;(5) the cavity’s shape is symmetrical with respect to the horizontal line of motion of the projectile, and the diameter of the cavity decreases with decreasing projectile velocity. The cavity radius at the tail end of the projectile isr. It is generally acknowledged that the cavity radius for moving body increases as the increase of velocity; therefore a linear relationship with the forward velocity x˙ of the projectile is introduced to simplify the prediction model of cavity radius:(1)rx˙=C1x˙+C2where C1 and C2 are undetermined coefficients. These coefficients are determined based on the given projectile velocity and cavitation parameters (generally obtained from simulations or experiments).In the above assumptions, the motion of the supercavitating projectile is simplified into a rigid body motion model containing only translation and oscillation. When the projectile tail is not in contact with the liquid, its motion is modeled as translation in the x-direction and rotation about the projectile’s nose at a constant angular velocity, as illustrated in Figure1. The horizontal distance from the fixed launching datum to the center of mass is set asx, and the projectile body is subject to drag force FD at its nose. A simplified model of supercavitating projectile motion is obtained according to these assumptions, as shown in Figure 1.Figure 1
Motion model of the supercavitating projectile.The displacement of the projectile’s center of mass due to small inclination angleα is ignored. The axis of the projectile and cavity centerline coincides in thex-direction and the following equation can be obtained according to the experimental results of Mayet al.[21] and Newton's second law:(2)mx¨=-FD=-12ρAx˙2kcos2⁡αwhereρdenotes the density of water,A is the cross-sectional area of the nose, x˙ is the forward speed as above, x¨ is the acceleration,m is the mass of the projectile, andk is the dimensionless parameter (generally set as 0.9).Settingε=ρAk/2m, when α tends to be 0, the above equation can be turned into(3)limα→0⁡-FDm=limα→0⁡-ρAk2mx˙2cos2⁡α=limα→0⁡-εx˙2cos2⁡α=-εx˙2From (3), we find(4)x¨=-εx˙2where x˙=dx/dt. Letting x˙0 denote the initial velocity of the projectile, the following equations can be obtained by solving the differential equation to give(5)xt=1εIn1+εx˙0t(6)x˙t=x˙01+εx˙0tBy substituting (6) into (1), we get(7)rt=C1x˙01+εx˙0t+C2For the above mathematical model, the initial conditions for homogeneous projectile are given as follows [10]: the projectile length isL= 0.12 m; the projectile mass is 0.15 kg; the projectile tail diameter isD= 0.015 m; the nose diameter is D0 = 0.002 m; the taper angle of nose is 25° and the corresponding cross-sectional area isA= 3.14159×10−6 m2; the distance from the tail of the projectile to its center of mass isa= 0.05577 m; and the moment of inertia of the projectile about its center of mass isI= 0.000777 kg m2. Suppose that the initial velocity of the projectile is x˙0=1500m/s with the diameter of the cavity at the projectile tail beingr= 0.05 m, and x˙=100m/s withr= 0.03 m. According to (1), C1 = 0.0001429 and C2 = 0.02857, i.e., rx˙=0.00001429x˙+0.02857. The MATLAB software package is used to simulate forward motion of the projectile, and results for velocity and displacement versus time are demonstrated in Figure 2. It is found that the present results are in good agreement with that got from Rand [10], as shown in Figure 2. The projectile’s velocity rapidly drops from 1500 m/s to 391.9 m/s within the first 0.2 s, during which the displacement of the projectile is 142.5 m. Then, the velocity slowly decreases, and the displacement slowly increases, reaching 51.2 m/s and 358.3 m at 2 s, respectively. In general, the minimum speed range required for the projectile to maintain a supercavitation (i.e., cavitation number σ ≤ 0.1) is about 40~50 m/s, so that the projectile may lose stability after 2 s of motion due to the collapse of supercavitation; its subsequent motion characteristics at lower velocity are beyond the scope of this article.Figure 2
Plot of velocity and displacement for the horizontal motion of the supercavitating projectile, and comparison to previous work.
(a) (b)


### 2.2. Modeling Tail Slap Motion

Based on the abovementioned forward motion model for the projectile, motion characteristics of its tail slap will be discussed next. When the forward motion is disturbed by a random and instantaneous perturbation from surroundings or the cavitation’s instability, the projectile is modeled as a rigid body whose nose is fixed and the tail swings around the nose based on previous. The tail will slap repeatedly on the supercavity interface around the projectile and a single tail slap is taken into account to study the slap details firstly.Ifδ indicates the maximum distance that the projectile tail immersed into the water layer through the cavity interface during slap (Figure 3), then(8)δ=Lsin⁡α+D2cos⁡α-rwhereL is the length of the moving body andD is the diameter of the projectile tail.Figure 3
The forces acting on the projectile during a single tail slap.According to the above model, thickness of the water layer disturbed by the projectile is defined asλδ (where λ is a coefficient less than 1) and the cross-sectional area of fluid layer is written asA1=λδ·D. The longitudinal and lateral forces of the cavity boundary acting on the projectile tail areRD andRL, respectively, as shown in Figure 3. The following equations can be obtained due to the conservation of momentum:(9)RD=ρA1x˙21-cos⁡α(10)RL=ρA1x˙2sin⁡αThen, the swing of the projectile body around the nose satisfies the following equation:(11)Iα¨=-RDasin⁡α-RLacos⁡αWhenα is small, sinα can be approximated as α and cos⁡α as 1. By substituting (9) and (10) into (11), we get(12)Iα¨+ρλLα+D2-rx˙Dax˙2α=0Let Φ be the relative movement angle of the projectile tail slap; then we haveΦ=α-2rx˙-D/2L>0. Eq. (12) can be written as(13)IΦ¨+ρλDax˙2Φ2+ρλDax˙2rx˙-D2Φ=0If the slap time is short, Φ is small andrx˙ can be seen as constant. Neglecting the term with Φ2 of (13), we can simplify the equation and end up with(14)Φ¨+ω02Φ=0where ω02=(ρλDax˙2/I)rx˙-D/2.The equation represents a process of the moment when Φ is 0 (the initial slap point of the tail passes through the cavitation wall and into the water layer) to the moment when Φ again crosses 0 (the initial slap point of the tail passes back into the cavitation bubble and exits the water layer). Therefore, the duration of the slap is half of the simple harmonic vibration period, that is,ΔT=T/2=π/ω0. Through solving the above simple harmonic equations, there is(15)Φ=A0cos⁡ω0t+φwhere A0=ωL/ω0 is the amplitude of the simple harmonic vibration of the projectile in the water layer and φ=-π/2 is the initial phase of the simple harmonic vibration.The forces on the projectile in the water layer can then be expressed as(16)F=-LmA0ω02cos⁡ω0t+φBased on the kinetic model and the above analysis, it is assumed that each tail slap of the supercavitating projectile is an instantaneous dynamic process and that the angular velocity is constant and the slap reflection angle is equal to the incident angle before and after a single tail slap. Let us define that the initial position of the projectile tail is located at the center line of the cavitation offset Δx (initial lateral deviation) and swings upward with the nose at the center. According to the change of the cavitation cavity size with time, the location of the next slap point can be predicted. Therefore, the movement of the projectile tail in the supercavitation is essentially a slap problem with shrinking boundary (Figure 4). The parameters of the first slap point are given: ω is the angular velocity of the projectile body around the nose (the angular velocity of the tail slap), θ1 is the incident angle, t1 is the time from the initial point to the first slap point, and r1 is the cavitation radius of the slap point. Then, there is r1′=r1-D/2,, so that(17)θ1=arctan⁡dxωLt1t1=r1-D/22-dx2ωLr1=dx2+ωLt12+D2Figure 4
Sketch of tail slap looking from behind the projectile.As shown in Figure4, θ1 is the angle between the center of the circle E at the symmetric position of the first slap point and the line OE at the center O of the cavity and the direction line EF at the first slap point. From the first slap point to the second slap point, the projectile's motion in the cavity is due to slaps between the projectile tail and the cavity boundary. When the projectile moves towards the second slap point, the cavity at the first slap point shrinks from E point along the EO direction. The cavity and projectile intersect at point A, and the central position of the projectile tail is point B. In the direction of the EF line, the following can be obtained:(18)r1-D2cos⁡θ1-r2-D22-r1-D2sin⁡θ12+ωLt2-t1=2r1-D2cos⁡θ1By analogy, the relationship between thenth slap and the (n+1)th slap is(19)rn-D2cos⁡θn-rn+1-D22-rn-D2sin⁡θn2+ωLtn+1-tn=2rn-D2cos⁡θnAccording to the iteration of the above equation, the cavitation radius and the tail slap reflection angle at the occurrence of the (n+1)th tail slap are as follows:(20)rn+1=C1x˙01+εx˙0tn+1+C2θn+1=arcsin⁡rn-D/2sin⁡θnrn+1-D/2By obtaining the above parameters, the state change of the projectile when the tail swings in the supercavitation can be obtained.


## 2.1. The Model of Forward Motion

A supercavitating projectile is affected by hydrodynamic force, vapor force in the cavitation, and external disturbances during movement. The movement is also closely related to the shape of the projectile and characteristics of the fluid medium. The actual dynamic characteristic of the projectile is very complicated. Considering that the supercavitating projectile is in horizontal motion, the following assumptions are made according to Rand [10]:(1) the effect of gravity on the trajectory of the projectile's center of mass is negligible;(2) the projectile is treated as a rigid body and its elastic deformation is not taken into account;(3) in the process of projectile motion, the projectile body can move forward steadily during the contact between the tail and the cavitation liquid-vapor interface, regardless of the influence of vapor in the supercavitation;(4) the projectile nose is always located at the central line of the cavity. The projectile body swings around the nose and does not rotate around its own axis of symmetry during its motion;(5) the cavity’s shape is symmetrical with respect to the horizontal line of motion of the projectile, and the diameter of the cavity decreases with decreasing projectile velocity. The cavity radius at the tail end of the projectile isr. It is generally acknowledged that the cavity radius for moving body increases as the increase of velocity; therefore a linear relationship with the forward velocity x˙ of the projectile is introduced to simplify the prediction model of cavity radius:(1)rx˙=C1x˙+C2where C1 and C2 are undetermined coefficients. These coefficients are determined based on the given projectile velocity and cavitation parameters (generally obtained from simulations or experiments).In the above assumptions, the motion of the supercavitating projectile is simplified into a rigid body motion model containing only translation and oscillation. When the projectile tail is not in contact with the liquid, its motion is modeled as translation in the x-direction and rotation about the projectile’s nose at a constant angular velocity, as illustrated in Figure1. The horizontal distance from the fixed launching datum to the center of mass is set asx, and the projectile body is subject to drag force FD at its nose. A simplified model of supercavitating projectile motion is obtained according to these assumptions, as shown in Figure 1.Figure 1
Motion model of the supercavitating projectile.The displacement of the projectile’s center of mass due to small inclination angleα is ignored. The axis of the projectile and cavity centerline coincides in thex-direction and the following equation can be obtained according to the experimental results of Mayet al.[21] and Newton's second law:(2)mx¨=-FD=-12ρAx˙2kcos2⁡αwhereρdenotes the density of water,A is the cross-sectional area of the nose, x˙ is the forward speed as above, x¨ is the acceleration,m is the mass of the projectile, andk is the dimensionless parameter (generally set as 0.9).Settingε=ρAk/2m, when α tends to be 0, the above equation can be turned into(3)limα→0⁡-FDm=limα→0⁡-ρAk2mx˙2cos2⁡α=limα→0⁡-εx˙2cos2⁡α=-εx˙2From (3), we find(4)x¨=-εx˙2where x˙=dx/dt. Letting x˙0 denote the initial velocity of the projectile, the following equations can be obtained by solving the differential equation to give(5)xt=1εIn1+εx˙0t(6)x˙t=x˙01+εx˙0tBy substituting (6) into (1), we get(7)rt=C1x˙01+εx˙0t+C2For the above mathematical model, the initial conditions for homogeneous projectile are given as follows [10]: the projectile length isL= 0.12 m; the projectile mass is 0.15 kg; the projectile tail diameter isD= 0.015 m; the nose diameter is D0 = 0.002 m; the taper angle of nose is 25° and the corresponding cross-sectional area isA= 3.14159×10−6 m2; the distance from the tail of the projectile to its center of mass isa= 0.05577 m; and the moment of inertia of the projectile about its center of mass isI= 0.000777 kg m2. Suppose that the initial velocity of the projectile is x˙0=1500m/s with the diameter of the cavity at the projectile tail beingr= 0.05 m, and x˙=100m/s withr= 0.03 m. According to (1), C1 = 0.0001429 and C2 = 0.02857, i.e., rx˙=0.00001429x˙+0.02857. The MATLAB software package is used to simulate forward motion of the projectile, and results for velocity and displacement versus time are demonstrated in Figure 2. It is found that the present results are in good agreement with that got from Rand [10], as shown in Figure 2. The projectile’s velocity rapidly drops from 1500 m/s to 391.9 m/s within the first 0.2 s, during which the displacement of the projectile is 142.5 m. Then, the velocity slowly decreases, and the displacement slowly increases, reaching 51.2 m/s and 358.3 m at 2 s, respectively. In general, the minimum speed range required for the projectile to maintain a supercavitation (i.e., cavitation number σ ≤ 0.1) is about 40~50 m/s, so that the projectile may lose stability after 2 s of motion due to the collapse of supercavitation; its subsequent motion characteristics at lower velocity are beyond the scope of this article.Figure 2
Plot of velocity and displacement for the horizontal motion of the supercavitating projectile, and comparison to previous work.
(a) (b)


## 2.2. Modeling Tail Slap Motion

Based on the abovementioned forward motion model for the projectile, motion characteristics of its tail slap will be discussed next. When the forward motion is disturbed by a random and instantaneous perturbation from surroundings or the cavitation’s instability, the projectile is modeled as a rigid body whose nose is fixed and the tail swings around the nose based on previous. The tail will slap repeatedly on the supercavity interface around the projectile and a single tail slap is taken into account to study the slap details firstly.Ifδ indicates the maximum distance that the projectile tail immersed into the water layer through the cavity interface during slap (Figure 3), then(8)δ=Lsin⁡α+D2cos⁡α-rwhereL is the length of the moving body andD is the diameter of the projectile tail.Figure 3
The forces acting on the projectile during a single tail slap.According to the above model, thickness of the water layer disturbed by the projectile is defined asλδ (where λ is a coefficient less than 1) and the cross-sectional area of fluid layer is written asA1=λδ·D. The longitudinal and lateral forces of the cavity boundary acting on the projectile tail areRD andRL, respectively, as shown in Figure 3. The following equations can be obtained due to the conservation of momentum:(9)RD=ρA1x˙21-cos⁡α(10)RL=ρA1x˙2sin⁡αThen, the swing of the projectile body around the nose satisfies the following equation:(11)Iα¨=-RDasin⁡α-RLacos⁡αWhenα is small, sinα can be approximated as α and cos⁡α as 1. By substituting (9) and (10) into (11), we get(12)Iα¨+ρλLα+D2-rx˙Dax˙2α=0Let Φ be the relative movement angle of the projectile tail slap; then we haveΦ=α-2rx˙-D/2L>0. Eq. (12) can be written as(13)IΦ¨+ρλDax˙2Φ2+ρλDax˙2rx˙-D2Φ=0If the slap time is short, Φ is small andrx˙ can be seen as constant. Neglecting the term with Φ2 of (13), we can simplify the equation and end up with(14)Φ¨+ω02Φ=0where ω02=(ρλDax˙2/I)rx˙-D/2.The equation represents a process of the moment when Φ is 0 (the initial slap point of the tail passes through the cavitation wall and into the water layer) to the moment when Φ again crosses 0 (the initial slap point of the tail passes back into the cavitation bubble and exits the water layer). Therefore, the duration of the slap is half of the simple harmonic vibration period, that is,ΔT=T/2=π/ω0. Through solving the above simple harmonic equations, there is(15)Φ=A0cos⁡ω0t+φwhere A0=ωL/ω0 is the amplitude of the simple harmonic vibration of the projectile in the water layer and φ=-π/2 is the initial phase of the simple harmonic vibration.The forces on the projectile in the water layer can then be expressed as(16)F=-LmA0ω02cos⁡ω0t+φBased on the kinetic model and the above analysis, it is assumed that each tail slap of the supercavitating projectile is an instantaneous dynamic process and that the angular velocity is constant and the slap reflection angle is equal to the incident angle before and after a single tail slap. Let us define that the initial position of the projectile tail is located at the center line of the cavitation offset Δx (initial lateral deviation) and swings upward with the nose at the center. According to the change of the cavitation cavity size with time, the location of the next slap point can be predicted. Therefore, the movement of the projectile tail in the supercavitation is essentially a slap problem with shrinking boundary (Figure 4). The parameters of the first slap point are given: ω is the angular velocity of the projectile body around the nose (the angular velocity of the tail slap), θ1 is the incident angle, t1 is the time from the initial point to the first slap point, and r1 is the cavitation radius of the slap point. Then, there is r1′=r1-D/2,, so that(17)θ1=arctan⁡dxωLt1t1=r1-D/22-dx2ωLr1=dx2+ωLt12+D2Figure 4
Sketch of tail slap looking from behind the projectile.As shown in Figure4, θ1 is the angle between the center of the circle E at the symmetric position of the first slap point and the line OE at the center O of the cavity and the direction line EF at the first slap point. From the first slap point to the second slap point, the projectile's motion in the cavity is due to slaps between the projectile tail and the cavity boundary. When the projectile moves towards the second slap point, the cavity at the first slap point shrinks from E point along the EO direction. The cavity and projectile intersect at point A, and the central position of the projectile tail is point B. In the direction of the EF line, the following can be obtained:(18)r1-D2cos⁡θ1-r2-D22-r1-D2sin⁡θ12+ωLt2-t1=2r1-D2cos⁡θ1By analogy, the relationship between thenth slap and the (n+1)th slap is(19)rn-D2cos⁡θn-rn+1-D22-rn-D2sin⁡θn2+ωLtn+1-tn=2rn-D2cos⁡θnAccording to the iteration of the above equation, the cavitation radius and the tail slap reflection angle at the occurrence of the (n+1)th tail slap are as follows:(20)rn+1=C1x˙01+εx˙0tn+1+C2θn+1=arcsin⁡rn-D/2sin⁡θnrn+1-D/2By obtaining the above parameters, the state change of the projectile when the tail swings in the supercavitation can be obtained.


## 3. Results and Discussion

Figure5 shows the curves of force and relative motion angle for projectile tail in water layer with time during repeated tail slaps composed of multiple single tail slaps. The force amplitude of the projectile tail in harmonic motion in the water layer decreases with the continuous occurrence of the tail slap and the attenuation trend gradually decreases with time. Among them, the force amplitude of the first tail slap is 10 N, while that of the second tail slap decreases to about 4 N, decreasing by 60 %. The relative motion angle which demonstrates the amplitude of the projectile tail in the water layer increases with the increase of the number of tail slaps, as shown in Figure 5(b). However, the maximum swing amplitude in 2 s is about 0.08°, which is far less than the absolute swing angle α of the projectile (generally, α > 14°). Figure 6 shows the variation of the contact time of the single tail slaps of the projectile in the cavity layer with the center of mass velocity and the moving distance of the projectile within 2 s. With the continuous occurrence of the tail slaps, the contact time of the projectile with the water layer gradually increases from less than 0.004 s. When forward speed of the projectile is lower than 250 m/s, the contact time rapidly increases to 0.018 s, but it is still far less than the tail slap interval (about 0.18 s). Within 0.018 s, the relative change in the radius of the supercavitation around the projectile is 0.03 %; that is, the previous assumption of regarding rx˙ as a constant in the above single tail slap is reasonable. When the contact time of the projectile tail in the water layer continues to increase dramatically (t > 2 s), the corresponding projectile velocity decreases to about 51.2 m/s. That is, supercavitation will eventually collapse and projectile motion instability may occur.Figure 5
Variation of the force and relative motion angle of projectile during tail slap in water layer with times.
(a) (b)Figure 6
Variation curve of contact time between a projectile and the liquid, velocity and horizontal displacement during projectile tail slap.
(a) (b)MATLAB is further used to simulate the whole process of projectile tail slap movement through a 2 s time interval. The specific working conditions and corresponding tail slap frequency are shown in Table1. Figures 7 and 8 show the schematic diagram of projectile tail motion relative to the central axis of the cavity at different angular velocities (ω = 1, 2 rad/s) and different initial lateral deviation conditions (Δx= 0.01 m, 0.02 m), respectively. Under the same initial side deviation Δx, the number of tail slaps increases with the increase of angular velocityω, and the effect of cavitation superposition gradually becomes obvious, as shown in Figure 7. When the angular velocityω of tail slap is constant, the number of tail slaps increases as Δx increases (Table 1), and the tail slap trajectory is closer to the cavitation bubble wall.Table 1
Tail slap frequency of projectiles with different conditions.
Statistical time Lateral deviation Slap frequency ω= 1 rad/s ω= 2 rad/s ω= 3 rad/s In 1 s Δx= 0 m 2 5 7 Δx= 0.01 m 3 5 8 Δx= 0.02 m 4 7 11  In 2 s Δx= 0 m 3 5 7 Δx= 0.01 m 3 6 8 Δx= 0.02 m 5 9 12Figure 7
The motion trajectory of the projectile tail over 2 s with different tail slap speeds.
(a)
Δx= 0.01 m,ω= 1 rad/s (b)
Δx= 0.01 m,ω= 2 rad/sFigure 8
Motion trajectory of the projectile tail over 2s with different initial lateral deviations.
(a)
ω= 3 rad/s, Δx= 0.01 m (b)
ω= 3 rad/s, Δx= 0.02 mTable2 shows the variation of cavitation radius rn, tail slap reflection angle θn, and tail slap occurrence time tn under typical conditions (Δx= 0.01 m, ω = 2 rad/s). During the tail slap sequence from 0 to 2s, the cavity radius decreases and the rate of speed loss decreases, as the projectile moves forward and the tail slap occurs. The radius decreases from 0.0368 m for the first slap to 0.0324 m for the second slap. The cavitation radius of the 5th tail slap is 0.0302 m, while the cavitation radius from the 6th to the 11th is stable at around 0.029 m. The reflection angle gradually increases with the continued tail slaps, and the increasing trend decreases constantly, which is basically consistent with the trend of the cavitation radius. The time interval between two adjacent tail slaps also decreases slowly from 0.2096 s, which is about 0.1777 s between the 10th and 11th tail slaps; that is, the time interval between tail slaps is generally stable in the range of 0.17 s to 0.21 s.Table 2
Slap data of projectile tail in 2s.
n r n (m) θ n (°) t n (s) 1 0.0368 19.9897 0.1145 2 0.0324 23.6683 0.3241 3 0.0312 25.0036 0.5151 4 0.0305 25.7275 0.7006 5 0.0302 26.1871 0.8833 6 0.0299 26.5062 1.0644 7 0.0297 26.7415 1.2444 8 0.0296 26.9222 1.4236 9 0.0295 27.0657 1.6022 10 0.0294 27.1823 1.7803 11 0.0293 27.2791 1.9580Figure9 shows the cavitation shrinkage curve and diagram of the tail slap process with different tail slap angular velocities (ω = 1, 2, 3 rad/s) and different initial lateral deviations (Δx= 0.01, 0.02, 0.03 m), which further explains the tail slap frequency and tail slap points depicted in Figures 7 and 8. Figure 10 depicts the variation of the reflection angle of slap points caused by the projectile’s lateral deviations. When the lateral deviation is zero, the projectile is reflected along the vertical direction of the cavitation center line, and its reflection angle is zero. The reflection angle gradually increases as the lateral deviation increases. In comparison to the curve with lateral deviations of 0.01 m and 0.02 m, when the angular velocity remains unchanged, slap points increase with the increase of lateral deviation.Figure 9
Diagram of variation cavity radius and tail slap points over time in difference conditions.
(a)
Δx= 0.01 m (b)
ω = 2 rad/sFigure 10
Variation of the reflection angle of slap points over time at difference lateral deviations (ω= 2 rad/s).


## 4. Conclusion

In this paper, a simplified model of the tail slap motion of a supercavitating projectile in water is established. In the case that the angular velocity energy loss is not considered, motion of the projectile is modeled as translation in thex-direction and rotation about the projectile’s nose at a constant angular velocity. The forward speed and the forward path of the projectile are inversely proportional functions and logarithmic functions with time as the independent variable, respectively. The projectile spends less time and has a relatively smaller relative motion angle during the tail slap process, and the angular velocity remains constant when it is not in contact with the cavitation bubble’s liquid surroundings. This makes the rebound an instantaneous process when the projectile tail slaps the cavitation bubble’s boundary. The problem of the projectile tail slap motion under this condition is a dynamic boundary slap problem in which the boundary is constantly changing, the variation law is known, and the incident angle is equal to the reflection angle. The position of the tail slap point is determined by the cavitation radius of the previous slap point, the time at which the tail slap occurs, and the incident angle. From the tail slap simulation, the following conclusions can be drawn:(1) The force of the projectile in the water layer decreases with the increase of tail slap time and number of slaps, resulting in an increase of the contact time between the tail and the liquid water, and relative motion angle of the projectile in the water layer. However, the contact time and relative swing angle of projectile in the water layer are relatively small.(2) Within the 2 s simulated time, the contact time of the projectile with the water layer is much smaller than the projectile’s tail slap harmonic period, and the slap of the projectile and the cavitation wall can be regarded as an instantaneous effect. When the stay time of projectile tail in the water layer continues to increase dramatically (t > 2 s), the projectile velocity drops below 50 m/s; that is, supercavitation possibly collapses and projectile motion instability may then occur.(3) With the initial lateral deviation of the projectile and the increase of the angular velocity of the projectile tail, the time of the first contact between the projectile tail and the cavitation boundary advances, and the number of slaps between the projectile tail and the cavitation wall surface also increases. Furthermore, the trajectory map of the projectile's tail slap becomes more complex, the reflection angle of the slap points gradually increases, and the trajectory of the projectile gets closer to the contour of the cavity. In the case of supercavitation, parameters such as the cavitation radius of the tail slap point, the reflection angle, and time intervals between two tail slaps tend to be stable as the number of tail slaps increases.



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*Source: 1290157-2019-05-21.xml*