Qian－Kun He，Ying－Jie Wei，Cong Wang，Jia－Zhong Zhang

(School of Astronautics，Harbin Institute of Technology，Harbin 150001，China)

1 Introduction

When the underwater vehicle moves at sufficient speed through water，the local fluid pressure may drop below a level which sustains the water as liquid phase，and a low-density gaseous‘cavity’can be formed.As the underwater vehicle is entirely enveloped by cavity，the flow is called‘supercavitating flow’.Since the moving body does not contact with the fluid through most of its length，skin drag is almost negligible.Therefore，the underwater vehicle can achieve extremely high submerged speeds［1］. Especially for small size projectiles，they may even achieve a speed more than 1000 m/s［2］.Besides，the presence of the cavity changes the nature of motion of the projectile.While moving in the forward direction，the projectile also starts rotating about its tip in the vertical plane.This rotational motion is imparted to the projectile due to disturbances that occur during firing.Because of this rotation，the tail of the projectile impacts periodically on the interior surface of the cavity till the diameter of the cavity becomes sufficiently small［3］.Also，this phenomenon has been observed experimentally［4］.Through these periodic impacts(“tail-slap”)，the hydrodynamic stability of supercavitating projectiles is achieved.However，the interactions between the body and the water/cavity interface are the sources of structural strains and vibrations，which may undermine the structural reliability of the projectile and affect its guidance［5］.

Although extensive efforts have devoted in the past to the related analysis of the hydrodynamic characteristics of supercavitating vehicles［6］a nd most of the previous and current studies on guidance，the control and stability have considered projectiles as rigid bodies［7］，but very little research has been dedicated so far to the evaluation of the structural behavior of slender elastic bodies traveling underwater in supercavitating regimesathigh speed.By simplifying boundary conditions and hydrodynamic loads， R uzzene［5］had studied the dynamic behavior of supercavitating vehicles stiffened by rings and constrained the translation of its tip. C hoi［8］had studied the dynamic behavior of supercavitating vehicles with free boundary.Kubenko［9］h ad investigated(within the framework of a plane problem)the process of impact interaction between a thin long cylindrical body and the surface of the cylindrical cavity in an ideal compressible fluid for the initial stage of interaction，so the hydrodynamic and rigid body dynamic characteristics are obtained.However，the tail-slap processes involve the interaction of fluid and structure，and very few papers have considered the interactions to the calculation.

In this paper，we study the dynamic behavior of supercavitating projectiles traveling in tail-slap conditions.By considering the influence of structural deflections on the impact forces，the Fluid/Structure Interaction(FSI)calculation procedure has been built.The system is composed of three parts which are related with each other(as shown in Fig.1).The dynamic equation of the projectile，with no constrained boundary conditions，has been established，and solved by Finite Element Method(FEM). Firstly， the influences of FSI to hydrodynamic forces and dynamic vibrations of projectile at different constant speeds are analyzed.Then，the decelerating projectile with an initial speed is solved in FSI calculation procedure，and the characteristics of hydrodynamic impact forces are analyzed as well as the characteristics of rigid body dynamic and elastic body dynamic responses of the projectile.

Fig.1 Flowchart of FSI simulation

2 Oscillation Equation of Supercavitating Projectile

First ofall，the supercavitating projectile is confined to move in the vertical plane.Since the projectile is entirely enveloped by cavity，it has no fixed constrains.The oscillation equation can be expressed as

where q is the displacement of the body;M，C and K are the mass，damping and stiffness matrix of the body.When the projectile vibrates freely(F=0)，its natural frequency{ω}and principal mode shape Φcan be defined.In this paper，we study a model obtained from a 17th-mode truncation.Since the vibration system has no fixed constrains，the body has rigid motion in plane and the first two mode frequencies are equal to 0 Hz.The last fifteen mode frequencies correspond to the elastic displacements of the body.

According to mode superposition method，Eq.(1)can be transformed by using modal coordinate ξ.Then multiplies ΦTon the left side，we can get

Since the first two modal frequencies of the system ω1= ω2=0，and based on the orthogonality of principal mode shapes，the system matrix can be expressed as

where Eq.(5)belongs to the rigid motion expression of the system;Eq.(6)is the elastic oscillation equation of the body.

Eq.(4)can be subdivided as

3 Impact Model

Any disturbances will make interactions between projectile and water/cavity interface，and the interaction will produce huge impact forces acting on the tail of projectile enveloped by a cavity.In this paper，we treat the fluid as ideal， weightless， and incompressible.The path of the center of mass of the body is assumed to be well-approximated by a straight horizontal line L.This assumption neglects gravity，which has been justified by experimental work which shows no effect of gravity at speeds greater than 8 m/s［11］，and the cavity is assumed to be approximately fixed in an orientation which remains symmetric about the horizontal line L.The real motion of the cavity which traces a serpentine form as the body oscillates about the line of travel is simplified［12－13］.The shape of the cavity，used in determining when the tail of the body impacts the cavity/water interface，is assumed to be a known function of the forward velocity of the body.To simplify the following discussion，we also restrict attention to the case in which the body strikes the cavity walls normally，and the motion of the body is two-dimensional motion.

By the assumptions above，the projectile is acted upon by a drag force applied to the tip，and by an impact force generated by the interactions between the tail and the water/cavity interface.A schematic representation of the external forces of the body is given in Fig.2.

The drag force applied to the tip is given by

where ρ is the density of water;A is the projected area of the tip of the body(or cavitator);V is the velocity of the body;Cxis the drag coefficient［11］.The impact force applied to the tailofprojectile can be decomposed into two components:one is considered as lift force RLwhich makes the projectile rotate around its mass center;the other is considered as drag force RDwhich decelerates the axial motion of the projectile.

Fig.2 Forces applied to supercavitating projectile

As the speed of the projectile is V，the water also moves at the speed V relative to the projectile.It is assumed that in a very short impact time dt，a water layer of thickness λδ is deflected from its original course parallel to the direction of flight，by the tail，through an angle θ(as shown in Fig.3).Based on the Newton＇s third law，water is acted upon by forcesˉRLandˉRD，equal to RLand RDnumerically but in opposite directions.The momentum of this water layer，whose mass is dm，is changed.Because the impact time is very short and the fluid is inviscid，the loss of speed can be neglected here.The velocity of the water layer is not changed numerically，but only for its direction.By the composition of vectors plotted in Fig.3，it can be seen that the momentums changed byˉRLandˉRDare dmVLand dm(V － Vt)respectively.

Fig.3 Momentum transfer

The impact forces on tail are expressed as follows［13］When δ≤ 0

Hence，we get According to

When δ＞0

where A1= λδd is cross-sectional area of water layer;λ=0.5 is the empirical constant;δ represents the maximum penetration distance of the tail into the surrounding fluid. Fig. 2 gives a schematic representation of δ;in the figure，δrrepresents the maximum penetration distance of the tail of rigid body motion.If the influence of bending deflection q of the body is considered，then δ= δr;whereas，there will be δ = δr－ q.

The rotation of the projectile around its mass center during impact is governed by the following equation

where Izis the transverse moment of inertia of the projectile about its center of mass(Iz=mL2/12 for cylinder);a is the distance between the tail and the mass center of projectile.

The projectile will rebound to the cavity after impact.Based on the conservation of angular momentum，we can get

where Mz=－RDasin θ－RLacos θ，θ˙tis the angular velocity of projectile after impact;τ is the duration of once impact.

4 Numerical Results and Discussion

The considered supercavitating projectile is modeled as a uniform steel beam of circular crosssection.The main dimensions for the beam and the material properties are listed in Table 1.

Table 1 Dimensions and material properties of the supercavitating projectile

4.1 Responses of Projectile at Different Speeds with and without Considering FSI

It is Supposed that the supercavtating projectile moves at a constant speed，and the responses with and without considering FSI(for convenience，we use FSI and No-FSI for short)are evaluated and compared for the sets of initial conditions listed in Table 2.

Table 2 Initial conditions assigned to rigid degrees of freedom

The impact loads of tail-slaps of projectile that moves at the speed of 650 m/s are shown in Fig.4，for two conditions(FSI and No-FSI).The plots clearly indicate the effects of the bending reflections on the impact forces of tail-slaps.The curve of impact load of FSI is not as smooth as the curve of No-FSI.Also，we can find that the effect of FSI reduces the amplitude of impact load but increases its cycle.

Fig.4 Tail-slap force(V=650 m/s)

The vibration responses in terms of elastic displacement of tail of FSI and No-FSI are shown in Fig.5.Both curves show that the elastic displacement oscillates periodically，and the maximum amplitudes of each cycle appear during the impacts.Additionally，the residual response of the body after each impact affects the amplitude of vibrations resulting from the following impact.By comparing the two curves，we can find the vibration response of FSI has bigger cycles and smaller amplitudes than No-FSI.

Fig.5 Dynamic responses(V=650 m/s)

At different speeds，the effect of FSI on vibration responsesofelastic displacementofprojectile is different.The cycles and amplitudes ofelastic displacement at different speeds are shown in Fig.6.With the increase of speed，the cycle of FSI decreases slowly than that of No-FSI;in contrast the amplitudes of FSI increase slowly than that of No-FSI.Especially for the range of high speed，the results of No-FSI change rapidly. Hence， the higher the speed of projectile is，the more apparent the influence of FSI is.Then the effect of FSI should be considered.

Fig.6 Cycles and maximum deflections at different speeds

4.2 Responses of Projectile Riding a Reducing Cavity

The dynamic behavior of the considered supercavitating projectile is also analyzed when the cavity size become smaller as the forward motion of the projectile decelerating.Based on the conservation of momentum of the axial motion of the projectile［14］

where RDis small relative to FD.Ignoring the effect of RDon speed and substituting Eq.(7)to Eq.(14)，it can be given

In this paper，the initial speed V0of projectile is 800 m/s.The simulating simulation time is 0.34 s，because the cavity’s diameter at in the tail’s location of projectile is equals to the diameter of the tail at this time.The result time histories of speed changing and passed distance S are shown in Fig.7.

It is assumed that the initial speed of projectile is V0，so V can be solved as follows

Fig.7 Time history of velocity and distance

Because of the decrease of speed，a more complicated dynamic response is expected as a result of the reduction of the cavity size，which will affect the frequency of the impacts，the motion of the body and the vibration induced in the body.

The influence of FSI on the dynamic behavior of the projectile is assessed by observing both rigid body motion and elastic displacement.The analysisis performed by considering the initial conditions in Table 2.The results of angular velocity of projecitleprojectile with and without considering FSI are shown and compared in Fig.8.As shown in the Figure，the magnitude of the angular velocity after impact increases with time.It is also seen that the amplitudes and frequencies of impacts of FSI is smaller than that of No-FSI.

The responses in terms of vibration components with and without considering FSI are displayed in Fig.9.Itshows thata reducing cavity causes the amplitudes of the elastic displacement to oscillate slightly at the beginning，and increases when the cavity size is close to the diameter of the tail of projectile.The effect of FSI reduces the amplitude and frequency of the oscillation.

Fig.8 Time history of angular velocity

Fig.9 Time history of deflections

5 Conclusion

A simple model for the Fluid/Structure Interaction vibration analysis of a high speed supercavitating projectile had been presented in this paper.The predicted motion of the projectile involves a series of impacts between the tail of the projectile and the cavity/water interface.The dynamic behaviours of the projectile operating in tail-slap conditions with and without considering Fluid/Structure Interaction were obtained and compared and the following conclusions were derived:

1)At different constant speed，the cycle of FSI decreases slowly than that of No-FSI with the increment of speed;in contrast the amplitudes of FSI increase slowly than that of No-FSI.Especially for the range of high speed，the results of No-FSI change rapidly.Hence，the higher the speed of projectile is，the more apparent the influence of FSI is.

2)For the decelerated motion of supercavitating projectile，it is observed that a reducing cavity leads to an increment in the magnitude of the angular velocity of projectile，a slight oscillation of the amplitude of the elastic displacement at the beginning，and finally an increase of the elastic displacement.

3)For the decelerated motion of supercavitating projectile，the effect of FSI reduces the amplitudes and frequencies of the impact loads and the vibration.The Fluid/Structure Interaction should be considered in the study on dynamic behavior of the supercavitating projectile.

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