EXPERIMENTAL AND THEORETICAL STUDY ON THE HIGH-SPEED HORIZONTAL WATER ENTRY BEHAVIORS OF CYLINDRICAL PROJECTILES*

GUO Zi-tao, ZHANG Wei, WANG Cong

Hypervelocity Impact Research Center, Harbin Institute of Technology, Harbin 150080, China,

E-mail: guozitao@hotmail.com

EXPERIMENTAL AND THEORETICAL STUDY ON THE HIGH-SPEED HORIZONTAL WATER ENTRY BEHAVIORS OF CYLINDRICAL PROJECTILES*

GUO Zi-tao, ZHANG Wei, WANG Cong

Hypervelocity Impact Research Center, Harbin Institute of Technology, Harbin 150080, China,

E-mail: guozitao@hotmail.com

In this article, the horizontal water-entry of flat-nose projectiles of two different lengths at impact velocities of 400 m/s-600 m/s is studied experimentally and theoretically. Based on the solution of the Rayleigh-Besant problem, a set of projectile dynamic equations are derived and a cavity model is built to describe the projectile’s water entry dynamics. A parameter in the cavity model is determined by employing the principle of energy conservation. The results indicate that the flat-nose projectiles enjoy a good stability of trajectory, the drag coefficient and the velocity decay coefficient are dependent on the cavitation number, and increase along the penetration distance but with a relatively small variation. The maximum cavity radius decreases monotonically with the penetration distance. Projectiles with the same nose shapes at different initial velocities have a basically consistent cavity dimension before the deep pinching off phenomenon occurs. Good agreements are observed between results obtained by the analytical model and the experimental results.

horizontal water entry, experiment, theoretical analysis, cavity dimension, projectile dynamics

Introduction

The water entry problem is an interesting topic in fluid mechanics with a long history of more than 80 years. The earliest work before World War II mainly was based on Von Karman’s physical picture of the water entry problem, which amounts to mathematical solutions of the impact forces acting upon the rigid bodies during water entry. After World War II, the research of water entry was extensively motivated by the demands to develop the low-speed underwater weapons and to determione the fluid dynamic characteristics of supercavitating vehicles. Thus besides the impact force system, the study was extended to the cavity growth, decay, and missile trajectory. The most important research progress in this period was the establishment of a framework for the study of water entry. However, until recently, projectile water entry is still a difficult problem and is not completely solved.

In recent years, the initial stage process of the low velocity vertical water entry of bodies such as disks, cylinders and wedges, etc. was much studied[1-4], besides, the cavity dynamics due to the vertical water entry by spheres or disks at very low velocities is another topic of interest[5-9]. For high-speed water entry problems, Lee et al.[10]presented a method for modeling the cavity formation and the collapse induced by high-speed vertical impact of a rigid projectile into water based on the energy conservation. Shi and Takuya[11,12]studied the vertical water entry of a highspeed blunt solid body and obtained the velocity decay coefficient when the projectile was treated equivalently as a sphere. Gu et al.[13]conducted an experimental and theoretical study on the penetration law for rotating pellets entering water.

However, the studies on the horizontal waterentry are quite rare as compared with the vertical water-entry studies. Though they are not essentially different in nature, a difficulty would arise in the determination of the pressure difference between the surrounding fluid and the cavity along the penetration distance in high-speed horizontal water-entry problems. In many vertical water entry problems, the pressure difference ΔP was considered as a function ofthe pressure caused by the projectile gravity irrespe-velocities. Duclaux et al.[5]and Aristoff and Bush[6]also employed the similar approaches. But in the cases of horizontal water entry where the effects of the projectile gravity and the surface tension are negligible, it might be difficult to determine the values of ΔP.

In this article, under the assumption that the pressure difference is constant and the drag coefficient is velocity-dependent in the horizontal water-entry process, a set of projectile dynamic equations are proposed to describe the projectile dynamics of water entry and a cavity model based on an extension of the method in Duclaux et al.[5]for solving the Rayleigh-Besant problem is built to describe the cavity dynamics of water entry. The digital high-speed photography provides the basis for the establishment of the mathematical models to describe the high-speed water-entry problems.

1. Experimental set-up

The sketch of the experimental device used for impact tests is shown in Fig.1.

Fig.1 Sketch of the experimental device

1.1 Gas gun

The projectile accelerator is a two-stage light-gas gun which is non-power-driven, installed in Harbin Institute of Technology. The gun is used for a waterfilled vessel. On one side-wall of the vessel is a 0.012 m thick transparent polycarbonate panel, through which the process of water-entry can be recorded by a high speed camera.

1.2 Rojectile

The projectiles launched against the tank are of two different lengths, 0.0254 m and 0.0381 m, with the same diameter of 0.0127 m and their average masses are 23.5g and 32.8 g. The projectile is made of 38CrMnSi steel and the yield strength of the material is about 1.5 Gpa. The projectiles are regarded as rigid bodies as compared with the water.

1.3 High speed Camera and lighting system

To record the process of the projectile penetrating the fluid and the development of the cavity, a Photron Ultima APX-RS digital high-speed camera is employed. The selected frame rate is 36 000 per second, so that a frame is taken every 27.78 μs with a resolution of 512×128 pixels. These settings are selected based on early testing as an optimal trade off between available lighting and the minimization of blur in the images. Lighting is provided by two 1 200 W lamps.

Fig.2 Water-entry process of projectile of the length of 0.0254 m at 454 m/s

Fig.3 Water-entry process of projectile of the length of 0.0381 m at 414 m/s

2. Experimental results

Figure 2 shows the photographs of the projectile of 0.0254 m in length penetrating the fluid at the speed of 454 m/s. The photographs of the projectile of the length of 0.0381 m traveling through the water at the velocity of 414 m/s are shown in Fig.3. Comparing Fig.2 and Fig.3, it is observed that for projectiles of different lengths and different velocities of water entry, the cavity-shapes are similar in the early stage of water entry.

The velocity and the position of the projectile inside the water can be determined through the pictures taken by the high-speed digital photography. The velocity attenuations of the projectiles of two different lengths at initial impact velocities of 400 m/s and 500 m/s are plotted in Fig.4. It can be seen that the velocity decays rapidly in a very short time and the velocity attenuation curves are very similar. The experimental time histories of the penetration distance for two projectiles are shown in Fig.5.

Fig.4 Comparison of velocity attenuations for projectiles of two different lengths

Fig.5 Experimental curves of penetration distance vs. time for projectiles of two different lengths

3. Theoretical analysis

3.1 Cavitation number and the drag coefficient

Cavitating flows are commonly described by the cavitation number, which is a non-dimensional quantity that represents the extent of cavitation. It is defined as

where p represents the pressure outside of the cavity, pcthe pressure inside the cavity, which is assumed to have a negligible variation in the penetration-direction in the process of water entry as suggested by the measurements in Ref.[14],ρwis the water density, and vpis the transient projectile velocity.

As in the initial stage of water entry, the cavity formed by high-speed projectile has no time to close up and a strong air entrainment occurs, the initial cavitation number σ0can be determined as

where ρa(bǔ)represents the air density and the coefficient of air pressure drop Ca=5-15[13].

For the horizontal water entries, the variation of the pressure difference ΔP along the penetration distance is difficult to determine. A common way is to assume ΔP to be constant in a horizontal water entry process. Considering the initial conditionsthe cavitation number in a whole water entry process can be expressed as

For the disk-type nose, the drag coefficient is determined by the relation

where the drag coefficient is defined as Cd= 2F/ρAv2, in which F stands for the drag force and A0denotes the projected frontal area of the projectile. The hydrostatic component of the drag coefficient C0=0.82-0.83.

3.2 Projectile dynamics

Disregarding the gravity distortion effects, the motion equation of the projectile can be obtained from Newton’s second law

where mpandmδ denote the mass and the added mass of flat projectiles, respectively, vpstands for the projectile velocity.

Assume that Cd(vp) is velocity-dependent, the velocity attenuation as a function of time t can be obtained by integrating Eq.(5),

Integrating Eq.(6), the penetration distance x is determined as

From Eq.(6) and Eq.(7), the relation between the projectile velocity and the penetration distance can be expressed as

Reconsider Eq.(3), the cavitation number as a function of penetration distance x can be determined from Eq.(8)

Substituting Eq.(9) into Eq.(4), yields

Thus there is a relation between the parameter k and the velocity decay coefficient β(vp), which is defined as

Fig.6 Comparison of the experiments and the analytical calculations for velocity attenuations

Fig.7 Comparison of the experimental penetration distance with time and the analytical calculations for projectiles of two different lengths

Fig.8 Comparison of experiments and calculations for projectiles of two different lengths

Fig.9 Variation of σ, Cdand β with the penetration distance for projectiles of two different lengths

In super-cavitation flows, the liquid phase does not contact the moving body over most of its length, the drag force of the projectile in water is dominated by the pressure drag component and the skin drag can be neglected. The added mass δmin the projectile head is a small quantity and could be neglected as compared with the projectile mass mp. According to the range of velocities considered, the value 0.8275 is chosen for C0. For the projectile of 0.0254 m in length, mp=23.5g , D0=0.0127m,A0=1.27× 10-4m2and k =2.236m-1. For the projectile of 0.0381 m in length, mp=32.8g andk=1.60m-1. If the initial cavitation number σ0is assigned a value of 0.01, the comparisons between the experiments and the analytical calculations for the projectiles of two different lengths are shown in Fig.6, Fig.7 and Fig.8, respectively. Noting that, the velocity values calculated as an average between two frames do not correspond exactly with the velocity at each point. Nevertheless, the calculations based on the model agree well with the experimental results. The variation of the cavitation number σ, the drag coefficient Cdand the velocity decay coefficient β along the penetration distance for the projectiles of two different lengths are shown in Fig.9, where the three parameters are found almost constant in the early stage of the water entry process before the cavity deep closure occurs.

3.3 Cavity dynamics

Due to the complexities involved in the water entry event, the mathematical analysis of the dynamics of the water-entry cavity is a relatively little explored area. Lee et al.[10]studied the high-speed vertical water-entry of spheres by employing the principle of energy conservation, where it is assumed that the kinetic energy loss of the projectile is equal to the total energy (kinetic plus potential) in a fluid section. Recently, Duclaux et al.[5], Aristoff and Bush[6]developed a theoretical model for the cavity evolution based on an extension of the method used to solve the Rayleigh-Besant problem: the collapse of a spherical cavity in a fluid of infinite extent. The above models actually included the same assumption that the cavity motion is a purely radial or perpendicular to the cavity axis as suggested by the experiments conducted by Birkhoff and Caywood[15].

This article makes the analysis and extends the approach of Duclaux et al.[5]to describe the cavity dynamics of the horizontal water entry of cylindrical projectiles. For convenience of discussion, the geometry of the problem is sketched in Fig.10.

Fig.10 The cavity growth model

The fluid motion is assumed to be irrotational everywhere so that the fluid motion can be described through a simple potential u=Δφ. Considering theunsteady Bernoulli equation

where A and B represent two different locations in the fluid. ChoosingA to be at the cavity boundary and B to be far from the cavity where there is no motion but at the same penetration distancexp, we obtain

where ΔP stands for the pressure difference between the surrounding fluid and the cavity in the water entry process.

As the projectile travelling in the water is approximated as a moving source, the potential φ can be expressed as φ=q(x,t)ln(r/r∞), where x denotes the penetration index, q(x,t ) represents the penetration distance-dependent source strength,r∞is a function of time related with the range of the disturbances caused by the water impact.

The source strength q(x,t) can be determined by considering the boundary condition at the cavity surface where r=R(x), that the local fluid radial velocity is equal to the cavity wall velocity.

This leads to q(x,t)=RR˙, thus the potential φ is expressed as

Substituting Eq.(15) into Eq.(13) and assuming N=ln(r∞/R), yields

where N is a dimensionless geometric parameter for the range of the disturbances caused by the water impact.

At a given penetration distance xp, Eq.(16) for the cylindrical cavity can be integrated once

where the initial conditionswith κ being a constant much smaller than 1. While Eq.(17) gives a solution for the cavity radius, the numerical integration should be required to obtain the evolution of the cavity shape accurately. To get an approximate analytical solution of the cavity evolution, Eq.(16) should be simplified. Note that

Then Eq.(16) is reduced to

Integrating Eq.(19) twice, the time evolution of the cavity can be obtained as

Equation (20) describes the time evolution of the cavity radius at x0, starting at the time t0when the projectile reachesx0. When the projectile reaches a placex＞x0, the radius of the cavity at x0can be calculated by this equation. This expression can describe the cavity evolution of flat nose projectiles where the nose radius is the same as the projected frontal radius of projectiles R0.

Fig.11 Cavity growth model att0

Determination of the parameter κ:

Note that in Eq.(17), the parameter κ is defined as the ratio of the initial cavity velocity to the pro-jectile velocity vpat time t0. In order to better understand the parameter, the definition of κ should be clarified. Assuming that the cavity motion is purely radial or perpendicular to the cavity axis, the growth model of the cavity at the penetration distancexpfrom time t0to t0+dt0is plotted in Fig.11, where the projectile travels a distancedxpand the cavity expands by an amount dR after an infinitesimal time dt0. From Fig.11, the definition of κ can be given as

where π/2≤θc≤π.

From the above definition, it can be seen that the larger the cavity radius, the larger the κ values will be. This observation implies that the magnitude ofκ is related to the projectile nose shapes. But the value of κ is difficult to be calculated simply by Eq.(21). However it can be done by using the principle of energy conservation that the loss of the kinetic energy Etof the projectile with respect to the penetration distancex is transformed to the kinetic energy andentry of spheres.

From Fig.10, the kinetic energy, Ekof the fluid in the section within a finite radius r∞, can be obtained by integrating the following expression

Considering a purely radial cavity motion andthe kinetic energy Ekis expressed as The change rate of the projectile kinetic energy Epwith respect to the penetration distance xpis determined by

In this article it is assumed that the initial kinetic energy of a fluid section of the cavity at the distance xptotally comes from the loss of the kinetic energy Epof the projectile with respect to penetration distance x. This leads to

According to the initial conditionsRt=t0=R0and=κvp, Eq.(25) can be reduced to

Thus the κ is determined and found to be a function of the drag coefficient associated with the projectile nose shapes.

From Eq.(10), the parameter κ as a function of the cavitation number can be expressed as

Note that the following relation can be obtained from Eq.(1)

Thus from Eq.(20), Eq.(26) and Eq.(28), a new form of the time evolution of the cavity can be expressed as

Considering the approximate relationship vp= (x-x0)/(t-t0), the cavity radius in Eq.(29) can be rewritten as

If t0and x0are considered as special values of t and x, Eq.(29) and Eq.(30) describe the cavity evolution along the straight trajectory at different times after being impacted by projectiles. Eq.(29) and Eq.(30) indicate that the cavity radius only depends on the projectile nose shape, the drag coefficient, the cavitation number and the value of N. This means that the projectiles of two different lengths consideredin this article will have the same cavity dimensions in the early water entry process before a deep pinching off occurs. If the drag coefficient Cdis chosen a value of 0.835, the corresponding cavitation number σ is equal to 0.0183 and N is assigned a value of 1.4, the calculations from formula (30) agree well with the experimental results as shown in Fig.12.

Fig.12 Comparison between results of the cavity model and the experimental data

From the cavity model, the maximum cavity radius along the penetration distance can be estimated as

The trends of the maximum cavityradius along the distance for the two projectiles of different lengths are shown in Fig.13. Though it is just a qualitative evaluation, it can be found that the maximum cavity radius is the maximum in the water-entry entrance and decreases monotonically with the penetration distance.

Fi g.13 Qualitative analysis of the maximum cavity radius along the distance for the two projectiles

4. Conclusions

In the present study, the horizontal water entry behaviors of cylindrical projectiles are studied theoretically and experimentally. Particular attention is paid to the projectile dynamics and the cavity dimensions before a deep pinching occurs. Based on the assumption of the constant pressure difference between the surrounding fluid and the cavity and the velocitydependent drag coefficient in the water process, a set of projectile dynamic velocity-dependent equations based on the unsteady potential flow theory are derived and a new cavity model is built to describe the projectile dynamics of the water entry. A parameter in the cavity model is discussed and determined through the principle of energy conservation. From the study, the following conclusions can be drawn:

(1) The flat projectiles enjoy a goo d underwater trajectory stability and strong velocity attenuations. Projectiles with different lengths have a distinct law of velocity attenuation.

(2) The drag coefficient and the velocity decay coefficient dependent on the cavitation number increase along the penetration distance but the variation is relatively small.

(3) The maximum cavity radius decreases monotonically with the penetration distance.

(4) The cavity dimensions for the same projectile of different initial velocities are basically consistent in the early water entry process before a deep pinching off occurs.

(5) The analytical results agree well with the experiments.

However, it should be noted that the accuracy of the cavity model and the attenuation equations that describe the underwater motion of projectiles are both very sensitive to the value of the cavitation number. For high velocity water-entry problems currently under discussion, the chosen values of the cavitation number are shown to be effective.

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September 4, 2011, Revised December 9, 2011)

* Biography: GUO Zi-tao (1979-), Male, Ph. D. Candidate Corresponding author: ZHANG Wei,

E-mail: Zhdawei@hit.edu.cn