ACTA MECHANICA SINICA (English Series), Vol.17, No.l, February 2001 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A.

ISSN 0567-7718

H Y D R O D Y N A M I C B E H A V I O R OF A N U N D E R W A T E R M O V I N G B O D Y A F T E R W A T E R ENTRY* Shi Honghui ( ~ )

(Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan) Takuya Takami ( i ~ i ~ $ t k )

(Space Engineering Division, Space Engineering Development Co., Ltd., Issei Build., Takezono 2-3-12, Tsukuba 305-0032, Japan) An experimental study was conducted to investigate the water entry phenomenon. A facility was de.signed to carry out the tests with the entry velocities of around 352 m/s. Visualization, pressure measurement, velocity measurement and underwater impact test were performed to investigate the hydroballistic behavior of the underwater moving body, the underwater flow field, the supercavitation, etc.. This study shows that the motion of a high-speed underwater body is strongly three-dimensional and chaotic. Furthermore, it is found that the distribution of the trajectory deflection of the underwater projectile depends on the depth of water. It is also found by measuring the deformation on a witness plate submerged in water, that the impact energy of an underwater projectile is reduced as it penetrates deeper into water. ABSTRACT:

K E Y W O R D S : water entry, supercavitation, underwater acoustic wave, underwater

projectile trajectory, projectile velocity

1 INTRODUCTION The study of water entry of a high-speed blunt solid body and its underwater motion has wide applications in industries, natural sciences and defense technologies, such as, the landing of the space vehicle and satellites on the sea surface, slamming between a high-speed ship and a water surface, the formation of the earth geometry due to the impact of meteorites with the sea, supercavitation around the rotating blade of hydraulic machines, torpedo and submarine launched ballistic missiles. Most recently, it has found applications in the penetration mechanics in the space study, where the penetration technology (sampling return to a space ship) is critical to explore whether water would exist at the Mar's underground. Because of the importance of the study, its history could be traced back to one century ago (Worthington and Cole, 1900)[ 1]. Received 28 February 2000, revised 5 June 2000 * The project supported by Japan Society for the Promotion of Science under a Grant-in-Aid for Scientific Research (C) (Grant No. 12650162)

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From 1948 to 1952, May published three important papers to try to find the drag coefficient and the scaling relationship in water entry of blunt solid bodies by considering the Reynolds and Froude numbers [2~4]. Using high-speed photography, May" found that the cavity behind the underwater projectile experienced the surface closure and the deep closure [4]. May's work was continued by Abelson in 1970 [5] and 1971D1, which indicated the importance of the variation of the cavity pressure and the cavity shape. Similar to the approach of finding a scaling relationship, Eroshin et al. [7] in 1988 proposed an experimental method of using a liquid with a low speed of sound so that the impact Mach number of the projectile w&s changed. Separately, the rapid growth of the ship buihling industry in the 1960's stimulated the research on the force of impact on blunt solid bodies in water entry is~l~ Then in the 1980's, the results of experimental measurements and theoretical analyses of the force in water-solid impact were still being reported [11'12]. For a two-dimensional or axisvmmetric water entry problem, the cavity shape and the drag coefficient can be well predicted (Kate, 1979)B3]. However, when the flow field becomes three-dimensional, there are no theoretical models available yet. This paper shows how the 3-D effect appears and why" it is necessary to take the 3-D effect into account in dealing with the water entry phenomenon. This paper presents an examination of the trajectory of the underwater projectiles and the other measurements of the flow field induced by the projectiles. The optical observation of the three-dimens~onal supercavitation has been made after the underwater trajectory deflection of the projectile occurs, which will be published in next paper. 2

EXPERIMENT"

Figure 1 shows briefly the experimental set-up [14J51. A projectile was fired downward from an Anschutz rifle (made in Germany). The projectile is 5.7ram in diameter, 12.3mm in total length, 2.67 g in mass and about 352 m/s in velocity. The water tank of 60 cm x 60 c m x 80 cm is made from 5 mm thick stainless plates. The water tank has two windows at each side for optical observation. The impact velocity before the water entry was measured by cutting two laser beams as shown in Fig.1 (a). As the projectile moves downward

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Shi & Takami: Hydrodynamic Behavior of an Underwater Moving Body

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through the laser beams, the photoelectric currents of the two photodiodes are shut off. Using a digital oscilloscope, the time interval between the photovoltage drops of the two photodiodes is measured and the velocity of the projectile is obtained. The measured results were calibrated by means of the method of cutting two electric wires. The projectile velocity in water after the entry was measured step by step with moving the two laser beams downwards (Fig.l(b)). Using an Xenon flash (NP-1A, 180ns exposure time, Sugahara Research Laboratory), the transient sequence of the entry, the motion of the underwater projectile, the splash and the cavity were photographed. The underwater acoustic wave caused by the impact of the projectile on the free surface was recorded by a submerged pressure transducer (Kistler 6205A). An application of the Kistler transducer in measuring the transient pressure pulse in water has been given by Shi and Itoh [16]. 3 RESULTS

May [4] presented photographic results of the sp]ash, the surface seal of the cavity, the formation of the jets following water entry from 9.75m/s to 27.52m/s. The phenomena that he photographed are among the most interesting things in water entry. The time for the surface seal of the cavity depends on the entry velocity. When the entry velocity is high, the surface seal occurs later on or even may not be observed in the observable distance of the experiment. Figure 2(a) is the sequence just before the projectile impacts on the water surface. Soon after the entry as shown in Fig.2(b), the lateral jets flowing along the free surface are formed. Later, the jets go upward to form a splash as shown in Fig.2(c). This process was observed by Shi [17] using high-speed photography o f a 95 m/s nylon ball entering water. As the projectile penetrates into deeper water, the splash slows down its speed of moving upward (Figs.2(d),--2(f)). Figures 2(g) and 2(h) are closer views by moving

Fig.2 Transient photographs of the water entry of 352 m/s projectiles at different stages of penetration. (a)~(f) are in the same scale. (g) and (h) are closer views taken by the camera which was put closer to the water tank. The cavitating streaks are visible on the wall of the cavity shown in (g) and (h). The streaks may also be composed of condensing droplets due to the negative pressure inside the cavity

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the cmnera closer to the water tank. In both cases of Figs.2(h)--~2(g), the projectiles have traveled about 60 m m from the surface. Helical cavitation streaks are found being formed on the wall surface of the cavity. Figure 3 gives the measured signals of the underwater acoustic wave at the position of 5 2 m m deep and 9 0 m m from tile impact center. The acoustic wave was generated by the impact of the projectile with the water surface. The over-pressure of the acoustic wave front is 1.4 M P a which is much less than the impact pressure on the surface, that is, P = p C V = 528 M P a if the projectile is considered to be rigid. Here p , C and i" are liquid density, acoustic velocity and impact velocity, respectively. Therefore, it is understood that the initial shock wave generated by the impact experiences a severe deceleration as it propagates into water because of the divergence of its energy and the effect of the expansion waves (Korobkin, 1992) [ls].

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Figures 4 and 5 show two examples of the velocity measurements of the underwater projectile using laser beams. It must be remembered that in front of the projectile, there is an underwater acoustic wave which has a sufficient strength (see Fig.3) and moves much faster t h a n the projectile. When the acoustic wave crosses the laser beams, it deflects the laser beams to cause the photovoltage drops (trigger points A and B in Fig.4). However, since the acoustic wave is very thin, when it passes the laser beams, the photovoltages of the photodiodes recover completely. On the other hand, when the underwater projectile crosses the laser beams, they are shut off totally so that the photovoltages drop to the bottom. In Fig.4, the trigger points of C amt B are at the position of half value of the each photovoltage. From the time delay between .4 and B, the acoustic wave velocity is measured as 1500 m / s and from the time delay between C and D, the velocity of the underwater projectile is measured as 256 m/s. The meanings of A, B, C and D in Fig.5 are the same as those in Fig.4. In the case of Fig.5, the acoustic wave velocity is measured as 1 5 1 5 m / s and the velocity of the underwater projectile is measured as 290 m/s. The reliability of the laser cutting method for velocity measurements can be verified by referring the standard value of the acoustic velocity of 1 500 m/s. It is known from the measured velocity of the acoustic wave that the measurement error is only 1%. F i g u r e 5 gives the results at the water depth of 174mm while Fig.4 gives the results at the water depth of 275 mm. If all the data are put into a velocity-water depth diagram, the relationship between the underwater projectile velocity and the depth of water can be obtained (Fig.6). The experimental data have been correlated with an expression of Vp = 352 e x p ( - 1 . 2 0 Z ) m / s

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Voi.17, No.1

Shi & Takami: Hydrodynamic Behavior of an Underwater Moving Body

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where Z is the d e p t h o f water in millimeter. It is seen t h a t the projectile is decelerated after water entry. Shi [17] has found t h a t for nylon balls, the initial high i m p a c t pressure P = pCV on the water surface m a y lead to a 10%.--15% reduction of the projectile velocity. For the 2.67g, 3 5 2 m / s lead projectile used in this experiment, the initial impact pressure on the surface m a y have a minor effect on the velocity reduction, while the effect of the drag in water Co = C o ( 0 ) ( 1 3- a) (Kate, 1979)[ 13] plays a m a j o r role ill the deceleration, where C O is the drag coefficient, CD(O) the drag coefficient w i t h o u t cavitation and a the cavitation number.

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The most important result of this paper is the examination of the trajectory of the underwater projectile. A 0.5 mm thick copper plate was put on the top of a box containing clay which was submerged at 510 mm deep ill water. The trajectory is determined from the impact position on the witness plate. Figure 7(b) is the photo of the impacted 0.5 mm copper plate. Figure 7(a) is a schematic ~]rawing where S represents the water depth. In Fig.7(b), the central hole in the photo was benetrated when the tank was empty. When the tank was filled with water, all the trajectories of the 10 projectiles were deflected from the center. This fact tells us that the supercavitation and the boundary layer aromld the projectile behave strongly three-dimensionally and chaotically, and exert a horizontal force on the projectile to cause it deflected (Fig.7(a)). Watanabe et al. [19] have nulnericaily indicated that the unstable vortices at the wake region of the projectile after it c,vertakes a shock wave in air, produce asymmetric forces to the projectile. The flow field around a high-speed underwater projectile is much complicated than that predicted by the gasdynamics of a projectile in air. When a 3-D phenomenon is concerned, the drag coefficient of Cr) = CD(0)(I + a) needs to be modified. The question is how to modify the 2-D fornmla into a 3-D one. To do this, more experimental results are required. A 3-D underwater supercavitation is often accompanied by the unsteadiness of the flow field.

Fig.7 Illustration of the experiment of finding the underwater trajectories of the projectiles. (a) Impact situation; (b) Impacted 0.5 mm thick copper plate when S = 510 mm. The white arrow shows the position of the central axis shown in (a) The underwater projectiles shown in Figs.2(a),,-2(h) are still in a vertical direction, because the projectiles penetrate into water for only about 60turn where the trajectory deflection has not made evident. This can be confirmed by an experiment of examining the impact sites on the witness plate at different water depth. Figures 8(a)--,8(f) show the distributions of the underwater impact sites from the water depth of S = 100mm to the depth of S = 600mm. At every water depth of Figs. 8(a),--8(f), 10 impacts of 3 5 2 m / s entry velocity were made. Obviously, it is found that the trajectory deflection occurs at a 200 m m water depth (Fig.8(b)). Then the deeper the water, the more deflected is the trajectory. This paper quantitatively indicates the dependence of the underwater trajectory on the

Vol.17, No.1

Shi & Takami: Hydrodynamic Behavior of an Underwater Moving Body I

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(Young and Holl)[2~ (3) the deep closure of the cavity and reentrant jet (Knapp et al. [2fl, Wade and Acosta[22]). From the observed helical streaks on the cavity wall in Figs.2(g) and 2(h), it is considered that the rotation of the projectile in water is not so strong. The rotation effect of the supercavitation is limited. This may be attributed to the fact that the drag in water is much greater than that in air. For the third possibility, the theoretical calculation [23] of the 352 m / s water entry shows that the deep closure of the cavity occurs at a water depth of 650 mm whereas Fig.8 shows that the deflection starts from a water depth of 200mm. Therefore, the deflection is most possibly caused by the 3-D flow field. Further experiments to observe the cavity's shape along a deflected trajectory of the underwater projectile has been carried out. The experimental data of Fig.8 are collected me a relationship between the deflection and the depth of water in Fig.9. An empirical formula is given by

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where T is the deflection in mm and Z is the depth in mm. The plus symbol in Eq.(2) corresponds to the upper solid line in Fig.9. The minus symbol in Eq.(2) corresponds to the lower solid line in Fig.9. Figure 10 shows the depth of depression of the 0.5 mm copper plate caused by the impact of the underwater projectile. The three vertical arrows in the figure mean that the plates are perforated. It is found that the depression depth decreases with the increase of the depth of water, which is in agreement with the results of the velocity measurement (see Eq.(1)). The depression depth starts to be largely reduced from 200,~300 mm depth. This distance roughly corresponds to the water depth where the trajectory deflection begins (Fig.8).

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AND DISCUSSION

In an incompressible flow, the stagnation point and streamlines around a blunt body can be described by Bernoulli's equation[ 24]. If the moving velocity of an underwater blunt body is very high, the fluid compressibility, the body deformation and the shock wave in the solid must be considered and G.I. Taylor's theory [2s] can still be used to describe the

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hydrodynamics. One aspect, which was not considered by G.I. Taylor, is t h a t cavitation at the interface between a solid body and the surrounding medium must occur if the interface's moving velocity is high. The cavitation will surely impose an extra force on the body. The second aspect is t h a t the turbulent boundary layer develops along the body's surface and the vortices detach from the body at the wake region. The boundary layer is a two-phase flow consisting of bubbles and droplets, which may transform the m o m e n t u m to the body under the convection of the vortices. With these effects, the pressure distribution on the solid body is unlikely to keep symmetric. In the experiment of the entry of solids into water by Gilbarg and Anderson in 1948[ 26] , they photographed the deflection of the trajectory of 10.67m/s for 6.35 m m diameter steel sphere and 15.24 m / s for 6.35 m m diameter duralumin cylinder in water. For both projectiles, the Reynolds numbers are close to 105 . Probably they thought that the deflection happened accidentally. At the time of sixty years ago, peoples' understandings to cavitation and turbulence were not sufficient, so that Gilbarg and Anderson did not explain why the trajectory was deflected. The modern fluid mechanics has revealed the complexity of cavitation and the statistical chaos of turbulence. For the problem that we studied in this paper, it is time to re-think the hydrodynamics of the phenomenon. It is now clear that much more information, such as the distribution of fluid velocity, the boundary-layer structure or the turbulence, the shape of the 3-D cavity are needed.

REFERENCES 1 Worthington AM, Cole RS. Impact with a liquid surface studied by the aid of instantaneous photography (part II). Phil Trans Roy Soc London Set A, 1990, 194:176~199 2 May A, Woodhull JC. Drag coefficients of steel spheres entering water vertically. J Appl Phys, 1948, 19:1109~1121 3 May A. Effect of surface condition of a sphere on its water-entry cavity. J Appl Phys, 1951, 22: 1219~1222 4 May A. Vertical entry of missiles into water. J Appl Phys, 1952, 23:1362,~1372 5 Abelson HI. Pressure measurements in the water-entry cavity. J Fluid Mech, 1970, 44:129~144 6 Abelson HI. A prediction of water-entry cavity shape. Trans ASME, Journal of Basic Engineering, 1971, 93:501,~504 7 Eroshin VA, Konstantinov GA, Romanenko NI, Yakimov YuL. Experimental determination of the pressure on a disk entering a compressible fluids at an angle to the free surface. Izv Akad Nauk SSSR, Mekh Zhid I Gaza, 1988, (2)(March-April): 21~25 8 Verhagen JHG. The impact of a fiat plate on a water surface. J Ship Resh, 1967, 11:211,~223 9 Lewison G, Maclean WA. On the cushioning of water impact by entrapped air. J Ship Resh, 1968, 12:116~130 10 Koehler BR, Kettleborauh CF. Hydrodynamic impact of a falling body upon a viscous incompressible fluid. J Ship Resh, 1977, 21:165~181 11 Moghisi M, Squire PT. An experimental investigation of the initial force of impact on a sphere striking a liquid surface. J Fluid Mech, 1981, 108:133~146 12 Cointe R. Two-dimensional water-solid impact. Trans ASME, Journal of O]~shore Mechanics and Arctic Engineering, 1989, 111:109~114 13 Kato Y. Cavitation. Tokyo: Maki Bookshop, 1979. 161,~162 14 Shi HH, Takami T, Itoh, M. Trajectory deflection of underwater high-speed moving body by three-dimensional supercavitation. In: Proc. 35th Joint Meeting of Chubu & Kansai Branches of JSASS, Nagoya, Japan, 1998. 35~36

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15 Takami T, Shi HH, Itoh M. Entry of fast blunt body into water. In: Proc. 47th General Meeting of Tokai Branch of JSME. Nagoya, Japan, 1998. 157,~158 16 Shi HH, Itoh M. Gener~;tion of high-speed liquid jet from a rectangular nozzle. Trans Japan Soc Aero Space Sci, 1999, 42:195~202 17 Shi HH. Fast water entry of blunt solid projectile. In: Proc. 74th JSME Spring Ann, Meeting, Tokyo, 1997, 6 : 1 ~ 4 18 Korobkin A. Blunt-body impact on a compressible liquid surface. J Fluid Mech, 1992, 244: 437~453 19 Watanabe R, Fujii K, Higashiura F. Computational analysis of the unsteady flow induced by a projectile overtaking a preceding shock wave. Trans Japan Soc Aero Space Sci, 1998, 41:65~73 20 Young JO, Holl JW. Effect of cavitation on periodic,wakes behind symmetric wedges. Trans ASME, J of Basic Engng, 1966, 88:163~176 21 Knapp RT, Daily JW, Hammitt FG. Cavitation. New York: McGraw-Hill, 1970. 153~217 22 Wade RB, Acosta AJ. Experimental observations on the flow past a plaslo-convex hydrofoil. Trans ASME, J of Basic Engng, 1966, 88:273~283 23 Shi HH, Takami T. Some progress in the study of water entry phenomenon. Experiments in Fluids, 2001, 30 (in print) 24 Ito H, Honda H. Fluid Mechanics. Tokyo: Maruzen Co., Ltd., 1981. 24,,~25 25 Taylor GI. The use of flat-ended projectiles for determining dynamic yield stress. Proc Roy Soc London Set A, 1948, 194:289~299 26 Gilbarg D, Anderson RA. Influence of atmospheric pressure on the phenmena accompanying the entry of spheres into water. J Appl Phys, 1948, 19:127~139