Cavity dynamics in hydrodynamic ram analysis of confined containers under ballistic impacts
Introduction
Hydrodynamic ram phenomenon (HRAM) induced by a high-speed projectile penetrating a liquid-filled tank is a fascinating and extremely complex problem studied for along time. The difficulty does not only lie in difficult mathematics involved, but also in the physical complexity of the system which involves motion of a rigid body, cavity formation, two-way fluid-solid interactions, etc. The process of energy transfer in an HRAM event is generally characterized by four stages: the shock stage, the drag stage, the cavity growth and collapse stage [Ball (1985), Disimile et al., 2009]. Each stage contributes to structural damage through a different mechanism and to a different extent. In the shock phase which is produced by the initial impact of the projectile with the fluid-filled structure, a high-pressure hemispherical shock wave moves out from the impact point and propagates through the fluid. During the drag and cavitation phase, a radial pressure field is created along the displacement of the fluid from the projectile path as the energy is imparted to the fluid through projectile drag. Compared with the pressure generated during the shock stage, the fluid is accelerated gradually rather than impulsively, which will causes less intense peak pressures, but greater temporal extent (Disimile et al., 2009). Thus the physical comprehension of the hydrodynamic effects that occur during an HRAM event is essential in the civilian domain as well as for the military aircraft design, which would in fact contribute to designing better structures with respect to this particular threat.
D.Varas [2009a,b, 2011; 2012] studied the hydrodynamic ram events in metallic and composite tubes filled with water experimentally and numerically. The numerical results were compared with the experimental results to evaluate the accuracy of finite element method in performing such a complex phenomenon. (Disimile et al., 2009, Disimile et al., 2011) examined the pressure transfer mechanism in each stage of an HRAM event by using a large-scale shadowgraph technique, and developed an attenuation technique that the HRAM effect can be effectively reduced with the appropriate design of triangular bars within the water-filled tank by the destructive interference between the original pressure wave and its reflections. However, depending on the studied impact conditions and the structure features in which the HRAM is generated, the shock pressure wave may not be the most destructive phase. Artero-Guerrero et al., (2018) proposed another HRAM attenuation method by placing honeycomb panels inside the fluid-filled structure that is subjected to the HRAM phenomenon, according to the previous studies where the cavity expansion was shown as the major cause of deformation and failure in the tanks during an HRAM event when cavity size is similar to the tank size [D.Varas et al. (2012), Artero-Guerrero et al. (2014)]. Deletombe et al. (2013) performed experiments of the impact in a small closed water-filled tank and a larger hydrodynamic pool which are denoted as ‘‘confined’’ and ‘‘infinite’’ water containers by tumbling projectiles at 850 m/s. They demonstrated the effect of the tumbling of projectiles and the container size on the cavity shape and dynamics, and observed higher pressures of shorter duration during the drag stage than during the cavity growth. Nevertheless, they concluded that none of these stages could be neglected for the sizing of structures because they could both carry significant amounts of energy. Based on their work, Thomas Fourest et al., 2014, Fourest et al., 2015a, Fourest et al., 2015b took container confinement effects into account and modified the Rayleigh-Plesset equation to simulate a single bubble dynamics created by an HRAM event induced by tumbling projectile entry of a water-filled confined geometry at ballistic speeds, as similarities in bubble behaviour between HRAM and underwater explosion situations were observed in previous tank penetration/water entry experiments [Deletombe et al. (2013)].
Besides, cavity characteristics under container constraints, and on the other hand, the hydrodynamic pressure caused by the cavity expansion due to regular projectile impact are also of great interests, although the available literature is very limited with only a few sporadic articles on these issues. May (1951) studied the wall effects in vertical water entry of spheres and concluded that wall effects can't be negligible until the tank width is at least five times the maximum diameter of the cavity. Mansoor et al. (2014) examined the influence of increasing wall effects on cavity shapes and noted the formation of surface undulations along the cavity interface, which produce multiple pinch-off points. They also reported the observations of “kinked” pinch-off points and the suppression of downward facing jets in the presence of wall effects.
Comprehensively, the cavity formation and development is a complex stage that occurs during an HRAM event or a projectile water entry. Truscott et al. (2014) summarized two types of cavity-formation including the low-speed air entrainment and high-speed supercavitation based on the physical mechanism responsible for cavity creation. To the present, the effect factors on generation and development of the low-speed air-entraining cavity during water entry were extensively studied, including but not limited to the atmospheric pressure [Gilbarg and Anderson (1948)], object surface treatment [Duez et al. (2007), Aristoff et al. (2008), Aristoff and Bush (2009), Techet and Truscott (2011), Truscott and Techet (2009, 2012), Marston et al. (2012), Mansoor et al. (2014), Korkmaz and Güzel (2017), Li et al. (2018), Li, D.(2019)], or liquid properties [Sun et al. (2019), Grumstrup et al. (2007), Le Goff et al. (2013), Tan and Thomas (2018), Tan (2019)]. In other works, Duclaux et al. (2007) presented a full characterizations of the sphere cavity dynamics, Yan et al. (2009) gave a numerical estimates of cavity formation. However, the majority of these investigations into water entry by spheres or other objects focused on the low speeds, where the inertial forces and surface tension are important factors that can't be ignored. For high-speed cases, determination of the unsteady cavity evolution becomes more complex as unsteady flows are involved. By high-speed cameras, May and Albert (1952) was the first to observe the cavity motion, surface closure and deep closure phenomena induced by steel spheres entering water at about 8–40 m/s, and found that at increasing depths more energy appears to get into cavity production than that is lost by the projectile. He suggested that the difference might be due to the radial motion of the fluid from the cavity wall, as observed by Birkhoff and Caywood (1949) in their photographic studies. By employing the principle of energy conservation, Birkhoff and Isaacs (1951) proposed a vertical water entry cavity model, where it is assumed that the kinetic energy loss of the projectile equals the total energy (kinetic plus potential) in a fluid section. Lee et al. (1997) presented a method for modeling the cavity formation and collapse induced by high-speed vertical impact of a rigid projectile into water by assuming that the kinetic energy lost by the projectile equals that fed into a fluid section. Shi and Kume (2001) studied the hydrodynamic behaviors of projectiles water entry with the velocity of 342 and 352 m/s and observed the cavity expansion using an optical technique. Guo et al. (2012) investigated the cavity characteristics formed by high-speed water entry of projectiles with different nose shapes. Chen (2019), Tan (2019) studied the mechanism of the trajectory stability by analyzing the evolution of the cavity and velocity attenuation of projectiles. In addition to these notable studies, other works on high-speed water entry problems involves the influence on cavity dynamics, such as oblique angle and compressibility in Chen et al., 2018a, Chen et al., 2018b.
Most recently, Guo et al. (2020) experimentally reported the confinement effect of container size on the cavity expansion characteristics induced by high-speed regular projectiles impacting water-filled containers with different radius, but the critical container size in presence of constraint effects was not considered. On that basis, this present paper has added the experiments of high-speed projectile hitting more larger water-filled container, and takes a theoretical effort to examine the container confinement effects on cavity evolution and the hydrodynamic behaviors generated in drag and cavity stages, with the objective of providing the theoretical foundation for studying the nature of hydrodynamic ram caused by the cavity expansion along the ballistic trajectory in subsequent work.
Section snippets
Experimental set-up
Fig. 1 illustrates schematic diagram of the experimental configuration and the adopted projectile dimension.
Projectile dynamics
Considering a projectile with an initial velocity v0 penetrating into water along a straight trajectory, the projectile motion can be described by Newton's second law [Lee et al. (1997), Guo et al. (2012)]where mp and vp denotes the projectile mass and velocity, respectively. A0 is defined as the projected frontal area of projectiles, Cd represents the drag coefficient which is associated with projectile nose shapes.
Assuming Cd is a constant over the water entry process, the
Concluding remarks
This paper presents a combined theoretical and experimental investigation on the cavity dynamics in high-speed water-entry of confined containers, particular attention is given to characterizing the container confinement effects on the cavity characteristics induced by regular projectiles impacting into different radius water-filled containers. Experimental results indicate that the confinement effect on the cavity evolution in a confined container is mainly reflected in the influences on the
CRediT authorship contribution statement
Zitao Guo: Supervision, Writing - review & editing, Conceptualization, Methodology. Tuo Chen: Investigation, Software, Data curation. Wei Zhang: Supervision, Validation, Methodology. Zhongcheng Mu: Investigation, Data curation.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The present work is financially supported by the National Natural Science Foundation of China (Grant Nos.: 11562008, 11962007, 11672092), and the Natural Science Foundation of Jiangxi Province, China (Grant No. 20181BAB201020).
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