## 1 Introduction  
To improve the penetration and destructive capabilities of projectiles, anti-armor weapons are widely equipped with rod projectiles with large length to diameter ratio (L/D ratio), strong anti-interference ability, good flight stability, high energy density and strong armor-piercing ability [1], whereas underwater weapons mostly adopt explosively formed projectiles (EFPs) [2, 3]. As the warship broadside can be easily attacked by anti-ship weapons due to its large area, liquid cabin structures are often set on the broadside to protect the inner structure against penetration or armor-piercing damage caused by fragments and debris generated by the warhead shell and the outer plate of the cabin [4]. However, before penetrating into the liquid cabin, the projectile inevitably interacts with the stiffened plate and other obstacles in the empty cabin, changing the ballistic characteristics such as attitude angle of the projectile or causing asymmetric deformation of the projectile, and thus affecting its penetration capability [5]; when attacking underwater targets, torpedoes and projectiles usually enter the water at an attack angle; thus, it is of great significance to study the non-ideal water entry of rod projectiles for warship protection. Generally, the water entry of projectiles can be divided into four stages: (1) Impact stage. As a projectile impacts the water at a high speed, causing a large impact force on the projectile, the projectile nose is prone to mushrooming deformation, and meanwhile the impact by the projectile leads to the formation of high-speed shock waves in the water, which propagates in a semicircle [6]. (2) Flow formation stage. After being impacted, the water will be separated by the projectile and flow away from its sides as the projectile moves forward, resulting in cavitation. Whip can be easily observed in this stage, where the pressure on the projectile is far less than that in the impact stage. (3) Open cavity stage. The cavity generated after the water entry of the projectile does not disappear immediately. It is still connected with the air at the water surface. As the projectile moves, the cavity gradually expands. When the projectile moves forward in the cavity, the uneven contact between the projectile nose and the water cause projectile swinging, resulting in ballistic instability. At the same time, due to tail swing, the tail contacts the cavity, causing tail flap, which further increases the uneven force on the projectile, leading to greater yaw [7,8]. (4) Closed cavity stage. After formation for some time, the cavity closes at a point on or below the water surface. Cavity closure may generate jet. When the jet strikes the cavity wall and causes its deformation, or hits the projectile and changes its trajectory, the projectile will detach from the cavity, leading to a significant change in the direction of projectile movement [9,10]. At present, the water entry problem of projectiles is generally studied from two perspectives: a projectile penetrating the free water, such as the water entry of aerial torpedo and spacecraft; a projectile penetrating the liquid tank, such as the penetration of a projectile into an aircraft fuel tank, and fragments penetrating the broadside liquid cabin. Researchers have conducted extensive research on the penetration of rod projectiles under ideal conditions.