Unsteady hydrodynamic forces of solid objects vertically entering the water surface

Abstract

We investigate the unsteady hydrodynamic force of solid objects vertically entering water with an air cavity behind the falling body. Physical models are proposed to represent the force components corresponding to the body acceleration, the gravity and the velocity of the body and the fluid particles. The theoretical or numerical solutions of the physical models are presented to understand the evolution of the force components. The body-acceleration force component is expressed as the high-frequency added mass times the body acceleration. Near the undisturbed free surface, the added mass grows strongly with increasing the submerged depth. It tends to be steady after the submerged depth is greater than a few characteristic lengths. The gravity force component consists of an upward hydrostatic term and a downward dynamic term. Generally, the hydrostatic term, which is obtained by integrating the gravity term in the Bernoulli's equation over the wetted body surface, is much larger than the gravity force component. For the threedimensional bodies, the gravity force component is found to vary as a power of the submerged depth, where the exponent is about 0.83. The velocity force component is represented as the drag coefficient defined by the V-squared law, which is characterized by the body geometry. The drag coefficient may experience three successive stages with increasing the submerged depth.

Gravity effects in two-dimensional and axisymmetric water impact models

“…Fully nonlinear potential flow solvers are less computationally demanding than CFD solvers while remaining at a high level of accuracy as the boundary conditions on the body and on the free surface are fully nonlinear. This approach has been used to investigate different water entry problems affected by gravity (Sun, Sun & Wu 2015;Yu et al 2018;Wang, Faltinsen & Lugni 2019).…”

“…This approach has been used to investigate different water entry problems affected by gravity (Sun, Sun & Wu 2015; Yu et al. 2018; Wang, Faltinsen & Lugni 2019).…”

Impact force reduction by consecutive water entry of spheres

“…One such example is shown in figure 1( a ), where a 50 mm sphere dropped from 0.72 m above the free surface results in an impulse with a peak impact acceleration of 8 g whereas the underwater acceleration is close to a constant value of 2 g (1 c ), indicating that at the moment of surface penetration the drag coefficient is four times the steady state underwater free fall. This initially high impact force is primarily due to the large rate of change of momentum of the added fluid mass (May 1975; Wang, Lugni & Faltinsen 2015; Wang, Faltinsen & Lugni 2019), which is the highest during a submergence depth of 10 %–20 % of the radius for spheres (figure 1 a ) (Shiffman & Spencer 1945; Moghisi & Squire 1981). Reducing this peak impact force is of significant interest because it presents structural failure risk to impinging bodies like aircraft landing on water, water landing spacecraft, underwater missiles, divers, base jumpers, etc.…”

Water impact of a surface-patterned disk

“…These complex fluid behaviours are closely related to the unsteady hydrodynamic force exerted on an impacting solid object (Truscott, Epps & Techet 2012; Wang, Faltinsen & Lugni 2019). Since the pioneering work of Wagner (1932), there have been significant efforts to analyse the force that arises shortly after the free-surface impact.…”