Asymptotic estimates of hydrodynamic loads in the early stage of water entry of a circular disk

Abstract

The hydrodynamic loads generated during the entry of a circular disk into deep water are evaluated with the help of the method of matched asymptotic expansions. It is assumed that the liquid is initially at rest and the disk is floating on the still liquid surface. Then the disk suddenly starts its downward motion. The study is carried out under the assumption of an ideal and incompressible liquid. Attention is focused on the initial stage of the entry process. The solution is sought in the form of an asymptotic expansion of the velocity potential with the non-dimensional displacement of the disk being a small parameter of the problem. Gravity and surface-tension effects are shown to be of minor significance. Owing to the flow singularity at the edge of the disk, an inner problem is formulated and its solution is matched with the second-order outer velocity potential to achieve a uniformly valid solution. It is shown that the initial asymptotics of the hydrodynamic loads involves terms with \({h^{-\frac{1}{3}}}\) and log h where h(t) is the non-dimensional displacement of the disk. Both terms are unbounded in the limit of small penetration depth of the disk. The theoretical estimates are validated versus fully nonlinear numerical simulations of the problem during the later stage of the process. It is shown that the derived asymptotic estimates remain accurate, even for moderate displacements of the disk. The relative difference between the theoretical estimate of the hydrodynamic force and its numerical prediction is less than 5% when the penetration depth is smaller than \({\frac{1}{20}}\) of the disk radius. A way to use the theoretical estimates for practical applications is proposed and comparisons with experimental data available in the literature are also presented.