The vertical mode method in the problems of flexural-gravity waves diffracted by a vertical cylinder

Abstract

The linear three-dimensional problem of ice loads acting on a vertical circular cylinder frozen in an ice cover of infinite extent is studied. The loads are caused by an uni-directional hydroelastic wave propagating in the ice cover towards the cylinder mounted to the see bottom in water of constant depth. There are no open water surfaces in this problem. The deflection of the ice cover is described by the Bernoulli-Euler equation of a thin elastic plate of constant thickness. At the contact line between the ice cover and the surface of the cylinder, some edge conditions are imposed. In this study, the edge of the ice plate is either clamped to the cylinder or has no contact with the cylinder surface, with the plate edge being free of stresses and shear forces. The water is of finite constant depth, inviscid and incompressible. The problem is solved by both the vertical mode method and using the Weber integral transform in the radial coordinate. Each vertical mode corresponds to a root of the dispersion relation for flexural-gravity waves. It is proved that these two solutions are identical for the clamped edge conditions. This result is non-trivial because the vertical modes are non-orthogonal in a standard sense, they are linearly dependent, the roots of the dispersion relation can be double and even triple, and the set of the modes could be incomplete. A general solution of the wave-cylinder interaction problem is derived by the method of vertical modes and applied to different edge conditions on the contact line. There are three conditions of solvability in this problem. It is shown that these conditions are satisfied for any parameters of the problem.

Diffraction of flexural-gravity waves by a vertical cylinder of non-circular cross section

“…where n V is the edge reaction, M  along the edge, see [12]. It is seen that the simply supported (8) and free edge conditions (9) are more complicated than the clamped edge conditions (7). The mixed boundary condition, in the case of ice cover is frozen on one portion of the cylinder and free on other portion, can be considered as more complicated edge condition.…”

“…The velocity potential ( , , , ) r z t  and the ice deflection ( , , ) w r t  are the solution of the boundary value problem (1) - (13). Following [9], these functions are decomposed into two parts:…”

“…correspond to the solution of the diffraction problem of hydro-elastic waves by a vertical circular cylinder (see, [9]).…”

“…Such a separation is only possible when both the water depth and the thickness of the ice sheet are constant, the bottom is horizontal and flow region has vertical solid boundaries, see [8]. Korobkin et al [9] solved the linear three-dimensional problem of hydro-elastic wave diffraction from a vertical circular cylinder by both the vertical mode method and the Weber integral transform. They proved that the two solutions are identical for the clamped edge conditions.…”

Numerical Simulation of a Polar Ship Moving in Level Ice Based on a One-Way Coupling Method

“…Experimental studies, including field tests [6][7][8] and model tests in ice tanks [9][10][11][12][13][14], are generally viewed as reliable; however, they are strict regarding the experimental facilities and methods used and are quite expensive so cannot be done easily. Analytical methods have a relatively long history from simple models to complex models [15][16][17][18][19][20]. However, they are usually based on many simplifications of body shape and ice model, which made them hard to extend to complex ship hulls and actual ice conditions.…”