Abstract

The two-dimensional nonlinear problem of steady flow with obstruction beneath an ice sheet is considered. The mathematical model of the flow is based on the velocity potential theory with fully nonlinear boundary conditions on the ice/liquid interface and on the nonlinear Cosserat plate model for an ice sheet, which are coupled throughout a numerical procedure which provide the same pressure distribution on the interface from the liquid and the elastic ice sheet sides. The integral hodograph method is employed to derive analytical expressions of the complex potential and the complex velocity of the flow both as functions of a parameter variable. The problem is reduced to a system of integral equations which are solved using the method of successive approximation and the collocation method. Case studies are conducted for a body submerged beneath the interface in the infinitely deep liquid and for the obstruction located on the bottom of the finite depth channel. For each case, both subcritical and supercritical flow regimes are studied. Results for interface shape, bending moment, and pressure distribution are presented for the wide ranges of Froude numbers and depths of submergence. In the case of infinite depth fluid, the dispersion equation predicts two waves of different lengths which may exist on the interface. The first longest wave is that caused by gravity located downstream of the body, and the second shorter wave is that caused by the ice sheet and is located upstream of the body. They exhibit a strongly nonlinear interaction above the submerged body near the critical Froude number such that occurs some range of submergences in which the solution does not converge. It is different in the case of the finite depth channel. The two waves may exist in the range of depth-based Froude Fcr