A uniform current passing bodies submerged beneath an ice sheet at critical Froude numbers

The problem of a uniform current interacting with bodies submerged beneath a homogeneous ice sheet is considered, based on linearised velocity potential theory for fluid and elastic thin plate theory for ice sheet. This problem is commonly solved by the boundary element method (BEM) with the Green function, which is highly effective except when the Green function becomes singular, and the direct solution of the BEM is no longer possible. However, flow behaviour, body force and ice sheet deflection near the critical Froude numbers are of major practical interest, such as in ice breaking. The present work successfully resolves this challenge. A modified boundary integral equation (BIE) is derived, which converts the singular Green function term to a far-field one and removes the singularity. The BIE is then imposed at infinity for additional unknowns in the far field. It is proved that the solution is finite and continuous at the critical Froude number F = Fc , where the body starts generating travelling waves, and finite but discontinuous at depth-based Froude number F = 1^± . Case studies are conducted for single and double circular cylinders and an elliptical cylinder with various angles of attack. A comprehensive analysis is made on the hydrodynamic forces and the generated flexural gravity wave profiles, and their physical implications are discussed. It is also concluded that the method developed in this paper is not confined to the present case but is also applicable to a variety of related problems when the BEM fails at the critical points.