Interaction of wave with a body floating on a wide polynya

A method based on wide spacing approximation is proposed for the interaction of water wave with a body floating on a polynya. The ice sheet is modelled as an elastic plate and fluid flow is described by the velocity potential theory. The solution procedure is constructed based on the assumption that when the distance between two disturbances to the free surface is sufficiently large, the interactions between them involve only the travelling waves caused by the disturbances and the effect of the evanescent waves is ignored. The solution for the problem can then be obtained from those for a floating body without an ice sheet and for an ice sheet/free surface without a floating body. Both latter solutions have already been found previously and therefore there will be no additional effort in solution once the wide spacing approximation formulation is derived. Extensive numerical results are provided to show that the method is very accurate compared with the exact solution. The obtained formulations are then used to provide some insightful explanations for the physics of flow behaviour, as well as the mechanism for the highly oscillatory features of the hydrodynamic force and body motion. Some explicit equations are derived to show zero reflection by the polynya and peaks and troughs of the force and excited body motion. It is revealed that some of the peaks of the body motion are due to resonance while others are due to the wave characters in the polynya

Interaction of wave with a body submerged below an ice sheet with multiple arbitrarily spaced cracks

The problem of wave interaction with a body submerged below an ice sheet with multiple arbitrarily spaced cracks is considered, based on the linearized velocity potential theory together with the boundary element method. The ice sheet is modeled as a thin elastic plate with uniform properties, and zero bending moment and shear force conditions are enforced at the cracks. The Green function satisfying all the boundary conditions including those at cracks, apart from that on the body surface, is derived and is expressed in an explicit integral form. The boundary integral equation for the velocity potential is constructed with an unknown source distribution over the body surface only. The wave/crack interaction problem without the body is first solved directly without the need for source. The convergence and comparison studies are undertaken to show the accuracy and reliability of the solution procedure. Detailed numerical results through the hydrodynamic coefficients and wave exciting forces are provided for a body submerged below double cracks and an array of cracks. Some unique features are observed, and their mechanisms are analyzed

Hydroelastic waves propagating in an ice-covered channel

The hydroelastic waves in a channel covered by an ice sheet, without or with crack and subject to various edge constraints at channel banks, are investigated based on the linearized velocity potential theory for the fluid domain and the thin-plate elastic theory for the ice sheet. An effective analytical solution procedure is developed through expanding the velocity potential and the fourth derivative of the ice deflection to a series of cosine functions with unknown coefficients. The latter are integrated to obtain the expression for the deflection, which involves four constants. The procedure is then extended to the case with a longitudinal crack in the ice sheet by using the Dirac delta function and its derivatives at the crack in the dynamic equation, with unknown jumps of deflection and slope at the crack. Conditions at the edges and crack are then imposed, from which a system of linear equations for the unknowns is established. From this, the dispersion relation between the wave frequency and wavenumber is found, as well as the natural frequency of the channel. Extensive results are then provided for wave celerity, wave profiles and strain in the ice sheet. In-depth discussions are made on the effects of the edge condition, and the crack

Wave diffraction by a circular crack in an ice sheet floating on water of finite depth

The problem of wave diffraction by a circular crack in an ice sheet floating on water of finite depth is considered. The fluid flow is described by the linear velocity potential theory, while the infinitely extended ice sheet is modeled as a thin elastic plate with uniform properties. At the crack, zero bending moment and shear force conditions are enforced. The solution starts from the Green function for ice sheet without the crack. This is then used to obtain an integral equation, in which the jumps of the displacement and slope across the crack are the unknowns. For a circular crack, the unknowns are expanded into the Fourier series in the circumferential direction. Through imposing the boundary conditions at the crack, a matrix equation is obtained for the unknowns, which is then truncated and solved. Convergence study is undertaken with respect to the truncation, and it has been found that the series converges fast. A far field identity is used to verify the solution procedure and is found to be satisfied very accurately. Extensive results are provided, and their physical implications are discussed. These include the jumps of the displacement and slope across the crack, resonant motion, far field diffracted wave amplitude, and the deflection of the ice sheet

Natural Modes of Liquid Sloshing in a Cylindrical Container with an Elastic Cover

Liquid sloshing and its interaction with an elastic cover in a cylindrical tank is considered. The velocity potential for the fluid flow is expanded into the Bessel-Fourier series as commonly used. An efficient scheme is then developed, which allows the plate deflection to use the same type of expansion as the potential. When these two series are matched on the interface of the fluid and the plate, the unknown coefficients in the two expansions can be easily obtained. This is much more convenient than the common procedure where a different expansion is used for the plate and upon matching each term in the series of the plate is further expanded into the series used for the potential. Through the developed method, an explicit equation is derived for the natural frequencies and extensive results are provided. The corresponding natural mode shapes and principal strains distribution of the elastic cover are also investigated. Results are provided and the underlining physics is discussed. To verify the obtained results, the problem is also solved through a different method in which the potential is first expanded into vertical modes. Another explicit equation for the natural frequencies is derived. While the equation may be in a very different form, through the residual theorem, it is found that the second equation is identical to the first one

Interactions of waves with a body floating in an open water channel confined by two semi-infinite ice sheets

Wave radiation and diffraction problems of a body floating in an open water channel confined by two semi-infinite ice sheets are considered. The linearized velocity potential theory is used for fluid flow and a thin elastic plate model is adopted for the ice sheet. The Green function, which satisfies all the boundary conditions apart from that on the body surface, is first derived. This is obtained through applying Fourier transform in the longitudinal direction of the channel, and matched eigenfunction expansions in the transverse plane. With the help of the derived Green function, the boundary integral equation of the potential is derived and it is shown that the integrations over all other boundaries, including the bottom of the fluid, free surface, ice sheet, ice edge as well as far field will be zero, and only the body surface has to be retained. This allows the problem to be solved through discretization of the body surface only. Detailed results for hydrodynamic forces are provided, which are generally highly oscillatory owing to complex wave–body–channel interaction and body–body interaction. In depth investigations are made for the waves confined in a channel, which does not decay at infinity. Through this, a detailed analysis is presented on how the wave generated by a body will affect the other bodies even when they are far apart

Wave diffraction by multiple arbitrary shaped cracks in an infinitely extended ice sheet of finite water depth

lexural-gravity wave interactions with multiple cracks in an ice sheet of infinite extent are considered, based on the linearized velocity potential theory for fluid flow and thin elastic plate model for an ice sheet. Both the shape and location of the cracks can be arbitrary, while an individual crack can be either open or closed. Free edge conditions are imposed at the crack. For open cracks, zero corner force conditions are further applied at the crack tips. The solution procedure starts from series expansion in the vertical direction based on separation of variables, which decomposes the three-dimensional problem into an infinite number of coupled two-dimensional problems in the horizontal plane. For each two-dimensional problem, an integral equation is derived along the cracks, with the jumps of displacement and slope of the ice sheet as unknowns in the integrand. By extending the crack in the vertical direction into the fluid domain, an artificial vertical surface is formed, on which an orthogonal inner product is adopted for the vertical modes. Through this, the edge conditions at the cracks are satisfied, together with continuous conditions of pressure and velocity on the vertical surface. The integral differential equations are solved numerically through the boundary element method together with the finite difference scheme for the derivatives along the crack. Extensive results are provided and analysed for cracks with various shapes and locations, including the jumps of displacement and slope, diffraction wave coefficient, and the scattered cross-section