Hydrodynamic force on a ship floating on the water surface near a semi-infinite ice sheet

The hydrodynamic problem of wave interaction with a ship floating on the water surface near a semi-infinite ice sheet is considered based on the linearized velocity potential theory for fluid flow and the thin elastic plate model for ice sheet deflection. The properties of an ice sheet are assumed to be uniform, and zero bending moment and shear force conditions are enforced at the ice edge. The Green function is first derived, which satisfies both boundary conditions on the ice sheet and free surface, as well as all other conditions apart from that on the ship surface. Through the Green function, the differential equation for the velocity potential is converted into a boundary integral equation over the ship surface only. An extended surface, which is the waterplane of the ship, is introduced into the integral equation to remove the effect of irregular wave frequencies. The asymptotic formula of the Green function is derived and its behaviors are discussed, through which an approximate and efficient solution procedure for the coupled ship/wave/ice sheet interactions is developed. Extensive numerical results through the added mass, damping coefficient and wave exciting force are provided for an icebreaker of modern design. It is found that the approximate method can provide accurate results even when the ship is near the ice edge, through which some insight into the complex ship/ice sheet interaction is investigated. Extensive results are provided for the ship at different positions, for different ice sheet thicknesses and incident wave angles, and their physical implications are discussed

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

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

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

Interaction of ocean wave with a harbor covered by an ice sheet

A domain decomposition method is developed to solve the problem of wave motion inside a harbor with its surface covered by an ice sheet. The shape of the horizontal plane of the harbor can be arbitrary while the sidewall is vertical. The entrance of the harbor is open to the sea with a free surface. The linearized velocity potential theory is adopted for fluid flow, and the thin elastic plate model is applied for the ice sheet. The domain is divided into two subdomains. Inside the harbor, the velocity potential is expanded into a series of eigenfunctions in the vertical direction. The orthogonal inner product is adopted to impose the impermeable condition on the harbor wall, together with the edge conditions on the intersection of the harbor wall and the ice sheet. In the open sea outside of the harbor, through the modified Green function, the velocity potential is written in terms of an integral equation over the surface of the harbor entrance, or the interface between the two subdomains. On the interface, the orthogonal inner product is also applied to impose the continuity conditions of velocity and pressure as well as the free ice edge conditions. Computations are first carried out for a rectangular harbor without the ice sheet to verify the methodology, and then extensive results and discussions are provided for a harbor of a more general shape covered by an ice sheet with different thicknesses and under different incident wave angles

Wave motions due to a point source pulsating and advancing at forward speed parallel to a semi-infinite ice sheet

The Green function, or the wave motion due to a point source pulsating and advancing at constant forward speed along a semi-infinite ice sheet in finite water depth is investigated, based on the linear velocity potential theory for fluid flow and thin elastic plate model for the ice sheet. The result is highly relevant to the ship motions near marginal seas. The ice edge is assumed to be free, or zero bending moment and shear-force conditions are used, while other edge conditions can be similarly considered. The Green function G is derived first through the Fourier transform along the direction of forward speed and then by the Wiener-Hopf technique along the transverse direction across both the free surface and ice sheet. The result shows that in the ice-covered domain, G can be decomposed into three parts. The first one is that upper ocean surface is fully covered by an ice sheet, and the second and third ones are due to the free surface and ice edge. Similarly, in the free-surface domain, G contains the component corresponding to that the upper water surface is fully free, while the second and third ones are due to the ice sheet and ice edge. In both domains, the latter two are due to the interactions of the free-surface wave and ice sheet deflection, which leads to the major complication. In-depth investigations are made for each part of G, and aim to shed some light on the nature of the wave motions induced by a ship advancing along a semi-infinite ice sheet at constant forward speed

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

Motion of a floating body in a harbour by domain decomposition method

A three-dimensional domain decomposition method is used to solve the problem of wave interaction with a ship floating inside a harbour with arbitrary shape. The linearized velocity potential theory is adopted. The total fluid domain is divided into two sub-ones: one for the harbour and the other for the external open sea. Boundary integral equations together with the free surface Green function are used in the both domains. Matching conditions are imposed on the interface of the two sub-domains to ensure the velocity and pressure continuity. The advantage of the domain decomposition method over the single domain method is that it removes the coastal surface from the boundary integral equation. This subsequently removes the need for elements on the coastal wall when the equation is discretized. The accuracy of the method is demonstrated through convergence study and through the comparison with the published data. Extensive results through the hydrodynamic coefficients, wave exciting forces and ship motions are provided. Highly oscillatory behaviour is observed and its mechanism is discussed. Finally, the effects of incident wave direction, ship location as well as the harbour topography are investigated in detail