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

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

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

Flexural-gravity wave interaction with multiple vertical cylinders of arbitrary cross section frozen in an ice sheet

The interaction problem of flexural-gravity wave with multiple vertical cylinders frozen in an ice sheet on the surface of water with finite water depth is considered. The linearized velocity potential theory is adopted for fluid flow, and the thin elastic plate model is applied for ice sheet deflection. Each cylinder is bottom-mounted, and the shape of its cross section can be arbitrary while remaining constant in the vertical direction. The velocity potential is expanded into an eigenfunction series in the vertical direction, which satisfies the boundary condition on the ice sheet automatically. The horizontal modes, which satisfy the Helmholtz equations, are then transformed into a series of boundary integral equations along the ice sheet edges or the intersection of the ice sheet with the cylinders. The problem is then solved numerically by imposing the ice sheet edge condition together with the impermeable condition on the cylinders. The solution is exact in the sense that the error is only due to numerical discretization and truncation. Computations are first carried out for single and multiple vertical circular cylinders, and good agreements are obtained with the semi-analytical solution. To resolve the difficulty of excessive computation at a large number of cylinders, the effect of the evanescent wave of a cylinder on those at large distance is ignored. This allows for the case of a large number of cylinders in different arrangements to be simulated. Extensive results are provided. Their physics and practical relevance 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

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

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