Linearized Velocity Potential Theory Research Articles

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 wave diffraction by a vertical circular cylinder standing in a channel with an ice cover

The problem of hydroelastic wave diffraction by a surface-piercing vertical circular cylinder mounted on the bottom of an ice-covered channel is considered. The ice sheet is modelled as an elastic thin plate with homogeneous properties, while the linearized velocity potential theory is adopted to describe the motion of the fluid. The solution starts from the Green function satisfying all other boundary conditions apart from that on the body surface. This is obtained through applying a Fourier transform in the longitudinal direction of the channel and adopting an eigenfunction expansion in the vertical direction. The boundary conditions on the side walls and ice edges are imposed through an orthogonal product. Through the Green function, the velocity potential due to a surface-piercing structure with arbitrary shape can be expressed through a source distribution formula derived in this work, in which only integrals over the body surface and its interaction line with the ice sheet need to be retained. For a vertical circular cylinder, the unknown source distribution can be expanded further into a Fourier series in the circumferential direction, and then the analytical solution of the velocity potential can be obtained further. Extensive results and discussions are provided for the hydrodynamic forces and vertical shear forces on the cylinder, as well as the deflection and strain of the ice sheet. In particular, the behaviour of the solution near one of the natural frequencies of the channel is investigated in detail.

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.

Three-dimensional interaction between uniform current and a submerged horizontal cylinder in an ice-covered channel

The problem of interaction of a uniform current with a submerged horizontal circular cylinder in an ice-covered channel is considered. The fluid flow is described by linearized velocity potential theory and the ice sheet is treated as a thin elastic plate. The potential due to a source or the Green function satisfying all boundary conditions apart from that on the body surface is first derived. This can be used to derive the boundary integral equation for a body of arbitrary shape. It can also be used to obtain the solution due to multipoles by differentiating the Green function with its position directly. For a transverse circular cylinder, through distributing multipoles along its centre line, the velocity potential can be written in an infinite series with unknown coefficients, which can be determined from the impermeable condition on a body surface. A major feature here is that different from the free surface problem, or a channel without the ice sheet cover, this problem is fully three-dimensional because of the constraints along the intersection of the ice sheet with the channel wall. It has been also confirmed that there is an infinite number of critical speeds. Whenever the current speed passes a critical value, the force on the body and wave pattern change rapidly, and two more wave components are generated at the far-field. Extensive results are provided for hydroelastic waves and hydrodynamic forces when the ice sheet is under different edge conditions, and the insight of their physical features is discussed.

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.

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

Abstract

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

Flexural-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.

A hybrid method for linearized wave radiation and diffraction problem by a three dimensional floating structure in a polynya

A hybrid method is developed to solve the interaction problem of wave with a three dimensional floating structure in a polynya. The linearized velocity potential theory is used for fluid flow, and the thin elastic plate model is adopted for the infinitely extended ice sheet. Because of sudden change of the upper boundary of the computational domain, namely from the ice sheet to the free surface, the domain is divided into two sub-domains, one below free surface and the other below the ice sheet. The solution method is divided into three components. The first component is the integral equation over the structure surface and the interface of the two sub-domains. In the second component, the velocity potential is expanded into a series of eigenfunctions in the vertical directions, which avoids the numerical difficulty in calculation of the fifth derivatives. This is coupled with a series of integral equations along the edge of the ice sheet. In the third component of the method, two orthogonal inner products are used to impose the continuity conditions of the velocity and pressure on the interface, as well as the boundary conditions on the ice edge. The developed method is verified through comparison with the analytical solution for a circular cylinder. Case study is then made for a FPSO in a polynya with different shapes and floating positions. The hydrodynamic coefficients, wave exciting force and wave elevation in polynya are provided and 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.

Interaction of wave with multiple wide polynyas

A method based on the wide spacing approximation is applied to the wave scattering problem in multiple polynyas. An ice sheet is modeled as an elastic plate, and fluid flow is described by the velocity potential theory. The solution procedure is constructed based on the assumption that the ice sheet length is much larger than the wavelength. For each polynya, of free surface with an ice sheet on each side, the problem is solved exactly within the framework of the linearized velocity potential theory. This is then matched with the solution from neighboring polynyas at their interfaces below the ice sheet on each side, and only the traveling waves are included in the matching. Numerical results are provided to show that the method is very accurate and highly efficient. Extensive simulations are then carried out to investigate the effects of the ice sheet number, ice sheet length, distribution of ice sheets, as well as polynya width. The features of wave reflection and transmission are analyzed, and the physical mechanism is discussed.

Interaction of uniform current with a circular cylinder submerged below an ice sheet

The problem of a uniform current passing through a circular cylinder submerged below an ice sheet is considered. The fluid flow is described by the linearized velocity potential theory, while the ice sheet is modelled through a thin elastic plate floating on the water surface. The Green function due to a source is first derived, which satisfies all the boundary conditions apart from that on the body surface. Through differentiating the Green function with respect to the source position, the multipoles are obtained. This allows the disturbed velocity potential to be constructed in the form of an infinite series with unknown coefficients which are obtained from the boundary condition. The result shows that there is a critical Froude number which depends on the physical properties of the ice sheet. Below this number there will be no flexural waves propagating to infinity and above this number there will be two waves, one on each side of the body. When the depth based Froude number is larger than 1, there will always be a wave at far upstream of the body. This is similar to those noticed in the related problem and is different from that in the free surface problem without ice sheet. Various results are provided, including the properties of the dispersion equation, resistance and lift, ice sheet deflection, and their physical features are discussed.

The Effects of Moving Load on the Hydroelastic Response of a Very Large Floating Structure

The hydroelastic response of a very large floating structure in regular waves suffering an external moving point load is considered. The linearized velocity potential theory is adopted to describe the fluid flow. To take into account the coupled effects of the structure deformation and fluid motion, the structure is divided into multiple segments and connected by an elastic beam. Then through adding a stiffness matrix arising from the elastic beam into the multiple bodies coupled motion equations, the hydroelastic response is considered. By applying the Fourier transform to the obtained frequency domain coefficients, the motion equation is transformed into the time domain and the external point load is further considered. The accuracy and effectiveness of the proposed method are verified through the comparison with experimental results. Finally, extensive results are provided, and the effects of the moving point load on the hydroelastic response of the very large floating structure are investigated in detail.

Diffraction of hydroelastic waves by multiple vertical circular cylinders

The diffraction problem of hydroelastic waves beneath an ice sheet by multiple bottom-mounted circular cylinders is considered. The elastic thin-plate theory is adopted to model the ice sheet, while the linearized velocity potential theory adopted for the fluid flow. The velocity potential corresponding to each cylinder is expanded into a series of eigenfunctions, and the total potential is expressed as a summation of these expansions over the entire NC number of cylinders. For each cylinder, the Green’s second identity is used outside its domain to obtain a set of linear equations. For each different cylinder, the domain used is different. NC cylinders give NC sets of coupled linear equations. Investigations are made for different arrangements of cylinders, piercing through ice sheets. Results for the wave forces on the cylinders with clamped and free conditions of the ice edge are obtained. Physical phenomena corresponding to cylinders arranged in square, in an array, in a double-array and in a staggered double array are discussed.

Wave diffraction and radiation by a vertical circular cylinder standing in a three-dimensional polynya

The wave diffraction and radiation problem of a body in a polynya surrounded by an ice sheet extending to infinity is considered through a vertical circular cylinder. The ice sheet is modelled through the elastic thin-plate theory and the fluid flow through the linearized velocity potential theory. In particular, when the polynya is of the circular shape, eigenfunction expansion method is applied to the two regions below the ice sheet and the free surface respectively, and the velocity and pressure continuity conditions are imposed on the interface of the two regions. The wave motion in the polynya, the hydrodynamic coefficients as well as the exciting forces on a body located arbitrarily in the polynya are calculated. The nature of highly oscillatory behaviour of the results is investigated and their physical implications are discussed.

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.

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 radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack

Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack are considered based on the linearized velocity potential theory together with multipole expansion. The solution starts from the potential due to a single source, or the Green function satisfying both the ice sheet condition and the crack condition, as well as all other conditions apart from that on the body surface. This is obtained in an integral form through Fourier transform, in contrast to what has been obtained previously in which the Green function is in the series form based on the method of matched eigenfunction expansion in each domain on both sides of the crack. The multipole expansion is then constructed through direct differentiation of the Green function with respect to the source position, rather than treating each multipole as a separate problem. The use of the Green function enables the problem of wave diffraction by the crack in the absence of the body to be solved directly. For the circular cylinder, wave radiation and diffraction problems are solved by applying the body surface boundary condition to the multipole expansion, through which the unknown coefficients are obtained. Extensive results are provided for the added mass and damping coefficient as well as the exciting force. When the cylinder is away from the crack, a wide spacing approximation method is used, which is found to provide accurate results apart from when the cylinder is quite close to the crack.

Wave excited motion of a body floating on water confined between two semi-infinite ice sheets

The wave excited motion of a body floating on water confined between two semi-infinite ice sheets is investigated. The ice sheet is treated as an elastic thin plate and water is treated as an ideal and incompressible fluid. The linearized velocity potential theory is adopted in the frequency domain and problems are solved by the method of matched eigenfunctions expansion. The fluid domain is divided into sub-regions and in each sub-region the velocity potential is expanded into a series of eigenfunctions satisfying the governing equation and the boundary conditions on horizontal planes including the free surface and ice sheets. Matching is conducted at the interfaces of two neighbouring regions to ensure the continuity of the pressure and velocity, and the unknown coefficients in the expressions are obtained as a result. The behaviour of the added mass and damping coefficients of the floating body with the effect of the ice sheets and the excitation force are analysed. They are found to vary oscillatorily with the wave number, which is different from that for a floating body in the open sea. The motion of the body confined between ice sheets is investigated, in particular its resonant behaviour with extremely large motion found to be possible under certain conditions. Standing waves within the polynya are also observed.

Wave radiation from a truncated cylinder in front of a vertical wall

In this paper an analytical model is presented for the three-dimensional wave radiation problem of a cylinder floating on a layer of water of finite depth in front of a vertical wall, based on the linearized velocity potential theory, image theory and an eigen-function expansion matching method. The image theory is utilized to transform the radiation problem from the floating cylinder beside a vertical wall of infinite length into the equivalent problem of wave radiation from two identical floating cylinders in unbounded fluid domain to derive expressions for added-mass and radiation damping. Expressions for velocity potentials are obtained by using the method of separation of variables, in which unknown coefficients are determined by applying the eigen-function expansion matching method. Added-mass and wave damping are then calculated directly from the radiated potentials. The model is validated by comparison of the present results with the numerical ones, and there is a good agreement. The influence of distance between the cylinder and the vertical wall, draft of the cylinder and water depth on hydrodynamic coefficients is given and discussed.

Wave diffraction and radiation by multiple rectangular floaters

ABSTRACTIn this paper, the diffraction and radiation problem of multiple two-dimensional rectangular bodies floating on a layer of water of finite depth with waves is studied. An analytical model for the analysis of wave excitation forces and added mass and damping coefficients is proposed, based on a linearized velocity potential theory. Expressions for velocity potentials are obtained by the method of separation of variables, in which unknown coefficients are determined by utilizing the eigen-function expansion matching method. The model is validated by a comparison of the simulation results with the available data. The validated model is then utilized to examine the effect of the structure width, structure draft, spacing between adjacent structures, and structure numbers on wave transmission coefficient of floating structures.