Radiation Of Water Waves Research Articles

감쇠판이 부착된 원기둥의 동유체력 특성

The radiation of water waves by a heaving truncated circular cylinder with damping plate is solved in the frame of the three-dimensional linear potential theory. The damping plate has a distinct advantage in reducing the motion response of a floating circular cylinder by increasing the added mass and the damping coefficient. Using the matched eigenfunction expansion method, the characteristics of hydrodynamic added mass and the damping coefficient are investigated with various system parameters, such as the radius and submergence depth of the damping plate. It is found that both added mass and the damping coefficient are significantly increased due to the arranged features of the larger damping plate with shallow submergence, which are positive factors as a motion reduction device of the floating offshore platform. Also the numerical results for an oscillating submerged disk show that the added mass is negative and that the damping coefficient has a peak value at resonant frequency when submergence depth is sufficiently small.

Water wave radiation by a heaving submerged horizontal disk very near the free surface

The radiation of water waves by heaving horizontal disk at a shallow submergence depth is considered. The problem is formulated as a Fredholm integral equation. The resonant frequencies originating from the shallow submergence are examined and their locations are computed by a modified Newton’s method. The hydrodynamic coefficients in a neighborhood of a resonance pole are found to be represented by circles in the complex plane. An asymptotic approximation is obtained for small values of the submergence depth. The relationship between the small-submergence first order problem and the dock problem is explored and found to be clearly represented by the added mass and damping coefficients.

Radiation of Water Waves by a Sphere in an Ice-Covered Two-Layer Fluid of Finite Depth

Radiation of Water Waves by a Sphere in an Ice-Covered Two-Layer Fluid of Finite Depth

Implicit formulation with the boundary element method for nonlinear radiation of water waves

An accurate and efficient numerical method is presented for the two-dimensional nonlinear radiation problem of water waves. The wave motion that occurs on water due to an oscillating body is described under the assumption of ideal fluid flow. The governing Laplace equation is effectively solved by utilizing the GMRES (Generalized Minimal RESidual) algorithm for the boundary element method (BEM) with quadratic approximation. The intersection or corner singularity in the mixed Dirichlet–Neumann problem is resolved by introducing discontinuous elements. The fully implicit trapezoidal rule is used to update solutions at new time-steps, by considering stability and accuracy. Traveling waves generated by the oscillating body are absorbed downstream by the damping zone technique. To avoid the numerical instability caused by the local gathering of grid points, the re-gridding technique is employed, so that all the grids on the free surface may be re-distributed with an equal distance between them. The nonlinear radiation force is evaluated by means of the acceleration potential. For a mixed Dirichlet–Neumann problem in a computational domain with a wavy top boundary, the present BEM yields numerical solutions for the quadratic rate of convergence with respect to the number of boundary elements. It is also demonstrated that the present time-marching and radiation condition work successfully for nonlinear radiation problems of water waves. The results obtained from this study concur reasonably well with other numerical computations.

Time-domain simulation for water wave radiation by floating structures (Part A)

Direct time-domain simulation of floating structures has advantages: it can calculate wave pressure fields and forces directly; and it is useful for coupled analysis of floating structures with a mooring system. A time-domain boundary integral equation method is presented to simulate three-dimensional water wave radiation problems. A stable form of the integration free-surface boundary condition (IFBC) is used to update velocity potentials on the free surface. A multi-transmitting formula (MTF) method with an artificial speed is introduced to the artificial radiation boundary (ARB). The method was applied to simulate a semi-spherical liquefied natural gas (LNG) carrier and a semi-submersible undergoing specified harmonic motion. Numerical parameters such as the form of the ARB, and the time and space discretization related to this method are discussed. It was found that a good agreement can be obtained when artificial speed is between 0.6 and 1.6 times the phase velocity of water waves in the MTF method. A simulation can be done for a long period of time by this method without problems of instability, and the method is also accurate and computationally efficient.

Water wave radiation by a sphere submerged in water with an ice-cover

Using the multipoles method, we formulate the problems of radiation (both heave and sway) of water waves by a submerged sphere in deep as well as in uniform finite depth water with an ice-cover, with the ice-cover being modelled as an elastic plate of very small thickness. In each case this leads to an infinite system of linear equations which are solved numerically by standard techniques. The added-mass and damping coefficients for a heaving and swaying sphere are obtained and depicted graphically against the wave number for various values of the radius of the submerged sphere and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover on these quantities. When the flexural rigidity is taken to be zero, the numerical results for the added-mass and damping coefficient for water with a free surface are recovered.

The radiation and diffraction of water waves by a bottom-mounted circular cylinder in a two-layer fluid

An analytical method was proposed to analyze the radiation and diffraction of water waves by a bottom-mounted circular cylinder in a two-layer fluid. Analytical expressions for added mass and damping coefficients, as well as the wave excitation forces of the circular cylinder were obtained by an eigenfunction expansion method. The hydrodynamic forces on the bottom-mounted circular cylinder in a two-layer fluid include not only the added mass and damping coefficients, but also the wave forces of the surface and internal-wave modes. This is different from the case of a homogenous fluid. Some examples were given, showing that density stratification can have a relative large effect on these hydrodynamic forces over a wide range of frequencies.

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.

Radiation and diffraction of linear water waves by an infinitely long submerged rectangular structure parallel to a vertical wall

The radiation and the diffraction of linear water waves by an infinitely long floating rectangular structure submerged in water of finite depth with leeward boundary being a vertical wall are analyzed in this paper by using the method of separation of variables. Analytical expressions for the radiated and diffracted potentials are derived as infinite series with unknown coefficients determined by the eigenfunction expansion matching method. The expressions for wave forces and hydrodynamic coefficients are given. A comparison is made between the results obtained by the present analytical solution and those obtained by the boundary element method. By using the present analytical solution, the hydrodynamic influences of the submergence, the width, the thickness of the structure, and the distance between the structure and the wall on the wave forces and hydrodynamic coefficients are discussed in detail.

On the radiation and diffraction of linear water waves by a rectangular structure over a sill. Part I. Infinite domain of finite water depth

The effects of a bottom sill on the hydrodynamic coefficients, wave force, and reflection and transmission coefficients of a rectangular structure floating on the free surface are analyzed in this paper by use of a semi-analytical method. Linearized water wave theory and the method of separation of variables are applied to obtain analytical expressions for the diffracted and radiated wave potentials. The unknown coefficients in the obtained expressions are determined by use of the eigenfunction expansion matching method. To verify the obtained expressions and the proposed method, a comparison of the analytical results with those obtained by the boundary element method is made and very good agreement is achieved, which shows that the analytical expressions for the radiated and diffracted potentials are correct. By use of the present analytical solution, several results for the hydrodynamic coefficients, wave force, reflection and transmission coefficients are given.

On the radiation and diffraction of water waves by a rectangular structure with a sidewall

A radiation and diffraction boundary value problem is investigated. It arises from the interaction of linear water waves with a freely floating rectangular structure in a semi-infinite fluid domain of finite water depth with the leeward boundary being a vertical wall. Analytical expressions for the radiated potentials and the diffracted potential are obtained by use of the method of separation of variables and the eigenfunction expansion method. The added masses and damping coefficients for the structure heaving, swaying and rolling in calm water are obtained by use of the corresponding radiated potentials and the wave excitation forces are calculated by use of the diffracted potential. To verify the correctness of the method, a boundary element method is used. A comparison of the analytical results with those obtained by the boundary element method is made and good agreement is achieved, which shows that the analytical expressions for the radiated and diffracted potentials are correct. By use of the present analytical solution, the added mass, damping coefficients, wave excitation force, together with the hydrodynamic effects of the draft, width of the structure and the clearance between the structure and the sidewall are also investigated.

On the radiation and diffraction of water waves by a rectangular buoy

An analytical method is presented to analyze the radiation and diffraction of water waves by a rectangular buoy in an infinite fluid domain of finite water depth. Analytical expressions for the radiated potentials and the diffracted potentials are obtained by use of the method of separation of variables. The unknown coefficients in the expressions are determined by use of the eigenfunction expansion matching method. The added masses and damping coefficients for the buoy heaving, swaying and rolling in calm water are obtained by use of the corresponding radiated potentials. Wave excitation forces are calculated by two different approaches, one is by use of the radiated potentials through Haskind’s theorem and the other is by the diffracted potential. It can be seen that the latter approach for wave forces on a rectangular buoy is much simpler than the former. To verify the correctness of the method, two specific examples in the past references are recomputed and the obtained results are in good agreement with those by use of other methods, which shows that the present method is correct.

Finite elements using a plane-wave basis for scattering of surface water waves.

A generalized, frequency-independent, finite-element method is proposed for the numerical simulation of combined diffraction, refraction, radiation, reflection and absorption of water waves. Plane waves are used as an external function space, exploiting the standard set of approximation functions defined by the finite element method to enforce conformity and to impose boundary conditions in a straightforward manner by applying the method to the plane waves on each element. An integration method originally devised only for diffraction of waves by Ortiz and Sanchez in 2001 is extended to combined refraction and diffraction of high-frequency waves, retaining its dramatic savings in the number of operations for a given error compared with standard numerical integration methods.

Radiation of water waves from an oscillating source located at a slope of angle π/4

An analytical investigation is performed of the linear radiation problem for water waves generated by an oscillating normal velocity distribution along a sloping beach with slope angle π/4. The distribution of normal velocity is arbitrary, and it oscillates with a given frequency. The solution is expressed in terms of the Green function which represents a source of unit oscillatory flux located at an arbitrary position along the slope. At infinity the radiation condition is applied to determine the outgoing wave. As a simple example of integrating the Green function, the reflection of an incoming sinusoidal wave is calculated.

Hydrodynamic coefficients of a submerged pulsating sphere in finite depth

By extending the work of Linton (Linton, C.M., 1991. Radiation and diffraction of waver waves by a submerged sphere in finite depth. Ocean Engineering 18 (1/2), 61–74), the problem of radiation of water waves by a submerged pulsating sphere in finite depth is formulated using the multipole method. As in Linton (1991), this leads to an infinite system of linear equations, which are easily solved numerically. Simple expressions are derived for the hydrodynamic characteristics of such a body. Results showing the effect of varying both the immersion depth and the water depth on the hydrodynamic coefficients of the pulsating sphere are given. The paper resumes the work presented in Lopes (Lopes, D.B.S., 1999. On the study of the Archimedes wave swing device for wave energy utilization (in Portuguese). MSc on the Management and Modelling of the Marine Environment, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa.).

Evaluation of Wave Amplitude for a Class of Water Wave Problems

In the present paper, a boundary value problem associated with scattering or radiation of water waves involving a vertical barrier present in deep water is reduced to dual integral equations. Noting that one of the integrals of the dual integral equations is bounded at both ends of the interval complementary to the one in which it is defined, it is found that a solution of the dual integral equations exists if and only if a certain solvability criterion is satisfied. Exploiting this solvability condition, the amplitude of the waves at infinity associated with scattering and radiation of water waves involving a vertical barrier have been evaluated.

On the analytic form in the frequency domain of water wave solutions

In this paper, an analytic result due to Ursell, which applies to the radiation of water waves by a heaving semi-immersed sphere, is extended to a wide class of bodies, and to the scattering problem. This result forms the basis of a numerical method, for the computation of the hydrodynamic coefficients, which has advantages over the conventional integral equation approach.

Scattering and radiation of water waves by permeable barriers

The two-dimensional problems of scattering and radiation of small-amplitude water waves by thin vertical porous plates in finite water depth are considered using the linear water wave theory. Applying the method of eigenfunction expansion, these boundary value problems are converted to certain dual series relations. Solutions to these relations are then obtained by a suitable application of the least squares method. For the scattering problem, four different basic configurations of the barriers are investigated, namely, (I) a surface-piercing barrier, (II) a bottom-standing barrier, (III) a totally submerged barrier, and (IV) a barrier with a gap. The performance of these types of barriers as a breakwater are examined by studying the variation of their reflection and transmission coefficients, hydrodynamic forces and moments for different values of the porous effect parameter defined by Chwang [J. Fluid Mech. 132, 395–406 (1983)], or the Chwang parameter. For the radiation problem, three types of wavemakers, which resemble types (I), (II), and (III) of the above-mentioned configuration, are analyzed. The dependence of the amplitude to stroke ratio on other parameters is also investigated to study the features of these wavemakers.

Discrete non-local absorbing boundary condition for exterior problems governed by Helmholtz equation

The finite element method is employed to approximate the solutions of the Helmholtz equation for water wave radiation and scattering in an unbounded domain. A discrete, non-local and non-reflecting boundary condition is specified at an artificial external boundary by the DNL method, yielding an equivalent problem that is solved in a bounded domain. This procedure formulates a boundary value problem in a bounded region by imposing a relation in the discrete medium between the nodal values at the two last layers. For plane geometry, this relation can be found by straightforward eigenvalue decomposition. For circular geometry, the plane condition is applied at the external layer and this condition is condensed through a structured annular region, resulting in a condition at an inner radius. Exterior problems with a bounded internal physical obstacle are considered. It is well-known that these kind of problems are well-posed, and have a unique solution. Numerical studies based on standard Galerkin methodology examine the dependence of the DNL condition with respect to the circular annular region width. The DNL condition is compared with local boundary conditions of several orders. Numerical examples confirm the important improvement in accuracy obtained by the DNL method over standard conditions. Copyright © 1999 John Wiley & Sons, Ltd.

Radiation of water waves by a heaving submerged horizontal disc

A thin rigid plate is submerged beneath the free surface of deep water. The plate performs small-amplitude oscillations. The problem of calculating the radiated waves can be reduced to solving a hypersingular boundary integral equation. In the special case of a horizontal circular plate, this equation can be reduced further to one-dimensional Fredholm integral equations of the second kind. If the plate is heaving, the problem becomes axisymmetric, and the resulting integral equation has a very simple structure; it is a generalization of Love's integral equation for the electrostatic field of a parallel-plate capacitor. Numerical solutions of the new integral equation are presented. It is found that the added-mass coefficient becomes negative for a range of frequencies when the disc is sufficiently close to the free surface.