Hydrodynamics of a submerged hydrofoil advancing in waves

The hydrodynamic problem of a hydrofoil travelling at constant speed in water waves has been investigated through velocity potential theory. The boundary conditions on the free surface have been linearized, and the effects are accounted for through the Green function. The overall problem is decomposed into the steady forward speed problem and periodic wave radiation and diffraction problems. Each of these problems is solved using the boundary integral equation over the hydrofoil surface together with a vortex sheet behind the trailing edge. The body surface boundary condition is imposed on its mean position. As a result the steady potential will contribute a well-known m_j term to the body surface boundary condition on the radiation problem. The numerical difficulty in dealing with this term is effectively resolved through a difference method. The effects of the thickness on the wave radiation and diffraction are investigated. The applicability of various reciprocity relationships in this problem is discussed.     

A numerical method for the solution of a line source under a free surface

A modified source-and-dipole type singularity panel method is proposed to calculate the flow properties for an oscillating arbitrary body in the presence of a free surface. The technique is based on Green's identity whereby the boundary value problem is expressed as a boundary integral equation which is solved numerically. The free-space Green function is used in the integral equation. To demonstrate the feasibility of the method, the problem of a pulsating submerged line source under a free surface is treated and results are compared with the exact solution.An excellent agreement with the theory is obtained for panel density of about ten panels per wavelength and paneled water surface length of two wavelengths with very low computing times, indicating the feasibility of the method for unsteady water wave problems.     

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.     

Fast computation of the transient motions of moving vessels in irregular ocean waves

A fast time-domain method is developed in this paper for the real-time prediction of the six degree of freedom motions of a vessel traveling in an irregular seaway in infinitely deep water. The fully coupled unsteady ship motion problem is solved by time-stepping the linearized boundary conditions on both the free surface and body surface. A velocity-based boundary integral method is then used to solve the Laplace equation at every time step for the fluid kinematics, while a scalar integral equation is solved for the total fluid pressure. The boundary integral equations are applied to both the physical fluid domain outside the body and a fictitious fluid region inside the body, enabling use of the fast Fourier transform method to evaluate the free surface integrals. The computational efficiency of the scheme is further improved through use of the method of images to eliminate source singularities on the free surface while retaining vortex/dipole singularities that decay more rapidly in space. The resulting numerical algorithm runs 2–3 times faster than real time on a standard desktop computer. Numerical predictions are compared to prior published results for the transient motions of a hemisphere and laboratory measurements of the motions of a free running vessel in oblique waves with good agreement.     

A potential based panel method for 2-D hydrofoils

A potential based panel method for the hydrodynamic analysis of 2-D hydrofoils moving beneath the free surface with constant speed without considering cavitation is described. By applying Green's theorem and the Green function method, an integral equation for the perturbation velocity potential is obtained under the potential flow theory. Dirichlet type boundary condition is used instead of Neumann type boundary condition. The 2-D hydrofoil is approximated by line panels which have constant source strength and constant doublet strength distributions. The free surface condition is linearized and the method of images is used for satisfying this free surface condition. All the terms in fundamental solution (Green function) of perturbation potential are integrated over a line panel. Pressure distribution, lift, residual drag and free surface deformations are calculated for NACA4412, symmetric Joukowski and van de Vooren profile types of hydrofoil. The results of this method show good agreement with both experimental and numerical methods in the literature for the NACA4412 and symmetric Joukowski profile types. The lift and residual drag values of the van de Vooren profile are also presented. The effect of free surface is examined by a parametric variation of Froude number and depth of submergence.     

A numerical modeling of hydrodynamic characteristics of various planing hull forms

A numerical algorithm based on the boundary element method (BEM) is presented for predicting the hydrodynamic characteristics of the various planing hull forms. The boundary integral equation is derived using Green's theorem on the wetted body surface and the free surface. The ventilation function at the transom is estimated with Doctor's empirical formula. This function is defined as the transom zone free surface boundary condition. The combined boundary integral equation and modified free surface boundary condition are simultaneously solved to determine the dipole on the wetted hull surface and the source on the free surface. The method is applied to investigate three examples of planing hulls, which include flat-plates, as well as wedge-shaped and variable deadrise planing hulls. Their hydrodynamic characteristics are calculated for different speeds. Computational results are presented and compared with existing theories and experiments. On the whole, the agreement between the present method and the selected experimental and numerical data is satisfactory.     

A novel decomposition of the quadratic transfer function (QTF) for the time-domain simulation of non-linear wave forces on floating bodies

In the present study, a novel method is proposed for the separation of the second-order sum- and difference-frequency wave forces—that is, quadratic transfer functions (QTFs)—on a floating body into three components due to wave–wave, wave–motion, and motion–motion action. By applying the new QTF components, the second-order wave forces on a floating body can be strictly computed in the time domain. In this work, the boundary value problems (BVPs) corresponding to the three kinds of QTF components were derived, and non-homogeneous boundary conditions on the free surface and the body surface were obtained. The second-order diffraction potentials were determined using the boundary integral equation method. In the solution procedure, the highly oscillatory and slowly converging integral on the free surface was evaluated in an accurate and effective manner. Furthermore, the application of the QTF components in the time domain was demonstrated. The second-order exciting forces in the time domain were divided into three parts. Each part of these forces was computed via a two-term Volterra series model based on the incident waves, the first-order motion response, and the QTF components. This method was applied to several numerical examples. The results demonstrated that this decomposition yields satisfactory results.     

A potential based panel method for 2-D hydrofoils

A potential based panel method for the hydrodynamic analysis of 2-D hydrofoils moving beneath the free surface with constant speed without considering cavitation is described. By applying Green's theorem and the Green function method, an integral equation for the perturbation velocity potential is obtained under the potential flow theory. Dirichlet type boundary condition is used instead of Neumann type boundary condition. The 2-D hydrofoil is approximated by line panels which have constant source strength and constant doublet strength distributions. The free surface condition is linearized and the method of images is used for satisfying this free surface condition. All the terms in fundamental solution (Green function) of perturbation potential are integrated over a line panel. Pressure distribution, lift, residual drag and free surface deformations are calculated for NACA4412, symmetric Joukowski and van de Vooren profile types of hydrofoil. The results of this method show good agreement with both experimental and numerical methods in the literature for the NACA4412 and symmetric Joukowski profile types. The lift and residual drag values of the van de Vooren profile are also presented. The effect of free surface is examined by a parametric variation of Froude number and depth of submergence.     

Numerical Investigation on Submerged Horizontal Plate

Hydrodynamic characters on a horizontal, thin, rigid plate located beneath the free surface are numerically investigated. Assuming a linear, time-harmonic potential flow and utilizing Green identity, the governing Laplace equation can be simplified into Fredholm integral equation ofthe second kind. Supposing linear-order discontinuous elements along intersecting vertical boundaries, and by use of the boundary element method, numerical solution about source strength distribution on the plate can be changed into a series of algebraic equations. The 3D Green function is introduced to set up the integral equations, and the GMRES solver is performed for solving the large dense linear system of equations. The added-mass, damping force and exciting force are evaluated directly from the equations. It is found that the added-mass coefficient becomes negative for a range of frequencies when the plate is sufficiently close to the free surface.     

Wave sloshing in corrugated bottom tanks with two-dimensional flow

The boundary integral element method based on Green's formula is applied to the analysis of transient flow problem in corrugated bottom tanks. The problem is formulated as a two-dimensional linear, initial boundary value problem in terms of a velocity potential. The Laplace equation and the boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by the application of Green's formula. Finite Difference discretization is applied timewise. Initially a triangular wave on the free surface is assumed to be formed. The height of the triangular corrugated bottom is varied between 1/10 and 1/5 of the tank depth. The form of the free surface and the equipotential lines for the flow in the tank are presented at different time steps. An accuracy analysis is performed and distortion in time is considered. Proper coefficients for solutions are derived and presented. The results show that utilization of triangular corrugated bottoms may help to regulate the flow in tanks.     

Stationary variational expressions for radiated and scattered acoustic power and related quantities

The construction of stationary expressions for quantities of physical interest such as radiated power and target strength is discussed broadly for acoustic problems involving radiation or scattering from finite objects of arbitrary shape. The Kirchhoff-Helmholtz integral corollaries of the wave equation, which express acoustic pressure at either interior or exterior points in terms of pressure and its normal derivative over any closed surface, yield for both interior and exterior problems two mathematically dissimilar but related functional relations between surface field quantities. One of these is the better known surface Helmholtz integral equation; the other is a differential-integral relation which involves the tangential derivatives of pressure on the surface. The four linear operators involved in these functional relations are studied and it is found that two are self-adjoint, while the other two are an adjoint pair. A general technique for constructing variational expressions recently developed by Gerjuoy et al. [28] is adapted to acoustic radiation and scattering problems with the functional relations taken as the primary governing relations. Included examples are stationary expressions for the radiated power when either the normal velocity or the pressure are specified on the surface (the other quantity being unknown) and the target strength for scattering from a rigid object. The adjoint relations allow simple physical interpretations for the Lagrange multipliers that arise in the theory, such that the guesses for good trial functions can take advantage of existing physical insight. It is demonstrated with a specific example (transversely vibrating disk) that the resulting estimate for radiated power is substantially more accurate than that of the trial function for surface pressure which was inserted into the stationary expression.     

Analysis of some key parametrizations in a beach profile morphodynamical model

A numerical process-based model to forecast beach profile morphodynamics has been developed. In the present paper, an analysis of various modelling approaches and key parametrizations involved in the estimation of the wave-driven current and the suspended sediment concentration is carried out.Several resolution techniques for the 1DV horizontal (i.e., in the x-direction perpendicular to coastline) momentum equation governing the Mean Horizontal Velocity (MHV) are analysed. In the first kind of techniques, the mean horizontal velocity is computed from the momentum equation, whereas the Mean Water Level (MWL) is computed using a parametrization of the depth-averaged momentum equation. Two boundary or integral conditions are thus needed. In the second kind, both mean horizontal velocity and mean water level gradient in the x-direction are the unknowns of the momentum equation, thus, three boundary or integral conditions are needed. Various additional conditions are discussed. We show that using a technique of the first kind is equivalent to imposing the difference between the surface and the bottom shear stresses in the 1D vertical equation. Both techniques lead to results that are in good agreement with the Delta Flume experimental data, provided the Stokes drift flow discharge is imposed as an additional condition. The influence of the breaking roller model and of the turbulent viscosity parametrization are also analysed.Suspended sediment transport by the mean current and wave-induced bedload transport are taken into account in the sediment flux. Three turbulent diffusivity parametrizations are compared for suspended sediment concentration estimations. A linear profile for the turbulent diffusivity taking into account the wave bottom shear stress and the surface wave breaking turbulence production is shown to give the best results. Using experimental data, we put forward the poor estimation of the bottom sediment concentration given by the three implemented parametrizations. We thus propose a new parametrization relying on a Shields parameter based on the breaking roller induced surface shear stress. Using this new parametrization, the bottom profile used in the tests keeps its two bars which disappear otherwise. However, the morphodynamical model still overestimates the bars offshore motion, a bias already observed in other models.     

一个新的破碎波统计模式

由运动方程和涡度方程所导得的两个首次积分估计给出了海波破碎的发生条件和破碎波的波面限制。由这两个条件所构造的新的破碎波统计模式，可导出海波的破碎面积率、破碎体积率和破碎能量损耗率的解析表达式，加之一个简单的白冠物理模型又导出了一个新的白冠覆盖率的解析表达式。     

Simulation of directional waves in a numerical basin by a desingularized integral equation approach

A numerical solution is developed to investigate the generation and propagation of small-amplitude water waves in a semi-infinite rectangular wave basin. The three-dimensional wave field is produced by the prescribed “snake-like” motion of an array of segmented wave generators located along the wall at one end of the tank. The solution technique is based on the boundary element approach and uses an appropriate three-dimensional Green function which explicitly satisfies the tank-wall boundary conditions. The Green function and its derivatives which appear in the integral equation formulation can be shown to be slowly convergent when the source and field points are in close proximity. Therefore, when computing the velocity potentials on the wave generators, the source points are chosen outside the fluid domain, thereby ensuring the rapid convergence of these functions and rendering the integral equations non-singular. Numerical results are shown which illustrate the influence of the various wavemaker and basin parameters on the generated wave field. Finally, the complete wave field produced by the diffraction of oblique waves by a vertical circular cylinder in a basin is presented.     

Dynamic and static analysis of a marine riser

A frequency domain normal mode solution is presented for the dynamic response of an unbuoyed marine riser subjected to periodic excitation from a surface vessel in the direction of wave propagation. The variable tension beam-column equation is solved in terms of normal modes of free vibration of the riser and the rigid body displacement. Drag forces on the riser are represented by Morison's formula taking account of the velocity of the riser and wave-induced fluid velocity. A periodic solution for the flexural motion of the riser and the bending stress is then obtained by means of an iterative solution of the frequency response function. The drag force induced stresses arising from a linearly varying current are also determined. The results presented compare favourably with those obtained by other methods.     

Hydroelastic analysis of cantilever plate in time domain

Numerical solutions for the hydroelastic problems of bodies are studied directly in the time domain using Neumann–Kelvin formulation. In the hydrodynamic part of problem, the exact initial boundary value problem is linearized using the free stream as a basis flow, replaced by the boundary integral equation applying Green theorem over the transient free surface Green function. The resultant boundary integral equation is discretized using quadrilateral elements over which the value of the potential is assumed to be constant and solved using the trapezoidal rule to integrate the memory or convolution part in time. In the structure part of the problem, the finite element method is used to solve the hydroelastic problem. The Mindlin plate as a bending element, which includes transverse shear effect and rotary inertia effect are used. The present numerical results show acceptable agreement with experimental, analytical, and other published numerical results.     

A three-dimensional model of the breeze-induced circulation of waters in the Kerch strait

We generalize the problem of evaluation of currents caused by the action of breeze winds to the three-dimensional case. In the “solid-lid” approximation, the problem is reduced to the numerical solution of a two-dimensional equation for the integral current function (with complex-valued coefficients) with subsequent evaluation of the components of current velocity according to analytic formulas. The breeze is specified as acting in a narrow coastal strip in the form of a zonal wind. The three-dimensional structure and the variations of currents near the west boundary of the Kerch Strait as functions of time are studied in detail.     

A fast convolution approach to the transformation of surface gravity waves: Linear waves in 1DH

The transformation of irrotational surface gravity waves in an inviscid fluid can be studied by time stepping the kinematic and dynamic surface boundary conditions. This requires a closure providing the normal surface particle velocity in terms of the surface velocity potential or its tangential derivative. A convolution integral giving this closure as an explicit expression is derived for linear 1D waves over a mildly sloping bottom. The model has exact linear dispersion and shoaling properties. A discrete numerical model is developed for a spatially staggered uniform grid. The model involves a spatial derivative which is discretized by an arbitrary-order finite-difference scheme. Error control is attained by solving the discrete dispersion relation a priori and model results make a perfect match to this prediction. A procedure is developed by which the computational effort is minimized for a specific physical problem while adapting the numerical parameters under the constraint of a predefined tolerance of damping and dispersion error. Two computational examples show that accurate irregular-wave transformation on the kilometre scale can be computed in seconds. Thus, the method makes up a highly efficient basis for a forthcoming extension that includes nonlinearity at arbitrary order. The relation to Boussinesq equations, mild-slope wave equations, boundary integral equations and spectral methods is briefly discussed.     

A NURBS-based high-order panel method for three-dimensional radiation and diffraction problems with forward speed

A high-order Rankine panel method based on Non-Uniform Rational B-Spline (NURBS) is developed for solving the three-dimensional radiation and diffraction problems with forward speed. A NURBS surface is used to precisely represent the body geometry. Velocity potential on the body surface is described by B-spline after the source density distribution on the boundary surface is determined. A collocation approach is applied to numerical computation and the integral equations are evaluated by applying Gauss–Legendre quadrature. The m_j-terms are evaluated by a desingularized method which utilizes NURBS technique. In order to verify the method proposed, it is firstly applied to the unbounded flow problem of a sphere and spheroids. The numerical results are found to be in good agreement with analytical solutions. Then the method is used to solve the radiation and diffraction problems of a sphere and the diffraction problem of a spheroid moving with a forward speed beneath the free surface in frequency domain. The numerical results are satisfactory in comparison with the published analytical results and experimental results.     

An improved numerical simulation mode of nonlinear wave with consideration of high order terms

In this paper the 0-1 combined BEM is adopted to subdivide the computational domain boundary,and to discretize the Green’s integral expression based on Laplace equation.The FEM is used to subdivide the wave surface and deduce the surface equation which satisfies the nonlinear boundary conditions on the surface.The equations with potential function and wave surface height as an unknown quantity by application of Taylor expansion approach can be solved by iteration within the time step.In m-time iteration within the computational process of time step(n-1)Δt to nΔt,the results of the previous iteration are taken as the initial value of the two-order unknown terms in the present iteration.Thus,an improved tracking mode of nonlinear wave surface is established,and numerical results of wave tank test indicate that this mode is improved obviously and is more precise than the previous numerical model which ignored the two-order unknown terms of wave surface location and velocity potential function in comparison with the theoretical values.