Linear Free-surface Conditions Research Articles

Forced surge motion of a floating bridge pontoon to evaluate the hydrodynamic damping properties

Dedicated model tests were carried out to investigate the hydrodynamic force coefficients in surge for a pontoon that is designed to support a floating bridge. The experiments were carried out by oscillating the pontoon at various Keulegan–Carpenter (KC) numbers and oscillation periods in forced motion. A pontoon designed for the floating bridge in Bjørnarfjorden, Norway, is investigated herein. Several numerical simulations are performed to supplement the experiments. The estimated damping coefficients in surge are of significant importance at the first horizontal resonant mode of the floating bridge, which may be excited due to swell sea in the fjord. The first horizontal resonant mode of the floating bridge is around 15–16 s. The model tests were conducted with and without turbulence triggers. A finding from the model tests were that turbulence triggers did not prove to be necessary to eliminate model scale effects in oscillatory flow.The KC-dependent added mass and damping coefficient in oscillatory flow is investigated. The added mass and damping coefficients are clearly KC-dependent. The wave radiation damping in surge is small at the tested oscillation periods, and the damping in surge is dominated by the damping due to flow separation.Numerical simulations using a hybrid potential flow-CFD solver with linear free-surface conditions, PVC3D, are carried out with both linear and nonlinear body-boundary conditions. The simulations with nonlinear body-boundary conditions predict the hydrodynamic force coefficients with reasonable accuracy, whilst the simulations with linear body-boundary conditions fail to predict the hydrodynamic coefficients with the desired accuracy — both used to investigate the viscous damping which arises due to flow separation along the pontoon body.

On Investigation of Liquid Sloshing in Cylindrical Tanks With Single and Multiply Connected Domains Using Isogeometric Boundary Element Method

Abstract This paper explores an isogeometric boundary element method (IGA-BEM) for sloshing problems in cylindrical tanks with single and multiply connected domains. Instead of the Lagrange basis functions used in the standard BEM, the nonuniform rational B-splines (NURBS) basis functions are introduced to approximate the geometries of the problem boundaries and the unknown variables. Compared with the Lagrange basis functions, NURBS basis functions can accurately reconstruct the geometric boundary of analysis domain with almost no error, and all the data information for NURBS basis functions can be directly obtained from the computer-aided design (cad) or computer-aided engineering (cae) commercial software, which implies the modeling process of IGA-BEM is more simple than that of the standard BEM. NURBS makes it possible for the IGA-BEM to realize the seamless connection between cad and cae software with relative higher calculation accuracy than the standard BEM. Based on the weighted residual method as well as the divergence theorem, the IGA-BEM is developed for the single and multiply connected domains, whose boundaries are separately defined in the parameter space by different knot vectors. The natural sloshing frequencies of the liquid sloshing in a circular cylindrical tank with a coaxial or an off-center circular pipe, an elliptical cylindrical tank with an elliptical pipe, a circular cylindrical tank with multiple pipes are estimated with the introduced method by assuming an ideal (inviscid and incompressible) liquid, irrotational small-amplitude sloshing, and the linear free-surface condition. The comparison between the results obtained by the proposed method and those in the existing literatures shows very good agreements, which verifies the proposed model well. Meanwhile, the effects of radius ratio, liquid depth, number, and location of internal pipe (pipes) on the natural sloshing frequency and sloshing mode are analyzed carefully, and some conclusions are outlined finally.

Comparative Study on Numerical Computation Methods for Radiation Forces on a Three-Dimensional Body With Edge in the Time Domain

AbstractA time-domain numerical model with the linear free surface condition and large amplitude body motions is developed based on the higher-order boundary element method. Five numerical formulations are used to compute the nonlinear radiation forces on a moving body in the domain. A series of submerged bodies that have sharp edges and rounded corners with small radii are considered to investigate their reliabilities. It is found that for the bodies with rounded corners the nonlinear radiation forces calculated by the five numerical formulations are close to each other, while large discrepancies exist for the body with sharp edges. Examination on the vertical nonlinear radiation forces due to the body surge motion reveals that the body edge has less influence on the method of the rate of change of the fluid momentum; however, it has a stronger effect on the other four formulations. The former applied a substitution of the spatial derivatives of the velocity potential, and the other four use a direct integration of the fluid pressure on the body surface.

On natural frequencies of liquid sloshing in 2-D tanks using Boundary Element Method

The paper investigates natural frequencies and mode shapes of liquid sloshing within two dimensional baffled tanks with arbitrary geometries. Without taking into account viscosity and compressibility, the fluid is considered to be potential. The vibration amplitude is presumed to be little and hence a linear free surface condition is used. The governing equations are solved using Boundary Element Method (BEM), where baffles are treated as two immersed adjacent layers. The resulted problem after discretization is transformed into a standard matrix eigenvalue issue, where only freedoms related to free surface are involved. The BEM approach is verified against analytical theories and existing numerical results. Natural frequencies of sloshing in tanks with a variety of shapes are studied. The effect of horizontal and vertical baffles is thoroughly investigated and empirical formulas are proposed to calculate natural frequencies of rectangular tanks with baffles.

Numerical study on local steady flow effects on hydrodynamic interaction between two parallel ships advancing in waves

Underway replenishment is an essential component of long-term naval operations. During underway replenishment, two ships travel in close proximity at a moderate forward speed. For this issue, frequency domain analysis methods with and without incorporation of local steady flow effects are developed to investigate wave loads and free motions of two parallel ships advancing in waves, which are based on analytical quadrature of the Bessho form translating-pulsating source Green function over a panel or a waterline segment and a direct velocity potential approach. The local steady flow effects were taken into consideration through m-terms in the boundary conditions; meanwhile, the Neumann-Kelvin linear free surface condition was combined. By comparing present added mass, damping coefficient and motion results of Wigley I to those of experiments and other numerical solutions; it is found that present computational results show good agreement. In order to verify these methods for hydrodynamic interaction between two parallel ships, two experiments are carried out respectively to measure the wave loads and free motions for adjacent parallel ship models advancing with an identical speed in head regular waves. Results obtained by the present solution are in favorable agreement with the model tests, meanwhile, results of methods taking the local steady flow effects into consideration show better agreement. Further, the component of wave loads and the interaction effects of speeds and different clearances are deeply investigated. It is found that the effect of clearance and the speed are important factors relating to the interaction effect.

Tsunami forcing by a low Froude number landslide

The waves generated by a submarine landslide, of great concern to coastal communities, exhibit strong dependence on the landslide motion along the sea floor. A series of two-dimensional physical experiments investigate the waves generated by a solid block landslide moving along a horizontal boundary, allowing measurement of both onshore- and offshore-propagating waves using the laser-induced fluorescence technique. This technique provides high-quality free surface measurements over the entire length of the experimental flume, and hence a data set that can be used to validate numerical models for this idealised scenario. The landslide motion is provided by a mechanical system, allowing testing of a range of landslide accelerations and terminal velocities. The landslide Froude number governs the overall behaviour of the wave field. At lower Froude numbers, the waves are almost entirely generated by the landslide acceleration and deceleration, and the offshore- and onshore-propagating wave groups contain approximately equal energy. Interactions between the landslide and the offshore-propagating waves become more important as the Froude number increases. Two inviscid-irrotational models demonstrate the importance of dispersive effects for tsunamis generated by a submarine landslide, and correctly predict the behaviour of the entire wave field at low Froude numbers. The predictions in the vicinity of the landslide worsen with increasing Froude number, due to the linear free surface conditions used by the models. Lower Froude numbers appear to be more representative of previous sloping-boundary experimental geometries, although rigid block landslides still represent an idealisation of a field scenario.

Wave-induced response of a floating two-dimensional body with a moonpool.

Regular wave-induced behaviour of a floating stationary two-dimensional body with a moonpool is studied. The focus is on resonant piston-mode motion in the moonpool and rigid-body motions. Dedicated two-dimensional experiments have been performed. Two numerical hybrid methods, which have previously been applied to related problems, are further developed. Both numerical methods couple potential and viscous flow. The semi-nonlinear hybrid method uses linear free-surface and body-boundary conditions. The other one uses fully nonlinear free-surface and body-boundary conditions. The harmonic polynomial cell method solves the Laplace equation in the potential flow domain, while the finite volume method solves the Navier-Stokes equations in the viscous flow domain near the body. Results from the two codes are compared with the experimental data. The nonlinear hybrid method compares well with the data, while certain discrepancies are observed for the semi-nonlinear method. In particular, the roll motion is over-predicted by the semi-nonlinear hybrid method. Error sources in the semi-nonlinear hybrid method are discussed. The moonpool strongly affects heave motions in a frequency range around the piston-mode resonance frequency of the moonpool. No resonant water motions occur in the moonpool at the piston-mode resonance frequency. Instead large moonpool motions occur at a heave natural frequency associated with small damping near the piston-mode resonance frequency.

An experimental and numerical study of heave added mass and damping of horizontally submerged and perforated rectangular plates

Forced harmonic heave motions of horizontally submerged and perforated rectangular plates are studied experimentally and numerically at both a deep and shallow submergence. The steady-state vertical forces are expressed in terms of added mass and damping coefficients. The numerical results are partly obtained by combining potential flow with linear free-surface conditions and a nonlinear viscous pressure loss condition at the mean oscillatory plate position. A domain decomposition technique is applied with a boundary element method in the inner domain and an analytical representation of the velocity potential in the outer domain. A drag term accounts for the vortex shedding at the outer plate edges. The numerically predicted Keulegan–Carpenter number dependent heave added mass and damping coefficients agree reasonably with experimental values, in particular for the deeper submergence.

A fluid impulse nonlinear theory of ship motions and sea loads

A new nonlinear seakeeping theory is presented for the prediction of the large amplitude motions and wave induced loads on ships in severe seastates. The hydrodynamic forces on the ship are evaluated using a new fluid impulse theory which expresses the nonlinear Froude–Krylov, radiation and diffraction forces as time derivatives of integrals of velocity potentials over the instantaneous ship wetted surface circumventing the time consuming evaluation of the pressure from Bernoulli's equation. Analogous expressions are derived for the structural loads. The free surface problems for the ship wave disturbance are solved on vertical planes fixed relative to an earth fixed frame using a 2D + t slender body theory. The linear free surface condition is enforced on the ambient wave profile assumed locally horizontal, a nonlinear boundary condition is imposed on the ship hull and the resulting boundary value problem is solved using the two-dimensional time-domain wave source potential. Computations are presented illustrating the performance of the new theory. The application of the theory to the evaluation of the extreme statistics of the ship responses and structural loads in a stochastic seastate is also addressed.

The Neumann–Michell theory of ship waves

The classical Neumann–Kelvin (NK) linear potential flow model of 3D flow about a ship hull steadily advancing in calm water is reconsidered, and a modified theory—called Neumann–Michell (NM) theory—is given. The main difference between the two theories is that the line integral around the ship waterline that occurs in the classical NK boundary-integral flow representation is eliminated in the NM theory. Specifically, the integrand of the waterline integral in the NK theory is $${G\phi_x-\phi G_x}$$ , where x is the coordinate along the ship length, $${\phi}$$ is the flow potential, and G is the Green function associated with the Kelvin–Michell linear free-surface boundary condition. It is shown that the term $${G \phi_x}$$ does not appear in a consistent linear flow model. Furthermore, the term $${\phi G_x}$$ can be eliminated using a mathematical transformation, which amounts to an integration by parts.

Linear free-surface effects on a horizontally submerged and perforated 2D thin plate in finite and infinite water depths

The wave diffraction, as well as the forced heave and roll oscillations, of a horizontally submerged two-dimensional plate of zero thickness are studied by a semi-analytical solution for both finite and infinite water depths. The flow caused by the plate is represented by a distribution of vortices along the plate that satisfy the linear free-surface, radiation and sea-bottom conditions in the frequency domain. The vortex density is expressed as the sum of a Fourier series and terms that account for the flow singularities at the plate edges. When the plate is perforated, a quadratic (nonlinear) pressure loss condition, which relates the pressure loss/difference to the square of the transverse velocity, is assumed in our detailed studies. A time-efficient iterative solution technique is applied, and less than 10 Fourier series terms are needed to obtain high accuracy. The added mass and damping, wave loads, wave transmission and reflection depend on a perforation-effect Keulegan–Carpenter (KC) number that combines the effects of the perforation ratio, discharge coefficient and KC number. The presented method is verified by comparing it with published results based on other solution methods. The influence of the wave number, submergence, water depth and perforation-effect KC-number is illustrated. Techniques to minimize wave transmission are discussed.

Steady-state liquid sloshing in a rectangular tank with a slat-type screen in the middle: Quasilinear modal analysis and experiments

Two-dimensional resonant liquid sloshing in a rectangular tank equipped with a central slat-type screen is studied theoretically and experimentally with focus on nonsmall solidity ratios of the screen (0.5≲Sn≲0.95), nonlarge number of slots (N≲50), and steady-state conditions. The tank is horizontally and harmonically excited with frequencies in a range covering the two lowest primary-excited natural sloshing resonance frequencies in the corresponding clean tank. The liquid depth is finite. Theoretical analysis is based on the multimodal method with linear free-surface conditions and a quadratic pressure drop condition at the screen expressing an “integral” effect of the screen-induced cross-flow separation (or jet flow). New experimental data on the maximum wave elevations at the wall are compared with the theoretical predictions. Very good agreement is shown for the smallest forcing amplitudes (the forcing amplitude-to-tank width ratio is ≈0.001). Increasing the nondimensional forcing amplitude to ≈0.01 leads to discrepancies due to secondary resonance causing the energy context from the two primary-excited antisymmetric modes to other, first of all, symmetric modes. A further increase of the nondimensional forcing amplitude to 0.03 leads to more complex secondary resonance effects. Specific surface wave phenomena, e.g., wave breaking, are experimentally observed and documented by photographs and videos.

A multimodal method for liquid sloshing in a two-dimensional circular tank

Two-dimensional forced liquid sloshing in a circular tank is studied by the multimodal method which uses an expansion in terms of the natural modes of free oscillations in the unforced tank. Incompressible inviscid liquid, irrotational flow and linear free-surface conditions are assumed. Accurate natural sloshing modes are constructed in an analytical form. Based on these modes, the ‘multimodal’ velocity potential of both steady-state and transient forced liquid motions exactly satisfies the body-boundary condition, captures the corner-point behaviour between the mean free surface and the tank wall and accurately approximates the free-surface conditions. The constructed multimodal solution provides an accurate description of the linear forced liquid sloshing. Surface wave elevations and hydrodynamic loads are compared with known experimental and nonlinear computational fluid dynamics results. The linear multimodal sloshing solution demonstrates good agreement in transient conditions of small duration, but fails in steady-state nearly-resonant conditions. Importance of the free-surface nonlinearity with increasing tank filling is explained.

Analytical modeling of liquid sloshing in a two-dimensional rectangular tank with a slat screen

Potential-flow theory is employed with linear free-surface conditions, multimodal method, and a screen-averaged pressure-drop condition to derive an analytical modal model describing the two-dimensional resonant liquid motions in a rectangular tank with a vertical slat-type screen in the tank middle. The tank is horizontally excited in a frequency range covering the two lowest natural sloshing frequencies. The model consists of a system of linear ordinary differential [modal] equations responsible for liquid sloshing in compartments, as well as a nonlinear ordinary differential equation describing the liquid flow between the compartments. New experimental model tests on steady-state wave elevations near the tank wall are reported for the solidity ratios 0.328 ≤ Sn ≤ 0.963 where Sn is the ratio between the solid area and the full area of the screen. The experiments generally support the applicability of the model. The discrepancy can be explained by the free-surface nonlinearity. The screen acts as a damping mechanism for low and intermediate solidity ratios, but it causes an increase in the lowest resonant sloshing frequency at higher solidity ratios as if the screen had been replaced by an unperforated wall.

Numerical simulation of transient hydroelastic response of a floating beam induced by landing loads

The dynamic response of a floating platform under the effects of impulsive and moving loads is of concern in various areas of engineering technology, such as floating airports, floating bridges and buoyant tunnels, among others. In the present work, the floating platform is simplified as a flexible beam floating in an infinite water domain. The water is assumed to be compressible and inviscid, whose governing equation is formulated in terms of the hydrodynamic pressure variable. The surface disturbance satisfies a linear free-surface wave condition and a radiation wave condition at an artificially truncated boundary. A time-domain finite element model is developed to analyze this fluid–structure interactive dynamical system, which exhibits strong coupling between the system’s components. The numerical results are compared with the experimental and numerical ones. The results of comparison confirmed the validity of the proposed approach. The influences on the dynamic responses of floating beam of some factors were studied.

Far-Field Ship Waves

The wave field of ships computed by a Rankine panel method is ‘extrapolated’ up to large distances from the ship. For deep water or shallow water of constant depth, a Fourier analysis of the waves within a small ‘analysis rectangle’ allows to determine the waves at other locations where the linear free-surface condition holds. The method is applicable also for a curved ship path. For shallow water of variable depth, a mild-slope wave propagation equation is solved by a finite-difference method separately for various wave frequencies.

An improved matching method to solve the diffraction—radiation problem of a ship moving in waves

On the underlying assumption that the disturbance of the far-field velocity potential caused by the motion of a ship is small and may be linearized, an improved matching method is developed. Two arrays of fundamental singularities are placed inside the ship hull, which satisfy the linear free surface condition outside the truncated fluid domain and the far-field radiation condition. The choice of fundamental singularity depends on the problem under investigation (e.g. pulsating, translating or translating and pulsating source). The unknown strength of these singularities and the near-field velocity potential are determined in a coupled manner. It is shown from numerical examples that the present method provides a more efficient and accurate representation of waves in the far field than conventional matching methods.

A Simple Green Function for Diffraction-Radiation of Time-Harmonic Waves with Forward Speed

Green functions that satisfy the radiation condition and the linearized free- surface boundary condition in the farfield – but only approximately in the nearfield – are given for diffraction-radiation of time-harmonic waves with forward speed and the special cases corresponding to zero forward speed and zero wave frequency. These Green functions are considerably simpler than the related Green functions employed in the literature, which satisfy the linearized free-surface boundary condition everywhere, i.e. in both the farfield (where the linear free-surface condition is valid) and the nearfield (where the linear free-surface condition is only an approximation, due to nearfield effects).

A numerical method to solve the m‐terms of a submerged body with forward speed

AbstractTo model mathematically the problem of a rigid body moving below the free surface, a control surface surrounding the body is introduced. The linear free surface condition of the steady waves created by the moving body is satisfied. To describe the fluid flow outside this surface a potential integral equation is constructed using the Kelvin wave Green function whereas inside the surface, a source integral equation is developed adopting a simple Green function. Source strengths are determined by matching the two integral equations through continuity conditions applied to velocity potential and its normal derivatives along the control surface. After solving for the induced fluid velocity on the body surface and the control surface, an integral equation is derived involving a mixed distribution of sources and dipoles using a simple Green function and one component of the fluid velocity. The normal derivatives of the fluid velocity on the body surface, namely the m‐terms, are then solved by this matching integral equation method (MIEM).Numerical results are presented for two elliptical sections moving at a prescribed Froude number and submerged depth and a sensitivity analysis undertaken to assess the influence of these parameters. Furthermore, comparisons are performed to analyse the impact of different assumptions adopted in the derivation of the m‐terms. It is found that the present method is easy to use in a panel method with satisfactory numerical precision. Copyright © 2002 John Wiley & Sons, Ltd.

Time-Domain Analysis of Large-Amplitude Vertical Ship Motions and Wave Loads

The vertical motions and wave induced loads on ships with forward speed are studied in the time domain, considering non-linear effects associated with large amplitude motions and hull flare shape. The method is based on a strip theory, using singularities distributed on the cross sections which satisfy the linear free surface condition. The solution is obtained in the time domain using convolution to account for the memory effects related to the free surface oscillations. In this way the linear radiation forces are represented in terms of impulse response functions, infinite frequency added masses and radiation restoring coefficients. The diffraction forces associated with incident wave scattering are linear. The hydrostatic and Froude-Krylov forces are evaluated over the instantaneous wetted surface of the hull to account for the large amplitude motions and hull flare. The radiation contribution for wave loads is also obtained in the time domain using convolution to account for the memory effects related to the free surface oscillations. Results of motions and wave loads for the S175 container ship are presented and analyzed. The results from the present method are compared with linear results.