Second-order Velocity Potential Research Articles

Second-order wave loads on a floating bridge of T-type cross section

An analytical model for the complete second-order wave loads on a floating bridge of T-type cross section is established within the framework of potential flow theory. The first-order problem is solved using the matched eigenfunction expansion method (MEEM). Meanwhile, the second-order wave loads are divided into two distinct parts which are computed separately. The first part is dependent on the quadratic products of the first-order velocity potential and its derivative. It has been evaluated through two different ways. One is based on a direct pressure integration, i.e., a near-field method. To ensure a high accuracy of the computation, a local expansion is applied to determine the spatial derivative of the first-order velocity potential at a sharp corner. The other is based on the momentum conservation over a control volume. The effect of the control surface location on the computation has been discussed. In addition, the two proposed methods have been applied to both the time-independent wave-load component as well as the second-harmonic component, respectively. The second part of the complete second-order wave loads is due to the second-order velocity potential. It is evaluated based on an indirect method by introducing an auxiliary potential. Detailed numerical investigation is then conducted. The influence law of the sectional geometric parameters on the second-order wave loads is investigated. The numerical results reveal that the cantilever plate imposes an apparent protection effect on the inner web beam. Such effect gets obviously enhanced as the section net width or section height decreases or as the plate width or draft increases.

Effects of the sound speed vertical profile on the evolution of hydroacoustic waves

We present a novel analytical model for the evolution of hydroacoustic waves in weakly compressible fluids characterised by depth variations of the sound speed profile. Using a perturbation expansion in terms of the small vertical variation of the sound speed, we derive a novel expression for the second-order velocity potential and show that this solution does not exist in the case of homogeneous sound speed. At the third order, we derive a linear Schr¨odinger equation governing the evolution of the wave envelope for large length and time scales, which features new terms depending on the sound speed distribution. We show that for generalised sound speed vertical profiles the frequency of the hydroacoustic signal can increase or decrease with respect to the constant sound speed case, depending on the profile. This has substantial implications on the speed of the wavetrain envelope. Our findings suggest the need to extend existing models that neglect the sound speed vertical variation, especially in view of applications to tsunami early warning.

A numerical study of second-order springing of an elastic body using higher-order boundary element method (HOBEM)

Second-order springing on an elastic body with forward speed is analyzed by numerical simulations. The boundary-value problem for the velocity potential is solved by means of the direct time-domain higher-order boundary element method (HOBEM). The free-surface boundary condition in the boundary-value problem is approximated on the mean surface up to second order by use of perturbation and Taylor-series expansion methods. The body boundary condition for an elastic body is derived with various quantities which are redefined in the generalized mode. These variables such as mode shape, normal vector, etc. are obtained by using directional derivative and continuum mechanics, and the same mathematical expressions are used to obtain several second-order generalized forces. To validate the numerical results, the second-order hydrodynamic force on the bottom-mounted rigid/elastic cylinders without forward speed is compared with other semi-analytic results. The property of second-order forces on an elastic ship is studied by changing the flexural rigidity and forward speed with elastic response. It is confirmed that the second-order velocity potential is important for a body with forward speed and investigation should be made more on numerical methods for accurate computation of the second-order velocity-potential force with forward speed.

A numerical investigation of second-order difference-frequency forces and motions of a moored ship in shallow water

A 3D time-domain Rankine source method is developed to study the hydrodynamic loads and motions of a moored ship in shallow water waves in head sea conditions. Both the wave steepness and ship motions relative to the ship’s draft are assumed small and the exact free-surface and body-boundary conditions are expanded about the mean surface by a Taylor series. A formulation correctly to second order in the wave steepness is adopted. A fourth-order Runge–Kutta method is used to time integrate the boundary conditions and the six degree of freedom motion equations. It is found that the water depth has significant effects on the hydrodynamic coefficients, especially on the vertical modes of motions. The linear horizontal motions of a moored ship have distinct increment in shallower water depth in the low-frequency domain. Further, the horizontal slow-drift excitation forces increase significantly with decreasing water depth and the second-order velocity potential gives dominant contribution in a frequency range of importance for moored ships in shallow water. Lastly, the slowly varying motions of an LNGC is simulated and the satisfactory agreements with experiments demonstrate that the present method can predict the slowly varying motions of a moored ship in finite-amplitude shallow water waves with acceptable results.

Second-order low-frequency drift motions of a floating body calculated by different approximation methods

The paper presents an experimental and numerical investigation of the low-frequency motions of a simple geometry floater subjected to bichromatic waves and to long crested irregular seas. The body is axis-symmetric about the vertical axis and it is restrained from drifting away by a linear mooring system. The investigation is carried out for three water depths representing deep water, intermediate water depth and shallow water. The objective is to assess the water depth influence on the second-order low-frequency motions. A second objective is to assess the results of different approximations for the second-order difference frequency wave exciting forces on the second-order motions. The quadratic transfer functions are calculated with a boundary element method using several levels of approximation for the second-order forces: (a) the most complete approximation solves the boundary value problem completely up to the second order, (b) the first-order approximation neglects the free surface forcing in the 2nd order boundary value problem solution, (c) Newman’s approximation is of zeroth-order with respect to the difference frequency, (d) a fourth method combines Newman’s approximation with a contribution from the second-order incident wave potential, (e) the fifth method is applied to the heave forces only and it combines Newman’s approximation corrected by an additional set down. This study shows that the contribution from the second-order velocity potential must be considered for shallow water calculations to achieve accurate results. For small difference frequencies, second-order scattering potential effects are small; therefore, in this case, a good practical approximation consists on considering the second-order potential is contributed by the incident waves only.

A numerical study of the second-order wave excitation of ship springing in infinite water depth

Assuming the wave steepness of the incident waves and the ship motions are small, the second-order weakly-nonlinear hydrodynamic problem of a ship moving with constant forward speed is studied numerically in a consistent way. The boundary value problem is formulated in a body-fixed coordinate system and the perturbation scheme is used. This formulation does not include any derivatives of the velocity potential on the right-hand side of the body-boundary conditions, and thus avoid the difficulties associated with the terms similar to the so-called mj-terms and their derivatives. The second-order sum-frequency wave excitation of ship springing is studied in both monochromatic and bichromatic head-sea waves. Different Froude numbers are considered. A time-domain Higher-Order Boundary Element Method based on cubic shape function is used as a numerical tool. An upstream finite difference scheme is used for longitudinal derivative terms in the free-surface conditions. For a modified Wigley hull in head-sea waves, it is found that the second-order velocity potential gives dominant contribution to second-order wave excitation of ship springing in the wave frequency region where sum-frequency springing occurs. Quadratic velocity terms in the Bernoulli equation have a relatively small contribution. The numerical results also demonstrate strong dependency of the second-order wave excitation of ship springing on the Froude numbers for small wave lengths. The effect of beam and draft is investigated.

The analytic form of Green’s function in elliptic coordinates

The purpose of this study is the derivation of a closed-form formula for Green’s function in elliptic coordinates that could be used for achieving an analytic solution for the second-order diffraction problem by elliptical cylinders subjected to monochromatic incident waves. In fact, Green’s function represents the solution of the so-called locked wave component of the second-order velocity potential. The mathematical analysis starts with a proper analytic formulation of the second-order diffraction potential that results in the inhomogeneous Helmholtz equation. The associated boundary-value problem is treated by applying Green’s theorem to obtain a closed-form solution for Green’s function. Green’s function is initially expressed in polar coordinates while its final elliptic form is produced through the proper employment of addition theorems.

Asymptotic estimates of hydrodynamic loads in the early stage of water entry of a circular disk

The hydrodynamic loads generated during the entry of a circular disk into deep water are evaluated with the help of the method of matched asymptotic expansions. It is assumed that the liquid is initially at rest and the disk is floating on the still liquid surface. Then the disk suddenly starts its downward motion. The study is carried out under the assumption of an ideal and incompressible liquid. Attention is focused on the initial stage of the entry process. The solution is sought in the form of an asymptotic expansion of the velocity potential with the non-dimensional displacement of the disk being a small parameter of the problem. Gravity and surface-tension effects are shown to be of minor significance. Owing to the flow singularity at the edge of the disk, an inner problem is formulated and its solution is matched with the second-order outer velocity potential to achieve a uniformly valid solution. It is shown that the initial asymptotics of the hydrodynamic loads involves terms with \({h^{-\frac{1}{3}}}\) and log h where h(t) is the non-dimensional displacement of the disk. Both terms are unbounded in the limit of small penetration depth of the disk. The theoretical estimates are validated versus fully nonlinear numerical simulations of the problem during the later stage of the process. It is shown that the derived asymptotic estimates remain accurate, even for moderate displacements of the disk. The relative difference between the theoretical estimate of the hydrodynamic force and its numerical prediction is less than 5% when the penetration depth is smaller than \({\frac{1}{20}}\) of the disk radius. A way to use the theoretical estimates for practical applications is proposed and comparisons with experimental data available in the literature are also presented.

Second-order hydrodynamic effects on an arrangement of two concentric truncated vertical cylinders

The second-order diffraction problem by a piston-like arrangement that consists of two concentric surface piercing cylinders is considered. The developed matched axisymmetric eigenfunction expansion solution methodology in cylindrical co-ordinates is based on the semi-analytical formulation of the velocity potentials in the various fluid regions which are defined by the geometry of the two-body arrangement. The main difficulty associated with the specific configuration originates from the fact that the geometry defines two fluid regions that extend up to the free surface in which the inhomogeneous second-order free surface boundary condition has to be fulfilled. To this end the associated velocity potentials in these regions are decomposed into a number of components defining the so-called ‘free’ and ‘locked’ waves. The latter are calculated by solving the resulting Sturm–Liouville problems. The seek second-order velocity potential in the whole fluid domain is then derived by enforcing matching conditions for the radial velocities and the fluid pressures at the cylindrical boundaries of adjacent fluid domains. Numerical results concerning the second-order hydrodynamic loading and the wave run-up on the cylinders are given, whereas special attention is given at incident wave frequency regions where the first-order exciting wave forces attain maximum values due to the resonant fluid motions in the moonpool.

Mechanics of nonlinear random wave groups interacting with a vertical wall

Nonlinear random wave groups interacting with a vertical wall are investigated. The analytical solution for the second-order free surface displacement and velocity potential when a high crest occurs at some fixed point on, or close to, the vertical wall is obtained. The solution is exact for any water depth, and it is given as a function of the frequency spectrum of the incident waves. It is obtained that the effects of nonlinearity strongly modify the linear structure of wave groups both in the space and the time domain. The maximum effect of nonlinearity occurs when the high wave hits the wall. Furthermore, it is shown that in finite water depth, the nonlinearity increases as the bottom depth decreases. Finally, a validation by means of Monte Carlo simulations of nonlinear random waves in reflection is given.

Second-order Wave Loads On Offshore StructuresUsing TheWeber’s Transform Method

The second-order wave loads are computed for the diffraction of monochromatic waves by a surface-piercing vertical cylinder in the finite and infinite fluid depths. The Weber transform method which is essentially a Hankel transform method with a more general kernel, is applied to compute the second-order force due to the second-order velocity potential. Suitable closed contours in the complex plane are used to derive the analytical solution of the improper integrals involved in this study. This makes the present solution distinct from the other available solution of the second-order forces.

Second-order interaction of irregular waves with a truncated column

A complete semi-analytical solution is obtained for second-order diffraction of plane bichromatic waves by a fixed truncated circular column. The fluid domain is divided into interior and exterior regions. In the exterior region, the second-order velocity potential is expressed in terms of ‘locked-wave’ and ‘free-wave’ components, both are solved using Fourier and eigenfunction expansions. The resulting ‘locked wave’ potential is expressed by one-dimensional Green's integrals with oscillating integrands. In order to increase computational efficiency, the far-field part of the integrals are carried out analytically. Solutions in both regions are matched on the interface by the potential and its normal derivative continuity conditions. Based on the present approach, the sum-and difference-frequency potentials are efficiently evaluated and are used to generate the quadratic transfer functions which correlates sum-frequency QTFs for a TLP column are present, which are compared for some frequency pairs with those from a fully numerical procedure. Satisfactory agreement has been obtained. QTF spectra for a case study TLP column, generated using the semi-analytical solution are presented. Also given are the results for nonlinear wave field around the column.

Diffraction of nonlinear random waves by a vertical cylinder in deep water

Abstract A solution, exact to second-order, is presented for the nonlinear diffraction of random waves by a fixed, surface-piercing vertical circular cylinder in deep water. The incident wave field is considered as a stationary random process, and the nonlinear diffraction problem is analyzed utilizing the Stokes perturbation expansion procedure combined with a Fourier-Stieltjes spectral representation of the stationary random wave kinematics. The second-order velocity potential is explicitly obtained by applying a modified form of Weber's Integral Theorem to invert the inhomogeneous second-order free-surface condition. Particular attention is directed towards the second-order diffraction forces on the cylinder. The spectral description of the second-order diffraction forces involves a complicated integral expression with highly oscillatory wave-wave interaction kernels and multiple convolutions of the linear wave spectrum. The present approach provides a complete spectral description of the second-order diffraction forces, and yields the spectral densities of the diffraction forces at the sum and difference frequencies. Numerical results are presented which illustrate the spectral content of the diffraction force due to an incident wave field represented by a superposition of waves described by band-limited white noise processes centered at different frequencies.

Second-order Long Period Wave Forces on an Offshore Structure in Shallow Water (2nd report)

In this paper, the second-order long period wave forces acting on a two-dimensional fixed obstacle of arbitrary shape in bichromatic waves in a finite depth water are investigated. In the evaluation, the second-order effect of a wave maker is included to satisfy the precise boundary condition in an experimental tank. The solution of the second-order diffraction problem is obtained by using hybrid type boundary integral equation method. One of the merit of this method is that the complicated second-order free surface condition can be treated easily by using the analytical form of second-order velocity potential which was derived in 1st report.Computations are done for a half immersed circular cylinder and the calculated results are compared with the results obtained from Green's second identity. These results are also compared with experimental values and the validity of the present method is discussed.

A hybrid boundary element method for second-order wave-body interaction

A new hybrid numerical-analytical method is developed for the evaluation of the second-order velocity potential associated with bichromatic incident waves attacking a floating long horizontal cylinder of arbitrary cross-section, in water of finite depth. The double sum or difference frequency velocity potentials may be obtained. An analytical treatment of the nonhomogeneous free surface condition makes the numerical solution accurate and efficient. The method is tested through a numerical example. Results showing the difference frequency sway force and sway response transfer function are presented for practical parameters of a moored vessel in a multi-buoy anchorage.

Second-Order Biharmonic Response of a Tension-Leg Platform

The procedure for predicting the second-order biharmonic response and tether forces of a tension-leg platform in regular waves is presented. This consists of evaluating the second-order wave exciting forces based on the second-order wave diffraction theory, and solution of the equation of motion including the effect of the second-order mooring tether forces. Model tests are carried out to measure the wave exciting forces on the fixed circular dock, as well as the response and tether forces of the tension-moored circular dock. Satisfactory agreement is observed between the theoretical and experimental predictions, confirming the validity of the proposed method. The conclusions arising from the present study are summarised as follows : 1) The vertical-plane motion and tether forces of the tension-leg platform in regular waves include the second-order biharmonic components which are comparable in magnitude with the first-order wave-frequency components.2) The nonlinear sum-frequency wave exciting forces provide an important source of excitation of such oscillation. These exciting forces can be predicted accurately based on the second-order wave diffraction theory.3) The contribution of the second-order velocity potential to the second-order wave exciting force is significant, in particular to the heave and pitch exciting forces. The simplified method neglecting this contribution is found to underestimate the forces drastically.4) The theoretical method proposed in this study provides an effective means for predicting the second-order sum-frequency response and tether forces of the tension-leg platform.

Computation of Slowly Varying Second-Order Hydrodynamic Forces on Floating Structures in Irregular Waves

An exact second-order formulation is presented for computing the slowly varying second-order hydrodynamic forces on floating structures in irregular waves. The near-field approach based on direct integration of the fluid pressure on the submerged body surface is employed in conjunction with numerical first-order solutions by means of the hybrid finite element technique. Green’s second identity is exploited to evaluate the second-order forces due to the second-order velocity potential. Numerical results are presented for the slow drift excitation forces on an articulated column and a semi-submersible platform. It is shown that the contribution from the second-order velocity potential is more significant to the roll moment than to the sway and heave forces on the semi-submersible.

半潜水式海洋構造物の不規則波中での長周期運動の数値シミュレーション

The paper reports the results of time-domain simulation of the slow-drift motions of a semi-submersible in irregular seas. The simulation is based on exact evaluation of the second-order wave exciting forces including the contribution from the second-order velocity potential. The validity of the simulation method is confirmed by comparing the simulated time histories of motions with the experimental data measured in model tests. The conclusions arising from limited simulation results reported herein are summarised as follows: (i) The sway and roll responses of the semi-submersible in irregular beam waves are dominated by the slow-drift resonance motions excited by the low-frequency component of the second-order wave forces. This is due to the very low natural frequencies and low damping possessed by the system. (ii) The slow-drift sway motion can be predicted fairly well with the knowledge of mean drift forces in regular waves, as suggested by Newman (Ref. 1)). (iii) The contribution of the second-order velocity potential to the slow-drift roll moment is significant. Exact evaluation of the second-order wave forces are therefore essential for correct prediction of the slow-drift roll motion of the semi-submersible. (iv) The present simulation method based on exact evaluation of the second-order wave forces can reproduce with reasonable accuracy the motions of the experimental model measured in a wave flume.

Radiation and diffraction of second-order surface waves by floating bodies

The paper studies the radiation and diffraction by floating bodies of deep-water bichromatic and bidirectional surface waves subject to the second-order free-surface condition. A theory is developed for the evaluation of the second-order velocity potential and wave forces valid for bodies of arbitrary geometry, which does not involve the evaluation of integrals over the free surface or require an increased accuracy in the solution of the linear problem. Explicit sum- and difference-frequency ‘Green functions’ are derived for the radiation and diffraction problems, obtained from the solution of initial-value problems that ensure they satisfy the proper radiation condition at infinity. The second-order velocity potential is expressed as the sum of a particular and a homogeneous component. The former satisfies the non-homogeneous free-surface condition and is expressed explicitly in terms of the second-order Green functions. The latter is subject to the homogeneous free-surface condition and enforces the body boundary condition by the solution of a linear problem. An analysis is carried out of the singular behaviour of the second-order potential near the intersection of the body boundary with the free surface.

Nonlinear wave loads on a vertical cylinder

Nonlinear contributions due to elevation of the free surface, the dynamic head, and the second-order velocity potential on the wave loads are presented in closed-form expressions. Such nonlinearities resulting from large-amplitude ocean waves are associated with the irrotational flow interacting with a fixed bottom-mounted vertical cylinder piercing the surface. These are expressed in the form of dynamic, waterline and quadratic forces all of which depend on the square of the wave amplitude. The appropriate modifications are made to both the classical Morison equation and the well-known linear diffraction theory of MacCamy and Fuchs for accounting the second-order effects. A limited comparative study is performed to verify the present theoretical derivations. In general, satisfactory agreements have been obtained with the test results from various laboratory studies by different researchers. However, under certain environmental conditions, some discrepancies still exist with the measured results.