Neumann Kelvin Research Articles

Numerical study on mean wave drift forces and moment of ships using Rankine panel method in frequency domain

A three-dimensional frequency-domain seakeeping model based on the Rankine panel method is developed to compute the mean wave drift forces and moment acting on ships with forward speeds. Two different linearization models (the double-body and Neumann–Kelvin models) are applied to evaluate the mean wave drift forces and moment using a near-field method. In these models, the Rayleigh’s artificial friction is adopted in the free-surface boundary condition to suppress longer wave components in computational regions apart from the ship. A novel upwind difference scheme is introduced to calculate the first-order and second-order derivatives of the velocity potential on the free surface. Wigley I, Wigley III and S175 containership are adopted in numerical simulations to validate the effectiveness of the developed model. In the present study, the motion response amplitude operators in head, beam and oblique seas obtained using the two models are compared, and their contributions to the mean wave drift forces and moment are analyzed. Results show that in head seas, the pitch motions have significant effects on the surge-directional wave drift force Fx. In beam seas, the heave motions have the most effect on the mean wave drift forces and moment (Fx, Fy and Mz).

Numerical investigations on seakeeping and added resistance in head waves based on nonlinear potential flow methods

A fully nonlinear potential flow (FNPF) method and a body-exact (BE) method based on the time-domain desingularized-Rankine panel (DRP) method have been developed to compute the motion responses and added resistance for ships traveling in regular head waves. In the present methods, the disturbed wave potential is separated from the incident potential, and the boundary value problem (BVP) for the disturbed potential is established in each method. To validate the present FNPF and BE methods, different hull forms are adopted, including two Wigley hulls and the S-175 containership. The results of radiation, diffraction, and free motion simulations are presented and systematically compared with the experimental data and linear solutions using the double-body (DB) and Neumann–Kelvin (NK) methods. Through comparison, we demonstrate that the present FNPF method can accurately predict the motion responses and added resistance of ships. For the ships with large flare angles, the FNPF and BE methods give better predictions of peak motion responses than those using DB and NK. In addition, the FNPF method can well predict the added resistance, and has significant advantages over the other numerical methods at high forward speed (Fn=0.3). The analyses also confirm that, compared with the heave and pitch motions, the added resistance and the total wave elevations are more sensitive to the incident wave amplitude. The dimensionless added resistance decreases with the increase of incident wave amplitude, which is mainly driven by the total wave elevations along the waterline in the bow region. As the incident wave amplitude increases, the 1st harmonic amplitudes of the heave and pitch motions and the total wave elevations in the bow region decrease, while the 2nd harmonic amplitudes increase.

Boundary-integral representation sans waterline integral for flows around ships steadily advancing in calm water

The boundary-integral flow-representation associated with the boundary-value problem, commonly called Neumann–Kelvin (NK) problem, that corresponds to linear potential flow around a ship steadily advancing in calm water involves an integral around the mean waterline of the ship. This ‘waterline integral’ is a notorious source of numerical difficulties and has been extensively studied. The waterline integral in the NK theory is largely – but not fully – eliminated in the modification, called Neumann–Michell (NM) theory, of the NK theory. Specifically, the NM theory includes a residual waterline-distribution of weak Rankine singularities, ignored in practical applications. A crucial element of the NM theory is a mathematical transformation that is based on a vector Green function, which is associated with the common scalar Green function used in the NK theory. This transformation is revisited in the present study. A rigorous analysis yields an answer to a fifty-year old puzzle: an exact boundary-integral flow representation that does not include a waterline integral. A remarkable feature of this new flow representation, which is a modification of the NM flow representation given previously, is that it explicitly determines the flow potential and the flow velocity at a ship hull surface in terms of the flow velocity at the hull surface, rather than in terms of the hull-surface potential as in usual boundary-integral flow representations obtained via Green’s classical identity.

Solvability of a three-dimensional free surface flow problem over an obstacle

ABSTRACT In this paper, we consider a problem of a three-dimensional free surface flow over an obstacle lying on the bottom of an infinite channel. The flow is supercitical, irrotational, stationary and the fluid is ideal and incompressible. We linearize the problem, then by a Fourier expansion of the velocity potential, we reduce the three-dimensional problem to a sequence of two-dimensional problems for the Fourier coefficients. We take into account of the gravity and we neglect the effects of the superficial tension. These problems are formulated by the Laplace operator and the Neumann–Kelvin condition on the free surface of the fluid domain. Some a priori properties of the solution are given, they allow us to construct a space, where we can use Lax–Milgram's theorem, to prove the existence and the uniqueness of the solution of the two-dimensional problem. Then, this sequence of solutions of 2D-problems, gives us the unique solution of the considered three-dimensional problem.

Rankine-type cylinders having zero wave resistance in infinitely deep flows

Abstract We prove the existence of a family of immersed obstacles that have zero wave resistance in the context of the 2D Neumann–Kelvin problem. We first build a waveless potential by superposing a source and a sink in a uniform flow for an appropriate choice of parameters. The obstacle is obtained by a combination of streamlines of the waveless potential. Numerical simulations show that the construction is valid for a large set of parameters.

A 3D direct coupling method for steady ship hydroelastic analysis

Asides from the influence of incoming waves, ships can experience steady motions, such as rigid-body sinkage and trim motions, and flexible-body vertical bending motions, due to a constant forward speed even under calm water conditions. In this paper, a novel approach to analyze steady-ship hydroelasticity, particularly for the steady-ship motions and surrounding steady-wave disturbances, is proposed using a three-dimensional (3D) direct coupling method, based on a higher-order boundary element method (HOBEM) and a higher-order shell finite element method (FEM). Within the linearized framework, a solution method is proposed based on a two-step procedure, using two types of Neumann–Kelvin (NK) linear flow models for the fluid part and a virtual work equilibrium equation for the structural part. The first step is to compute a mean position wave-resistance problem using the modified NK equation, the second step is to solve a perturbed position wave-resistance problem, by employing a classical NK model and a virtual work equation based on the first step’s solution. Detailed mathematical formulation and numerical procedures are described, and a few numerical results are illustrated. These include both rigid and flexible steady-ship motions, Von-Mises stress distributions, and wave-resistance coefficients for Froude numbers ranging from 0.15 to 0.5. Furthermore, the numerical results obtained using the present direct coupling method and a modal-based one are compared.

Numerical Study on Hydrodynamics of Ships with Forward Speed Based on Nonlinear Steady Wave

In this paper, an improved potential flow model is proposed for the hydrodynamic analysis of ships advancing in waves. A desingularized Rankine panel method, which has been improved with the added effect of nonlinear steady wave-making (NSWM) flow in frequency domain, is employed for 3D diffraction and radiation problems. Non-uniform rational B-splines (NURBS) are used to describe the body and free surfaces. The NSWM potential is computed by linear superposition of the first-order and second-order steady wave-making potentials which are determined by solving the corresponding boundary value problems (BVPs). The so-called mj terms in the body boundary condition of the radiation problem are evaluated with nonlinear steady flow. The free surface boundary conditions in the diffraction and radiation problems are also derived by considering nonlinear steady flow. To verify the improved model and the numerical method adopted in the present study, the nonlinear wave-making problem of a submerged moving sphere is first studied, and the computed results are compared with the analytical results of linear steady flow. Subsequently, the diffraction and radiation problems of a submerged moving sphere and a modified Wigley hull are solved. The numerical results of the wave exciting forces, added masses, and damping coefficients are compared with those obtained by using Neumann–Kelvin (NK) flow and double-body (DB) flow. A comparison of the results indicates that the improved model using the NSWM flow can generally give results in better agreement with the test data and other published results than those by using NK and DB flows, especially for the hydrodynamic coefficients in relatively low frequency ranges.

Comparative study on ship motions in waves based on two time domain boundary element methods

This paper presents a comparative study on the motions of a ship advancing in waves using two different three-dimensional time domain boundary element methods: a Transient Free surface Green's Function (TFGF) method and a Rankine Higher Order Boundary Element Method (HOBEM). Three models based on the HOBEM are also considered to identify the dominant factors affecting the accuracy of solutions: (i) the N–K model using Neumann–Kelvin (N–K) linearization, (ii) the N–K+DB model using the N–K linearization for the free surface boundary conditions and the double-body (DB) m-terms in the body boundary condition, (iii) the Rankine HOBEM based on DB linearization. The Wigley I and the Series 60 (CB = =0.7) are taken as study objects. The comparisons show that the Rankine HOBEM based on DB linearization is generally more accurate than the Rankine HOBEM with other two models and the TFGF method. The hydrodynamic coefficients are mainly affected by the m-terms, especially at low frequencies; while the influences of different linearizations of the free surface boundary conditions are negligible. Both the m-terms and the leading terms of steady velocity potential reserved in the free surface boundary conditions are important for motion responses, while the influence of the former is stronger than the latter.

Seakeeping computation of two parallel ships with Rankine source panel method in frequency domain

Present paper aims to develop a three dimensional frequency domain panel method to investigate the hydrodynamic performance of two parallel ships in close proximity with forward speed. As for present method, the Rankine source is adopted; meanwhile, the linearized free surface condition based on a double body or uniform basis flow, which can be shorten as “DB” or “Neumann–Kelvin (NK)”, is derived. As for the radiation condition at infinity, a simple combined evaluation method (SCM) is proposed, which is validated through comparison with numerical results from combined boundary-integral equation method (CM, the whole fluid domain is decomposed into the far-field and the near-field, and the Rankine source & 3DTP is used for the far-field and near-field domain, respectively, to satisfy the radiation condition exactly) and experimental results. Furthermore, present method is applied to evaluate hydrodynamic performance of two parallel ship models. Meanwhile, numerical solutions evaluated by “NK” method and “LFS-DB” method were also given for comparison. Results for present various numerical method agree with those of model experiments. Generally speaking, results from“DB” method show better agreement.

Numerical prediction of wave added resistance using a Rankine Panel method

Based on a time domain Rankine panel method, a computer code was developed to predict wave added resistance. The unknown variables of the discretized flow field boundaries were represented using the quadratic B-spline basis function; the temporal derivative of the free surface conditions were approximated using an Euler scheme. The added resistance was computed via a near-field method. Three numerical approaches were introduced, namely, choosing the boundary end condition, neglecting the second order derivative of the disturbed potential, and computing the water line integral. Numerical computations were conducted in short and long waves for a Wigley hull, the S-175 container ship, and the KVLCC2 tanker. The double-body linearization and the Neumann–Kelvin linearization were tested, and numerical results were compared to published experimental measurements and numerical data. Fairly good agreement was found between the results of double-body linearization and the published data. Effects of wave radiation and diffraction on added resistance were analyzed, demonstrating that the interaction between radiation and diffraction should not be omitted in long waves.

Boundary-integral representations for ship motions in regular waves

Alternative boundary-integral representations of linear potential flow around a ship that travels at a constant speed in regular (time-harmonic) waves in deep water are considered. In particular, the Neumann–Michell (NM) boundary-integral representation of steady flow around a ship traveling in calm water is extended to ship motions in regular waves. This extension is straightforward, but points to a complicated and unresolved fundamental issue that is essentially related to the formulation of a linear flow model. Indeed, the NM boundary-integral flow representation for ship motions in regular waves given in the study suggests that the classical Neumann–Kelvin (NK) flow model may be inadequate at a ship waterline. Alternative NM boundary-integral representations of time-harmonic ship waves that are consistent with the previously given NM representation of ship waves in calm water, and yield Fourier–Kochin representations of ship waves that are well suited for numerical computations, are given.

Boundary integral representations of steady flow around a ship

The classical Neumann–Kelvin (NK) linear potential flow theory of ship waves in calm water and the related Neumann–Michell (NM) theory are considered. Five alternative boundary integral representations are given: (1) the classical NK integro-differential representation, called “classical NK formulation”, which corresponds to an inconsistent linear flow model, (2) a modification of the classical NK flow formulation that corresponds to a consistent linear flow model and is called “consistent NK formulation”, (3) a flow representation, called “NM potential and velocity formulation”, that involves the flow potential ϕ and the velocity components ϕd and ϕt along two unit vectors d and t tangent to the ship hull surface, and yields an integro-differential equation for determining ϕ, (4) a flow representation, called “NM velocity formulation”, that only involves ϕd and ϕt and yields a pair of coupled integral equations for determining (ϕd,ϕt), and (5) a flow representation called “NM potential formulation” that only involves the flow potential ϕ and yields an integral equation for determining ϕ. The two NK formulations involve both a surface integral over the ship hull surface and a line integral around the ship waterline, whereas the three NM formulations do not involve a waterline integral. All flow representations other than the classical NK representation are based on a consistent linear flow model.

Ship-hull shape optimization with a T-spline based BEM–isogeometric solver

In this work, we present a ship-hull optimization process combining a T-spline based parametric ship-hull model and an Isogeometric Analysis (IGA) hydrodynamic solver for the calculation of ship wave resistance. The surface representation of the ship-hull instances comprise one cubic T-spline with extraordinary points, ensuring C2 continuity everywhere except for the vicinity of extraordinary points where G1 continuity is achieved. The employed solver for ship wave resistance is based on the Neumann–Kelvin formulation of the problem, where the resulting Boundary Integral Equation is numerically solved using a higher order collocated Boundary Element Method which adopts the IGA concept and the T-spline representation for the ship-hull surface. The hydrodynamic solver along with the ship parametric model are subsequently integrated within an appropriate optimization environment for local and global ship-hull optimizations against the criterion of minimum resistance.

Viscous and potential forces on an advancing surface-piercing flat plate with a fixed drift angle

Transverse force and yaw moment acting on a surface-piercing flat plate with yaw angle and forward speed are studied. Following the Froude hypothesis, the problem is decomposed into a tail-separated forward flow and a bottom-tip-separated cross-flow parts. The objective is to quantify the role of different components in the resulted force and moment acting on the plate. A 3D potential flow code using distribution of Rankine sources and dipoles is used to solve the linear potential flow problem. The free-surface boundary condition is linearized based on the undisturbed flow velocity (Neumann–Kelvin linearization). The plate’s trailing-edge flow separation is modeled using a dipole distribution behind the plate on a linearized vortex sheet. The force and moment due to the cross-flow separation from the plate’s bottom tip are calculated by means of a slender body cross-flow method with a rigid free-surface boundary condition. Hence the free-surface effects are confined in the forward flow problem and neglected in the cross-flow problem. Simulations are carried out for different aspect ratios and drift angles. The plate’s thickness is taken into account. Convergence and sensitivity studies are performed carefully. Results are compared with previous experimental and numerical studies. The importance of the second-order forces and validity of the linear assumptions are touched upon. The overall agreement of the results is acceptable. The relative importance of the two viscous and potential components are studied. It is shown that the decomposed problem could follow the experimental results up to a relatively high Froude number.

Isogeometric boundary-element analysis for the wave-resistance problem using T-splines

In this paper we couple collocated Boundary Element Methods (BEM) with unstructured analysis-suitable T-spline surfaces for solving a linear Boundary Integral Equation (BIE) arising in the context of a ship-hydrodynamic problem, namely the so-called Neumann–Kelvin problem, following the formulation by Brard (1972) and Baar and Price (1988). The local-refinement capabilities of the adopted T-spline bases, which are used for representing both the geometry of the hull and approximating the solution of the associated BIE, in accordance with the Isogeometric concept proposed by Hughes et al. (2005), lead to a solver that achieves the same error level for many fewer degrees of freedom as compared with the corresponding NURBS-based Isogeometric-BEM solver recently developed in Belibassakis et al. (2013). In this connection, this paper makes a step towards integrating modern CAD representations for ship-hulls with hydrodynamic solvers of improved accuracy and efficiency, which is a prerequisite for building efficient ship-hull optimizers.

A BEM-isogeometric method for the ship wave-resistance problem

In the present work isogeometric analysis is applied to the solution of the boundary integral equation associated with the Neumann–Kelvin problem and the calculation of the wave resistance of ships. As opposed to low-order panel methods, where the body is represented by a large number of quadrilateral panels and the velocity potential is assumed to be piecewise constant (or approximated by low degree polynomials) on each panel, the isogeometric concept is based on exploiting the same NURBS basis, used for representing exactly the body geometry, for approximating the singularity distribution (and, in general, the dependent physical quantities). In order to examine the accuracy of the present method, numerical results obtained in the case of submerged and surface piercing bodies are compared against analytical solutions, experimental data and predictions provided by the low-order panel or other similar methods appeared in the pertinent literature, illustrating the superior efficiency of the isogeometric approach. The present approach by applying isogeometric analysis and boundary element method to the linear NK problem has the novelty of combining modern CAD systems for ship-hull design with computational hydrodynamics tools.

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 seakeeping and added resistance analysis by means of body-fixed coordinate system

This paper presents an alternative formulation of the boundary value problem for linear seakeeping and added resistance analysis based on a body-fixed coordinate system. The formulation does not involve higher-order derivatives of the steady velocity potential on the right-hand side of the body-boundary condition, i.e., the so-called m j -terms in the traditional formulation when an inertial coordinate system is applied. Numerical studies are made for a modified Wigley I hull, a Series 60 ship with block coefficient 0.7, and the S175 container ship for moderate forward speeds where it is thought appropriate to use the double-body flow as basis flow. The presented results for the forced heave and pitch oscillations, motion responses, and added resistance in head-sea waves show good agreement with experiments and some other numerical studies. A Neumann–Kelvin formulation is shown to give less satisfactory results, in particular for coupled heave and pitch added mass and damping coefficients.

A discussion on Neumann–Kelvin's model for progressive water waves

This article focuses on the role of the capillarity in the modelling and the analysis of progressive waves at the surface of the sea. It is proved that the classical Neumann–Kelvin's model is fully unstable and should be modified in order to be mechanically reliable.

Numerical study on added resistance of ships by using a time-domain Rankine panel method

This paper considers the numerical computation of added resistance on ships. As a method of solution, a higher-order Rankine panel method is applied in time domain. Furthermore, two different formulations, Neumann–Kelvin and double-body linearizations, are applied to investigate the difference between computational results. The quantities of added resistance are obtained by integrating the second-order pressure on the body surface. Computational results are validated by comparing with experimental data and other computational results on several models, and fair agreements are found for all the models. The components of added resistance are observed with regard to integral terms, radiation and diffraction components. This study continues to the computation of added resistance in irregular waves. This kind of computational approach can provide some important information to understand the uncertainty level of towing-tank measurement of added resistance. In this study, a series of long numerical simulations are carried out, and their statistical variations are observed for different time windows. Based on this observation, an appropriate criterion and averaging method are proposed for the prediction of added resistance in irregular waves.