Wave-resistance Problem Research Articles

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.

Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem

A high-order finite-difference method solution of the linearized, potential flow, seakeeping problem for a ship at steady forward speed was recently presented by Amini-Afshar et al. [1,2]. In this paper, we provide a detailed matrix-based eigenvalue stability analysis of this model, highlighting the sources of instability and the effects of possible remedies. In particular, we illustrate how both boundary treatment and grid stretching are important factors which are not typically captured by a von Neumann-type analysis. The new analysis shows that when grid stretching is used together with centered finite difference schemes, the method is generally unstable. The source of the instability can in some cases be traced to an effective downwinding of the convective terms. Stable solutions can be obtained either by introducing upwind-biased schemes for computing the convective derivatives on the free-surface, or by application of a mild filter at each time-step. A second source of instability is associated with the treatment of the convective derivatives of the free-surface elevation at points close to the domain boundaries. Here it is necessary to consider whether the surrounding fluid points lie in an upwind or a downwind direction. For upwinded points, ordinary one-sided differencing can be used, but for downwinded points we instead impose a Neumann-type boundary condition derived from the body and free-surface boundary conditions. As an example application to complement those already given in [1,2], the method is applied to solve the steady wave resistance problem and comparison is made to reference solutions for a two-dimensional floating cylinder and a submerged sphere. Estimates of the wave resistance of the Wigley hull are also compared with experimental measurements.

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.

Semi-Analytical Methods for Different Problems of Diffraction-Radiation by Vertical Circular Cylinders

As in the other fields of mechanics, analytical methods represent an important analysis tool in marine hydrodynamics. The analytical approach is interesting for different reasons : it gives reference results for numerical codes verification, it gives physical insight into some complicated problems, it can be used as a simplified predesign tool, etc. This approach is of course limited to some simplified geometries (cylinders, spheres, ...), and only the case of one or more cylinders, truncated or not, will be considered here. Presented methods are basically eigenfunction expansions whose complexity depends on the boundary conditions. The hydrodynamic boundary value problem (BVP) is formulated within the usual assumptions of potential flow and is additionally simplified by the perturbation method. By using this approach, the highly nonlinear problem decomposes into its linear part and the higher order (second, third, ...) corrections. Also, periodicity is assumed so that the time dependence can be factorized i.e. the frequency domain formulation is adopted. As far as free surface flows are concerned, only cases without or with small forward speed are sufficiently simple to be solved semi-analytically. The problem of the floating body advancing in waves with arbitrary forward speed is far more complicated. These remarks are also valid for the general numerical methods where the case of arbitrary forward speed, even linearized, is still too difficult from numerical point of view, and "it is fair to say that there exists at present no general practical numerical method for the wave resistance problem" [9], and even less for the general seakeeping problem. We note also that, in the case of bluff bodies like cylinders, the assumptions of the potential flow are justified only if the forward speed is less than the product of wave amplitude with wave frequency.

Sailing Boat Performance Prediction

The inviscid, steady, hydro-aerodynamic flow around a sailing boat fully equipped with sails and crew was investigated using two different solvers. The hydrodynamic problem was solved based on Green functions for the linearised wave resistance problem, using a simple plane wake model. The aerodynamic problem is fully non-linear with the sails wakes discretised by means of vortex carrying particles. The boat speed was obtained as a result of the balance between hydro- and aerodynamic forces. Only heel and drift angles were free. The balance problem was solved for two cases: crew position or heel angle prescribed.

The Subcritical Motion of a Semisubmerged Body: Solvability of the Free Boundary Problem

We discuss existence and regularity of the solutions of the wave-resistance problem for a thin semisubmerged body moving at uniform subcritical velocity in a heavy fluid (e.g., water) of constant depth. The main assumption (on the geometry of the body) is that the flow is two-dimensional; i.e., it can be completely described in the vertical plane containing the direction of the motion. Then the problem can be formulated in terms of a boundary value problem for a holomorphic function (the complex velocity field) satisfying a nonlinear condition (the Bernoulli condition) on a free boundary (the free surface of the fluid). By a hodograph transformation and choosing an appropriate functional setting, we first reduce the problem to the resolution of a nonlinear functional equation depending on two unknown parameters, which are related to the positions in the hodograph plane of the points of contact between the free surface and the body. The main result of this paper is the proof of the existence, under mild as...

On unique solvability and regularity in the linearized two-dimensional wave resistance problem

We discuss existence, uniqueness, and regularity of the solutions of a boundary value problem in a strip, which is obtained by linearization of the equations of the wave-resistance problem for a cylinder semisubmerged in a heavy fluid of constant depth H H and moving at uniform velocity c c in the direction orthogonal to its generators. We show that the problem has a unique solution, rapidly decreasing at infinity, for every c > g H c > \sqrt {gH} , where g g is the acceleration of gravity. For c > g H c > \sqrt {gH} , we prove unique solvability provided c ≠ c k c \ne {c_k} , where c k {c_k} is a known sequence monotonically decreasing to zero. In this case, the related flow has in general nontrivial oscillations at infinity downstream.

A Genetic Algorithm Approach to the Problem of Minimum Ship Wave Resistance

A genetic algorithm-based optimization method is proposed for an optimization of a ship hull from a hydrodynamic point of view. In the optimization procedure, the wave resistance has been selected as an objective function. The genetic algorithm is coupled with a computer program for solving the three-dimensional potential flow around a ship hull. The potential flow solver is based upon the well-known Dawson method. The optimization procedure has been applied to the Series 60 CB = 0.60 hull taken as a basis hull. The computational examples show the optimization ability of the proposed method.

The forward motion of an unsymmetric surface-piercing cylinder: the solvability of a nonlinear problem in the supercritical case

We consider the wave-resistance problem for a ‘slender’ cylinder semisubmerged in a heavy fluid and moving at uniform, supercritical speed in the direction orthogonal to its generators. By a hodograph transformation, the problem (originally set up in a domain with a free boundary) reduces to the determination of a function, holomorphic in a fixed domain, satisfying some nonlinear boundary conditions depending on two (unknown) parameters. The problem in the hodograph plane is solved via the implicit function theorem; then, the two parameters are fixed by the requirement that the free boundary and the cylinder profile (which is assumed convex and reasonably smooth) form a single smooth (C1) streamline. Furthermore, the free boundary is monotone increasing downstream, monotone increasing upstream and lies under the level of calm water.

On Solvability of the Nonlinear Wave Resistance Problem for a Surface-Piercing Symmetric Cylinder

We solve the wave-resistance problem for a "slender" cylinder semisubmerged in a heavy fluid and moving at uniform, supercritical speed in the direction orthogonal to its generators. We prove that the free boundary and the cylinder profile form a single smooth streamline; moreover, the free boundary is monotone increasing downstream and lies under the level of calm water.

The Neumann–Kelvin Problem for a Beam

We discuss existence, uniqueness and regularity of the solution of the plane Neumann–Kelvin problem in a strip, in the case of a beam. The problem is obtained by linearization of the equations of the wave-resistance problem for a “slender” cylinder semisubmerged in a heavy fluid and moving at uniform, supercritical speed in the direction orthogonal to its generators. The results obtained may be relevant for the resolution of the nonlinear wave-resistance problem.

Exact Solution of the Wave-Resistance Problem for a Submerged Cylinder. II. The Non-linear Problem

. We study the problem of wave resistance for a “slender” cylinder submerged in a heavy fluid of finite depth with the cylinder moving at uniform supercritical speed in the direction orthogonal to its generators. We look for a divergence‐free, irrotational flow; the boundary of the region occupied by the fluid (consisting of the free surface, the bottom and the obstacle profile) is assumed to belong to streamlines and the Bernoulli condition is taken on the free surface. The problem is transformed, via the hodograph map, into a problem set in a strip with a cut. By using a “hard” version of the inverse function theorem and by taking account of the results obtained in Part I (which we recall here), we prove the existence of a complex velocity function satisfying all the requirements of the problem. In particular, this function is continuous up to the surface of the obstacle, and the only possible singularities appear at the end‐points where the boundary is not smooth. Moreover, two stagnation points appear near to the extremities of the submerged body.

Exact Solution of the Wave-Resistance Problem for a Submerged Cylinder. I. Linearized Theory

. We consider the problem of finding a holomorphic function in a strip with a cut \({\cal A}= \{(x,y) : \, x\in\RE,\,\,0<y<H\}\backslash \{(x,y) : \, |x|\le x_0,\,\,y=y_0\}\) satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for \(|x|\to\infty\). A special solution can be selected by fixing the value of the circulation around the cut. The problem is obtained by linearization of the equations of the wave‐resistance problem for a “slender” cylinder submerged in a heavy fluid and moving at uniform speed in the direction orthogonal to its generators. The results obtained, besides their own interest, are a crucial step for the resolution of the non‐linear problem.

Algebraic Discrete Nonlocal (DNL) Absorbing Boundary Condition for the Ship Wave Resistance Problem

An absorbing boundary condition for the ship wave resistance problem is presented. In contrast to the Dawson-like methods, it avoids the use of numerical viscosities in the discretization, so that a centered scheme can be used for the free surface operator. The absorbing boundary condition is “completely absorbing,” in the sense that the solution is independent of the position of the downstream boundary and is derived from straightforward analysis of the resulting constant-coefficients difference equations, assuming that the mesh is 1D-structured (in the longitudinal direction) and requires the eigen-decomposition of a matrix one dimension lower than the system matrix. The use of a centered scheme for the free surface operator allows a full finite element discretization. The drag is computed by a momentum flux balance. This method is more accurate and guarantees positive resistances.

Fully three-dimensional ship seakeeping computations with a surge-corrected Rankine panel method

A 3-d seakeeping code uses first-order Rankine panels with special numerical integration on the ship's hull and Rankine point source clusters above the free surface. The code computes the motions of the ship in regular waves of small height (linearized). The steady flow is captured without simplification by solving the fully nonlinear wave-resistance problem first. A special treatment of the surge motion considers the influence of periodic quantities on thrust and resistance, and improves surge motion predictions. Radiation and open-boundary conditions are enforced by staggered grids. Results for the ITTC standard test case S-175 containership agree well with experiments except for very long waves. The importance of capturing the three-dimensional steady flow contributions is also demonstrated.

A numerical solution of the nonlinear wave resistance problem for simple shaped submerged bodies

This paper presents a numerical method to compute the nonlinear wave resistance of a body, submerged in a free stream of infinite depth. The solution is given in the form of a potential of single layer and the nonlinear boundary conditions on the free surface are fulfilled by means of an iterative procedure. Results are shown for the 2-D case of an elliptical cylinder and for the 3-D case of prolate spheroids. All the presented results are discussed taking into account some experimental data and analytical solutions available in literature.

A study of nonlinear wave resistance using integral equations in Fourier space

An attempt is made in this paper to tackle the problem of nonlinear wave resistance by formulating it in Fourier space and by deriving a nonlinear integral equation for the wave amplitude by an approach similar to the one leading to the Zakharov equation.The procedure is illustrated for two simple examples of two- and three-dimensional travelling pressure distributions. A regular perturbation solution up to third-order terms in the slenderness parameter shows that the expansion is not uniform for small Froude numbers. A uniform, generalized, expansion is then constructed, with its leading term satisfying a new nonlinear integral equation. This rather simple integral equation, of a Volterra type, is solved numerically. The generalized wave drag is shown to be significantly larger than the one predicted by the regular perturbation expansion at small Froude number. The method adopted here has the advantage of singling out in a systematic manner the terms of the free-surface conditions which cause the small-Froude-number non-uniformity, and it is applicable to both two- and three-dimensional flows. The results are compared with existing approximate methods of computing wave drag at low Froude numbers. It is found that quasilinearized approximations may be quite accurate for the examples considered here.

Fundamental Studies on Hull Form Design by Means of Nonlinear Programming (6 th Report)

The author has investigated the application of the wave resistance theory to the hull form design by means of nonlinear programming. In this study two problems are discussed, one of which is the problem of hull form improvement, and the other is the problem of minimum wave resistance. The present paper concerns the concluding result of these series, and the feasibility of this approach has been proved by showing the high performance hull form of SR 138 container ship being improved further as an example.We can regard the hull form design as a kind of the problem of nonlinear programming, because the objective function is generally specified as the nonlinear functional of design variables and many constraints of equalities or inequalities are needed in the practical design stage. The nonlinear programming technique provides a rational method for the determination of lines of practical hull forms.

Fundamental Studies on Hull Form Design by Means of Nonlinear Programming (3rd Report)

The minimum wave resistance problem based on Michell's thin ship theory has been investigated by means of variational approach so far. In this paper, a kind of the fairing constraint to hull form is introduced into the minimum wave resistance problem, and the optimization technique is applied instead of the variational approach.Two formulations are shown for the present problem and optimized by means of SUMT (Sequential Unconstrained Minimization Technique) as in the 1st & 2 nd reports. Numerical examples are given under several kinds of design conditions. As the fairing constraint is considered, the minimum wave resistance hull forms without swan necks are obtained.