Ship Wave Resistance Research Articles

The DNL absorbing boundary condition: applications to wave problems

A general methodology for developing absorbing boundary conditions is presented. For planar surfaces, it is based on a straightforward solution of the system of block difference equations that arise from partial discretization in the directions transversal to the artificial boundary followed by discretization on a constant step 1D grid in the direction normal to the boundary. This leads to an eigenvalue problem of the size of the number of degrees of freedom in the lateral discretization. The eigenvalues are classified as right- or left-going and the absorbing boundary condition consists in imposing a null value for the ingoing modes, leaving free the outgoing ones. Whereas the classification is straightforward for operators with definite sign, like the Laplace operator, a virtual dissipative mechanism has to be added in the mixed case, usually associated with wave propagation phenomena, like the Helmholtz equation. The main advantage of the method is that it can be implemented as a black-box routine, taking as input the coefficients of the linear system, obtained from standard discretization (FEM or FDM) packages and giving on output the absorption matrix. We present the application of the DNL methodology to typical wave problems, like Helmholtz equations and potential flow with free surface (the ship wave resistance and sea-keeping problems).

On nonlinear ship waves and wave resistance calculations

A finite difference method based on the Euler equations is developed for computing ship waves and wave resistances. Time marching is carried out using a time-splitting fractional-step method. The second-order central difference is used to discretize the spatial differentials, while the convection terms are discretized by the QUICK scheme. A body- and free-surface-fitted grid system with a cell-centered stencil is used. A Poisson equation for the pressure increment at each time step is solved to enforce mass conservation. The method is validated by comparing the numerical results with the experimental data for a Wigley parabolic hull. The characteristics of ship waves, such as the wave profile along the hull, the wave pattern on the free surface, the pressure distribution on the hull surface, and the wave-making resistance are reasonably predicted. The calculated results are in good agreement with the experimental data.

Computing ship wave resistance from wave amplitude with a non-local absorbing boundary condition

A method for computing ship wave resistance from a momentum flux balance is presented. It is based on computing the momentum flux carried by the gravity waves that exit the computational domain through the outlet plane. It can be shown that this method ensures a non-negative wave-resistance, in contrast with straightforward integration of the normal pressure forces. However, this calculation should be performed on a transverse plane located far behind the ship. Traditional Dawson-like methods add a numerical viscosity that dampens the wave pattern so that some amount of momentum flux is lost, and resulting in an error in the momentum balance. The flow field is computed, then, with a centred scheme with absorbing boundary conditions. Copyright © 1998 John Wiley & Sons, Ltd.

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.

A Simplified Method to Predict the Added Resistance of a SWATH Ship in Waves

A simplified method called phase transfer is used to calculate the added resistance of a SWATH ship advancing in regular waves. Using the method, the total added resistance of the twin-hull body can be obtained by calculating the corresponding hydrodynamic forces of a single body only. There is no need to calculate the force on each body twice, and the hydrodynamic interactions are not included. The strip theory is still used and the nonlinear force prediction is based on the theory for a monohull ship advancing in waves. Two additional assumptions are also made:the hull body is a "weak scatterer" andthe second-order potential is small and can be neglected. Practically, the present method is suitable for predicting the corresponding hydrodynamic properties of a SWATH ship with high speed because the hydrodynamic interactions are not significant. The results obtained in the paper mostly agree with the experimental data and show that the present idea is reasonable. It is also found that the inclusion of the interaction between two hulls may lead to some unreasonable negative added resistance in bow waves. Therefore, the method developed here can be considered as a simpler tool to analyze the added resistance for SWATH ships advancing in waves, especially for higher speed.

Resistance Characteristics of High Speed Hull Form with Air-ship Form Bulb

In recent years, many ideas are proposed in order to pursue the high speed vessel. Many of those ideas are based on unconventional concepts such as the application of hydrodynamic lift generated by wings. In those cases, however, we have many problems for the purpose of sizing up and long range operation. As well known, conventional ship, that is a displacement type, has more merits than unconventional ones at least from point of view of buoyancy. Therefore it can be considered as one of the effective ways to pursue the high speed vessel based on the concept of displacement type.In the present study, minimization of wave resistance of a special type ship in high speed range is discussed according to the concept of wave making interaction between a main hull and an air-ship form bulb. This bulb is formed by the combination of a hydrodynamic point source and a line sink, semi-splitted from the main hull of displacement type and supported by the strut. Wave resistance of this special type ship is minimized under some design conditions by means of nonlinear programming. In this paper, experimental investigations are carried out for 2 selected models which have the air-ship form bulb and the mathematical water lines and frame lines based on the present theory. After showing the model test results of fixed trim conditions and free trim conditions, some discussions about the direction of model improvements are given. In order to adopt the air-ship form bulb as the practical bulbous bow, the effect of an entrance angle of the main hull, the control of the trim and so on are investigated experimentally.

On the theory of the Kelvin ship-wave source: the near-field convergent expansion of an integral

The velocity potential of the Kelvin ship-wave source is fundamental in the mathematical theory of the wave resistance of ships, but is difficult to evaluate numerically. We shall be concerned with the integral term F ( x, ρ, ∝ ) = ∫ ∞ -∞ exp {— 1/2 ρ cosh (2 u — i ∝ )} cos ( x cosh u )d u in the source potential, where x and ρ are positive and —1/2 π ≼ ∝ ≼ 1/2 π , which is difficult to evaluate when x and ρ are small. It will be shown here that F ( x, ρ, ∝ ) = 1/2ƒ( x, ρ, ∝ ) + 1/2ƒ( x, ρ, ─∝ ) + 1/2ƒ( ─x , ρ, ∝ ) + ½ƒ( ─x, ρ, ─∝ ), where ƒ( x, ρ, ∝ ) = P 0 ( x, ρ e -i ∝ ) Σ g m ( x, ρ e i ∝ ) c m ( x, ρ e -i ∝ ) + P 1 ( x, ρ e -i ∝ ) Σ g m ( x, ρ e i ∝ ) b m ( x, ρ e -i ∝ ) + Σ g m ( x, ρ e i ∝ ) a m ( x, ρ e -i ∝ ) In this expression each of the functions g m ( x, ρ e i ∝ ), a m ( x, ρ e -i ∝ ), b m ( x, ρ e -i ∝ ), c m ( x, ρ e -i ∝ ), satisfies a simple three-term recurrence relation and tends rapidly to 0 for small x and ρ when m → ∞, and the functions P 0 ( x, ρ e -i ∝ ) and P 1 ( x, ρ e i ∝ ) are simply related to the parabolic cylinder functions D v (ζ) respectively, where ζ = — i x (2 ρ ) -1/2 e 1/2 i ∝ .

On the theory of the Kelvin ship-wave source: asymptotic expansion of an integral

The Kelvin ship-wave source is important in the mathematical theory of the wave resistance of ships but its velocity potential is difficult to evaluate numerically. In particular, the integral term F ( x ,ρ,α) = ∫ ∞ -∞ exp{-½ρ cosh (2 u - iα)} cos ( x cosh u ) d u in the source potential is difficult to evaluate when x and ρ are positive and small, and when -½π ≤ α ≤ ½π. In this work we are concerned with the asymptotic expansion of this integral when x 2 /4ρ is large while x and ρ are not large, for which case the asymptotic expansion F ( x ,ρ,α) ~ - π I 0 (½ρ) Y 0 ( x ) - 2π Ʃ ∞ 1 I m (½ρ) Y 2 m ( x ) cos m α in terms of Bessel functions was proposed by Bessho (1964). This expansion has recently been shown to have great computational advantages but has never been proved. (The standard asymptotic theory of integrals, based on Watson’s lemma, is not applicable, and the expansion is not of standard form.) In this paper it is shown that the expansion is valid except near α = ±½π where an additional term is needed.

A modified method for calculating the wave resistance of SWATH ships

The determination of SWATH ship resistance through model tests alone would be expensive and because such ships are relatively new, even empirical data is mostly lacking. The author therefore develops a method for calculating their wave resistance, based on the linearized theory. Numerical results for two SWATH ships compare well with towing tank results, demonstrating the computational method's utility for practical purposes.

A Geometrically Consistent Linearization Method

The study of irrotational incompressible flows about thin geometries can be carried out using the well known perturbation procedures. In two-dimensional flows exact solutions based on mappings can be used to compare the accuracy of first order solutions. For most airfoil sections a first order perturbation solution is not sufficiently accurate in representing the pressure and velocity distribution, especially about the leading edge. For three-dimensional flows exact solutions are rare and for more complex problems such as ship wave resistance formulations an exact solution does not exist for comparison of results. In this last case second-order solutions exist but are very difficult to calculate. Therefore, it would appear advantageous to improve first-order calculations. To this end a perturbation method that incorporates the geometric properties of the body is studied. This method is applied to a symmetric Joukowski airfoil and to an elipse. This method, here called the “geometrically-consistent linearization method” predicts the leading edge pressure variations correctly in the two cases studied and appears to be superior to the classical first order solutions. An iterative solution following this procedure further improves the calculation especially for thicker foils. The method discussed and the following iteration procedure seem to form an efficient numerical solution to airfoil flow problems.

The effect of hull cross section on the theoretical wave resistance of a fast ship in shallow water

On a utilise la theorie des ondes linearisees existantes pour la resistance d'un navire a la fois des eaux profondes et peu profondes pour calculer l'influence des eaux peu profondes sur un navire rapide et se deplacant a des vitesses sub-critiques et supercritiques. Dans cette etude la representation de la coque est amelioree dans le cas d'une coque planaire la section transversale rectangulaire de la coque est remplacee par une forme pointue et dans le cas d'une coque circulaire la section transversale est parabolique. La comparaison des resultats calcules avec les donnees disponibles montre que la variation dans la section transversale de la coque entraine une difference notable dans la region du pic de resistance sub-critique

Explicit Approximations for Calculating Potential Flow About a Body

Several explicit approximations for calculating nonlifting potential flow about a body in an unbounded fluid are studied. These approximations are shown to be exact in the particular cases of flows due to translations of ellipsoids, and they are compared with the exact potential for two-dimensional flows about ogives in translatory motions. Two approximations, given by formulas (31) and (32) in the conclusion, appear to be of particular interest for practical applications, and they can be extended to free-surface flow problems, for example, ship wave resistance, and radiation and diffraction of regular waves by a body.

Alternative integral representations for the Green function of the theory of ship wave resistance

Three alternative single-integral representations for the Green function of the theory of ship wave resistance are derived in a unified manner from a basic double-integral representation. These alternative single-integral representations, which essentially are modifications of well-known double-internal representations due to Michell, Havelock, and Peters, are compared and discussed. Another object of this study is to examine the field equation and the boundary condition satisfied by the Green function in the limiting case when the singular point is exactly at the free surface.

Simple Calculation of Sheltering Effect on Ship-Wave Resistance and Bulbous Bow Design

The sheltering effect on the ship wave resistance is treated by the free-surface pressure distribution inside the ship. The theory is developed and the numerical values of wave resistance are obtained for parabolic and sinusoidal hulls. The corrected wave resistance is shown to be reduced more than 20 percent from the Michell's wave resistance even for a Wigley hull with beam-length ratio 0.1. However, the humps and hollows of wave resistance still remain with the magnitude closer to the experimental results. The optimum volume of bulb for sinusoidal ships is also shown to be reduced from 7.6 percent to 18 percent of the values without sheltering effect for the range of Froude numbers from 0.15 to 0.35.

The Low Speed Wave Resistance Theory Imposing Accurate Hull Surface Condition

The low speed wave resistance theory developed by Baba4) and Maruo5) seems to be a promising theory to predict the wave resistance of conventional ships, because it includes the nonlinear effect of the free surface condition. The wave resistance curves calculated by the low speed theory, however, have large humps and hollows, and they do not agree with the measured7) 10) .In this paper, a refinement of the low speed theory is discussed by imposing the more accurate hull surface condition. The wavy source distribution added to satisfy the hull sruface condition is obtained numerically, and the wave resistance, the hull side wave profiles and the velocities around the hull surface are evaluated. The results coincide fairly well with the measured values, especially the humps and the hollows of the calculated wave resistance curves are remarkably reduced.It is ascertained that the added source distribution plays an important role to improve the low speed wave resistance theory, in spite that its strength is one-order smaller than that of the double model source.

Added Resistance of Ships in Waves

A new method is presented for computing the added resistance of a ship advancing at constant mean speed and heading in regular waves. It is shown that the added resistance, even though it is a second-order quantity, can be expressed as a product of first-order terms that are all computed by computer programs presently in use for predicting the linear heave and pitch motions. Comparisons between the present theory and other theories, as well as experimental results, are presented for head waves. It is found that the discrepancies in the experimental results can be as large as the differences between different theories. A difference of 30% between experimental values for the maximum added resistances seems to be typical. It is concluded that the present theory predicts the added resistance with better accuracy for a wider range of ships forms and speeds than any other existing theory. It is also shown how the mean added resistance in a short-crested seaway can easily be obtained from the regularwave results.

Wave Resistance and Wave Patterns of Thin Ships

The wave resistance R and wave height h(x, z) are evaluated asymptotically for small Froude number F = U(gL)−½ for a slender hull of any shape. Michell's theory for a thin ship of length L moving with constant speed U along a straight line is the starting point. It is found that asymptotically R and h depend only upon four properties of the ship—the slope of the hull and the slope of the profile curve of the hull at the waterline at bow and stern. Simple formulas are obtained for R and h in terms of these slopes. The wave pattern consists of four waves—a longitudinal and a transverse wave from the bow and a similar pair from the stern. Their phases are the same as those of Kelvin waves due to pressure points at the bow and stern, and they also decay with distance like cylindrical waves. However, their amplitudes have different angular variations from those of Kelvin waves.

Introductory Note to Companion Papers on Wave Resistance

The following two papers were submitted independently by the authors in April 1974 to the JOURNAL OF SHIP RESEARCH. Both studies were motivated by the desire to establish a sound theoretical foundation for Guilloton's method, which has been found by Gadd (1978) to be particularly successful in predicting the wave resistance of ship models. Noblesse, F., " A Perturbation Analysis of the Wavemaking of a Ship, with an Interpretation of Guilloton's Method"; see page 140. Dagan, G., " A Method of Computing Nonlinear Wave Resistance of Thin Ships by Coordinate Straining"; see page 149.

A Method of Computing Nonlinear Wave Resistance of Thin Ships by Coordinate Straining

A systematical procedure is derived for determining the distribution of sources along the centerplane of a thin ship which generate a flow (and the associated wave resistance) satisfying the free-surface and body boundary conditions at second order. The method relies on a mapping of the flow domain onto a new domain by slightly straining the coordinates. The horizontal straining is selected such that the nonlinear solution should be easily derived from the first-order (Michell) solution in the strained domain. The results are shown to be similar to those obtained by Guilloton's method, but new features, such as the forward displacement of the bow singularities, are present.

On the Principal Particulars of Ship Hull Form and Wave Pattern Resistance (III)

In the previous papers, the characteristics of directly measured wave resistance, amplitude functions and wave spectra were discussed in comparison with Michell's theory, and some examinations were made how closely the principle of linear superposition of free waves is realized. In the former part of this paper, using the results described in the previous papers, some practical applications are shown for the estimation of wave resistance of ship of given principal particulars, and a method to determine the optimum principal particulars of high speed ships is established. The latter part of this paper is devoted mainly to the investigations of the effect of bow bulb on wave resistance which are examined by the tests of series models with airship type bulbs. Furthermore, a method for applying the results of the series tests to the hull form improvement is considered, according to the principle of linear superposition of free waves, which is confirmed by the experimental fact.