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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 66 New Green-Function Method to Predict Wave-Induced Ship Motions and Loads X-B Chen, L.Diebold (Bureau Veritas—DTA, France) Y.Doutreleau (Bassin d'Essais des Carènes, France) ABSTRACT A three-dimensional numerical method based on the new Green function given in Chen (1999) and the higher-order description of ship hull by bi-quadratic patches has been developed in order to predict wave-induced ship motions and loads. The development of this new method and its validations through a comprehensive comparison with semi-analytical solutions and results of experimental measurements, are described in the paper. Recent work on the ship-motion Green function in both analytical and numerical aspects provides an accurate and efficient way to evaluate the influence coefficients involved in integral equations. The use of bi-quadratic patches gives a precise representation of the ship geometry and a continuous representation of the velocity potential over the ship hull. Furthermore, application of the Galerkin procedure yields a square-matrix system and improves the accuracy of solutions. The excellent level of agreement with known semi-analytical solutions and experimental measurements shows that the present numerical method is reliable and practical in a number of applications. INTRODUCTION Very important in practice, notably for design, several 3D panel methods making use of the Green function, which satisfies the linearized free-surface boundary condition corresponding to time-harmonic flows observed from a translating system of coordinates, have been developed in recent decades for computing wave-induced ship motions and loads. However, the fairly small number of panels used in these methods and the large discrepancies observed among numerical predictions indicate the formidable numerical burdens. In fact, the substantial difficulties associated with the numerical evaluation of the Green function and its gradient, and their subsequent integrations over ship-hull panels and waterline- segments, have been a major stumbling block hindering the development of reliable and practical methods. These difficulties have been addressed in various ways in the numerical solution procedures developed by Chang (1977), Inglis & Price (1981), Guével & Bougis (1982), Wu & Eatock Taylor (1989) and Iwashita & Ohkusu (1992), and in the development of methods for computing the ship-motion Green function by Hoff (1990), Jankowski (1990), Bougis & Coudray (1991) and Ba & Guilbaud (1995), and very recently confirmed by the work in Chen (2000). It is shown formally in this work by an asymptotic analysis that the source potential is singular and highly-oscillatory for a field point approaching the track of the source at the free surface. To circumvent the above-mentioned difficulties encountered in previous studies, a new numerical method, based on the recent results in both theoretical and numerical aspects of ship-motion Green functions and the higher-order description of ship-hull geometry and fluid kinematics, has been developed. The newly-obtained important results for ship-motion Green functions given by Chen (1999) include several innovative features. First, new formulations of the free-surface component are developed based on the basic decomposition of the double Fourier integral obtained by Noblesse & Chen (1995) and new expressions of the wavenumber integrals. The resultant wave and local components are both expressed by simple integrals. The asymptotic analysis of the wave component gives analytical expressions of farfield ship waves and reveals their direct relationship with the dispersion relation. The singular and highly-oscillatory properties of potential flows generated by a source located at the free surface are analyzed and expressed in a closed form. Analysis of the line integrals on the free surface shows that they can be evaluated in an analytical way. Furthermore, efficient numerical developments have been realized to evaluate ship-motion Green functions in all configurations accurately including the most critical case for which the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 67 the free-surface effects are most important as both source and field points are close to or at the free surface. In fact, this recent work gives a new lease of life to the Green-function approach and provides the essential elements to construct a sound and solid basis to solve ship-motion problems. Further to the treatise presented in Chen (1999) on the Green functions, a number of new developments have been achieved. The work consists of the formulation for the gradient of ship-motion Green functions and the appropriate way to evaluate efficiently the influence coefficients involved in the kernel of integral equations. In fact, the influence coefficients associated with the free-surface effects are evaluated by quadrature integration of the wave and local components of ship-motion Green functions over the ship's hull and along the waterline. The formulation of the local component presented in Chen (1999) is used as it is well suited for the numerical evaluation. Indeed, the computation of the local component expressed by the single integral whose integrand is a smooth function is much more efficient than the double Fourier integral given in Chen & Noblesse (1998). On the other hand, the single integral for the wave component is a bit troublesome since the integrand can be highly- oscillatory when both field and source points are close to the free surface. The Fourier-Kochin approach presented in Noblesse & Yang (1995) and Chen & Noblesse (1998) was first adopted to evaluate this part of influence coefficients (Chen, 1998). These interesting studies show that the influence coefficients associated with a smooth distribution of singularity over a higher-order panel are not highly-oscillatory due to the cancellation of oscillatory terms at large wavenumber. Another study in Doutreleau & Chen (1999) on the analytical integration of the highly-oscillatory term leads to the same conclusion. However, both the Kochin approach and that of analytical integration need more complex calculations and are not so efficient as expected from the experience of the first author. The option to design a smooth function depending on the smoothness of distribution of singularities for damping out high oscillations is thus retained in the computation of influence coefficients associated with the wave component. Furthermore, the special algorithms given in Chen (1999) for oscillatory integrals improve further the efficiency of the wave-term computing. This elaborated approach allows us to evaluate accurately and efficiently the influence coefficients associated with the free-surface part of ship- motion Green functions, in all configurations including the case of two panels at the free surface. Another feature of the new Green-function method is the higher-order description of ship-hull geometry as well as singularity distributions on the hull and along the waterline. The ship hull is subdivided into a number of 9-nodes bi- quadratic patches which are defined by the coordinates at 9 nodes and the corresponding shape functions. Accordingly, the velocity potential and distributions of singularities are expressed by the classical series form associated with the shape functions which represent the bi-quadratic variation on each patch. To complete the work on influence coefficients, the Rankine term is evaluated by using the so-called Gaussian subdivision approach, which consists of subdividing the patch into a number of quadrilateral panels following the coordinates defined by the Gauss-Legendre rule. The quadrature integration of Gauss-Legendre is performed when the field point is at a large distance from the patch (or panels) relative to the size of patch (or panels). For a field point close to or on the patch, the analytical formulation is then applied to evaluate the Rankine term on the smaller panels. Unlike the methods adopted in previous studies using flat panels and constant singularities (discontinuous across panels) in which a large number of panels are required to achieve accurate representation of ship-hull geometry and velocity potential, the use of the bi-quadratic patches reduces the number of unknowns and thus the computation time. The continuous representation of the velocity potential in the present bi-quadratic patch method makes it possible to approximate correctly the velocity field within each patch, which is necessary in establishing the integral equation. Furthermore, application of the Galerkin procedure yields a square-matrix linear system which can be solved by either the Gauss-Elimination method or an iterative approach, and improves the accuracy of solutions. The new Green-function method solves three classes of problems including the wave diffraction-radiation at zero forward speed, Neumann-Kelvin steady flows and the wave diffraction-radiation with forward speed. It can then offer a wide range of applications in ship designs and offshore industries. The validation of this method is made through a comparative study in a systematic and comprehensive way. The numerical results obtained by using the present method are confronted with the classic analytical solution to the wave radiation around a floating hemisphere given in Hulme (1982), the semi-analytical results of Neumann-Kelvin steady flow around an ellipsoid studied by Farell (1973) and the semi- analytical solutions to wave diffraction and radiation of a sphere the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 68 advancing in water waves obtained by Wu (1995). An excellent level of agreement is found in all configurations. Furthermore, most results are presented in tabulated form which may be useful to readers. Finally, the comparison with experimental measurements on Wigley hulls made by Journée (1992) further confirms the validity of the present method. BOUNDARY VALUE PROBLEM AND NEW GREEN FUNCTION The reference system moving with the ship at the mean forward speed U along the positive x-axis is defined by letting (x, y) plane coincide with the mean free surface and z-axis be positive upward. Based on the assumptions of perfect fluid, irrotational flow and small wave steepness, the velocity potential Φ (x, y, z, t) is written as the sum of a steady and a time-harmonic unsteady potentials: (1) Both the steady potential and unsteady potential satisfy the Laplace equation in the fluid domain and an appropriate radiation condition at infinity. In particular, the steady potential (called Neumann-Kelvin flow) satisfies (2) on the mean free surface (z=0) due to the linearization and on the body surface H, respectively. In (2), g is the mean respectively the first derivatives of with respect to z and n, and the second acceleration due to gravity and derivatives of with respect to x. The normal vector n=(n1, n2, n3) on H is defined to point toward fluid. The time-harmonic potential is further expressed as the sum of various components: (3) are radiation potentials corresponding to 6 degree of freedom oscillations of the body and a1,2,···,6 are in which amplitude of corresponding motions. is the potential of incoming waves and given by (4) associated with the frequency ω0 of incoming waves and β the wave heading. with the wavenumber is the potential due to the diffracted waves associated with the amplitude a0 of incoming waves. The components of the radiation and diffraction potentials satisfy the linearized condition given by the equation (3.23) in Newman (1978): (5) due to basic flow which is steady and represented by W (Eq. 3.15 in Newman, over the deformed free surface 1978). In (5), is a linear operator depending on UW applied to If the uniform flow is chosen as the basic flow, i.e. W=− ex, the condition (5) is reduced to (6) applied on z=0. In (6), τ=Uω/g is the Brard number characterizing the flow, and k= ω2/g the wavenumber associated with the encounter frequency ω which is defined by the frequency ω0 and heading β of incoming wave, and the forward speed U through (7) On the body surface H (8) with the generalized vector (9a) and so-called mj terms defined by (9b) (9c) in which X=(x, y, z) is the position vector of a point on H and the basic flow (10) in the boundary condition (8) on H as well as over the Consistently, the basic potential is to be chosen as free surface (5). However, few studies have involved one other than the uniform flow due to the complexity of (5) and the computation of the steady flow such as that satisfying (2) and its higher-order derivatives needed on the free surface and in the mj terms (9). Only recently, the mj terms derived from the Neumann-Kelvin flow are used in the computation of the authoritative version for attribution. time-harmonic potential by Wu (1995), while the same condition as (6) on the free surface was retained. By assuming small forward speed, the consistence was respected in Chen & Malenica (1998) since in both conditions on H and over the free surface, the double-body flow satisfying (11)

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 69 on z=0 and H respectively, is used as the basic flow. Despite its limitations, the double-body potential is used in the present study to evaluate the mj terms while the classic condition (6) on z=0 is retained. In fact, the same method as used in Chen & Malenica (1998) is adopted in the computation of the double derivatives of and then mj terms. ( with j= 1, 2,···, 7) satisfying (6) and (8) can be determined by the integral The time-harmonic potential equation (12) for and with (13a) and (13b) The line integral along the waterline W in (13) arises from applying Stokes' integral theorem on the free surface, and the following identity (14) is used in (13) with (cu, cv, cn) depending on the geometry of H and given by (36c). In (13), ty is the y-component of the unit vector tangent to the waterline and oriented clockwise while are two tangential derivatives of over the ship hull H and is the normal derivative given by (8). The Green function involved in (13) represents the velocity potential of the flow generated at a field point by a source of unit strength located at the point is expressed by (15) Here GS is the simple singularity part (16) the mirror image of with respect to the mean free surface plane z=0. The free-surface part GF is with defined by the Fourier superposition of elementary waves (Noblesse & Chen, 1995) (17) where is the wavenumber and with L being the ship length. Furthermore, D in (17) is the dispersion function (18) and sign(Df)=sign(∂D/∂f)=−sign (Fα−f). The dispersion function (18) is associated with the same boundary condition as (6) on the free surface. is the nondimensional frequency and the Froude number. The free-surface part GF defined by (17) is expressed by Chen (1999) in the form (19) where H(τ−1/4) is a unit step function equal to 0 for τ≤1/4 and equal to 1 for τ>1/4. The free-surface part GF is then decomposed into three different components in (19) with GW the wave component, GN the local component and GC the complementary component for τ>1/4. The wave component GW is expressed as a single integral (20a) along the dispersion curve defined by the dispersion relation D=0, with Σ 1=sign(Df) and along the dispersion curves k=k ±(θ) given by (20b) respectively. Furthermore, erf(·) is the error function defined in Abramowitz & Stegun (1967). The local component in (19) is defined by (21a) the authoritative version for attribution. in which the function N ±= (21b) In (21b), we have used (21c) where erfc(·) and E1(·) are the complementary error function and the exponential integral function defined in Abramowitz & Stegun (1967).

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 70 The values of θc are equal to 0 for τ ≤1/4 and for τ >1/4. The complementary component GC in (19) exists for τ>1/4 and is defined by (22a) The wavenumber integral function KC is given by (22b) with (22c) and the unit step function H(kR) is equal to 0 for kR≤0 or equal to 1 for kR>0. Furthermore, Z± in where (22c) are defined by (22d) with and and (22e) (22f) The gradients of ship-motion Green functions with respect to the coordinates (x, y, z) of the source point, can be obtained by directly performing the space derivatives to (16) for the simple singularity part GS (23) and to (19) for the free-surface part G F: (24) in which ∇GW = (25) for the wave component, ∇GN= (26a) for the local component with N′ ± defined as (26b) and finally • GC= (27a) for the complementary component with (27b) In summary, the free-surface part GF and its gradients ∇ GF of ship-motion Green functions can be numerically evaluated by using the above formulations (20a) and (25) for GW and ∇GW , (21a) and (26a) for GN and ∇GN, and (22a) and (27a) for GC and ∇ GC. Once the integral equation (12) is solved, the time-harmonic pressure is given by Bernoulli's equation (neglecting second-order terms) (28) The added-mass coefficients (akj) and damping coefficients (bkj) due to radiation waves are defined by (29) for k, j=1, 2, ···, 6. The wave exciting loads are (30) for k= 1, 2, ··· 6 and then the amplitude of ship motions is evaluated by solving (31) for k=1, 2, ···, 6. In (31), mkj and ckj for k, j= 1, 2, ···, 6 are the inertia and stiffness matrices respectively. In the same way, solution of the Neumann-Kelvin problem (2) gives the steady potential and the hydrodynamic the authoritative version for attribution. steady forces by integrating the pressure obtained from the Bernoulli's equation (32) In particular, the wave resistance the lift forces and trim moments are important components in practice. BI-QUADRATIC PATCH METHOD The hull surface of ship is subdivided into NH curved patches. Among the N H patches, N W patches touch the free surface so that the waterline is constituted by N W curved segments corresponding to the upper

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 71 side of N W patches on the hull. Each patch is represented by the coordinates at 9 nodes with (j=1, 2, ···, 9) 8 of which are located along the boundary sides in the anticlockwise order, and the ninth one is at the center of the patch (Figure 1). The surface of patch is defined by a parametric representation of the form (33) where Nj(u, v) are shape functions given by (34) over the surface of patch is quadratic in both u and v curvilinear coordinates so that it The basic variation of is called bi-quadratic patch. Such a description of geometry is quite common in finite-element analysis. A number of classic results concerning its fundamental properties are directly used in the present study and not described here in detail as they can be found in Zienkiewicz & Taylor (1989). Figure 1: Quadratic patch and its subdivision One of particularities concerns the subdivision of the surface into smaller sub-patches (panels) necessary for the purpose of computing influence coefficients. As shown on Figure 1, the patch is subdivided into 5×5 panels whose reference point (close to the centroid of each panel) is determined by the coordinates following the 5×5 Gauss-Legendre rule. Furthermore, the coordinates in the 3×3 Gauss-Legendre rule are used in the definition of panels' nodes. The velocity potential as well as other flow values is expressed over the surface by its values at 9 nodes in the same way as (33): (35a) and the two tangential derivatives by (35b) The first derivatives of in global coordinate system are determined by the identity (36a) where the superscript (·)t means the transpose of the vector and the transformation matrix is given by (36b) In (36b), (x, y, z)u,v are derivatives of the coordinates (x, y, z) with respect to (u, v). Thus, the values of (cu, cv, cn) involved in (13) are written by (36c) on the lth patch, the potentials on the left side of (12) Now we consider the integral equation (12) at the point become: the authoritative version for attribution. (37a) and that on the right hand side: (37b) with the coefficients defined by (37c)

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 72 (37d) (37e) (37f) In above equations, sk and wk stand for the surface of kth patch and the waterline segment of kth patch which touches the free surface. Now we perform an integration on both sides (37a) and (37b) of the integral equation after multiplying the 9 shape functions within the lth patch. This domain collocation of Galerkin type yields (38) for l=1, 2, ···, N H and i=1, 2, ···, 9. The influence coefficients are written as (39a) (39b) (39c) (39d) with δlk=1 for l=k otherwise δlk=0. Finally, we note (40) to represent the connection between the jth node within the kth patch (ith node within the lth patch) and the Jth point (Ith point) in the general numbering of a total of N P points on the hull surface. This connection permits to arrange the equation (38) into (41a) with (41b) (41c) The matrix MIJ in (4la) is square and of size equal to the number of points NP used in NH patches to represent the ship hull. The linear system of equations (41a) can be solved by either the usual Gauss-Elimination method or an iterative approach. The influence coefficients defined by (39) involve two folds of integration on the influencing (kth) patch or waterline segment (of kth patch) and on the influenced (lth) patch. The integrand includes the jth shape function within kth patch and the ith shape function within lth patch, and the Green function as well as its normal gradient. Following (15) and (19), the Green function is known to be composed of a Rankine component, wave and local components, and complementary one for τ>1/4 (the sum of last three components accounting for free-surface effects). The integration of the local and complementary components is in general an easy task as they are not oscillatory. However, some numerical difficulties arise from the integration of the Rankine and wave components. The Rankine component is only involved in the two-fold integration on the influencing and influenced patches since it is nil along the waterline. The integration on the influencing patch is performed by a compound way. It consists of using a numerical integration of Gauss-Legendre rule when two patches are separated at a distance larger than the patches' dimension. When two patches are close to each other, the subdivision illustrated on Figure 1 is used to carried out the the authoritative version for attribution. numerical integration on smaller panels. When the influenced point is located on the influencing patch, the analytical integration of the singular Rankine component is performed over panels. The second fold of integration on the influenced patch is performed numerically by using 2×2, 3×3 and 5×5 rules of Gauss-Legendre depending on the distance between two patches. The integration of the wave component is not as easy as expected since the integrand can be highly oscillatory especially along the waterline. The influence coefficients involving the wave component have been performed in several manners as described in the introduction. Finally, the method adopted in the present study is to multiply the integrand of (20a) and (25) by a smoothing function of type

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 73 (42) with n≥1, a positive real constant C satisfying e−C 1 and km being determined by the size of the patches and smoothness of distribution of singularities. A number of numerical tests show the convergence and stable results which are few affected by the choice of the values of (C, km). NUMERICAL RESULTS The above-described numerical method is now tested through a comprehensive comparison with a series of semi- analytical results as well as experimental measurements. They include the classical analytical results for a floating hemisphere, semi-analytical solutions to the Neumann-Kelvin flow around an ellipsoid and semi-analytical results for an submerged sphere advancing in water waves. The results obtained by the present method are also confronted with those of experimental measurements on Wigley hulls. Furthermore, most results are presented in tabulated form which may be useful to readers. Hulme's he misphere The first set of numerical results obtained by using the new Green function method include added-mass and damping coefficients and wave exciting forces on the floating hemisphere studied by Hulme (1982). The analytical solution to the radiation of waves by a floating half-submerged sphere in deep water is found in Hulme's study by constructing an expansion for the velocity potential in terms of series of spherical harmonics. The added-mass and damping coefficients are then evaluated and believed to be accurate to 4 decimal places. Hulme's analytical results at zero forward speed provide naturally the first benchmark for the present numerical method. A mesh composed of a total of 66 bi-quadratic patches representing the hull of a hemisphere of radius R=1 is constructed and illustrated by a perspective view (Figure 2). Figure 2: Mesh used to represent Hulme's hemisphere Table 1: Added-mass and damping coefficients for Hulme's hemisphere kR a11 b11 a33 b33 0.02 0.5049 0.0000 0.8652 0.0448 0.05 0.5113 0.0001 0.8768 0.1037 0.10 0.5234 0.0011 0.8631 0.1817 0.20 0.5526 0.0082 0.7942 0.2794 0.30 0.5860 0.0256 0.7160 0.3255 0.40 0.6187 0.0558 0.6455 0.3411 0.50 0.6452 0.0989 0.5863 0.3391 0.60 0.6599 0.1519 0.5383 0.3271 0.70 0.6594 0.2096 0.5001 0.3098 0.80 0.6433 0.2659 0.4700 0.2899 0.90 0.6137 0.3152 0.4467 0.2690 1.00 0.5749 0.3542 0.4287 0.2483 1.10 0.5312 0.3817 0.4151 0.2284 1.20 0.4867 0.3985 0.4050 0.2095 1.30 0.4438 0.4062 0.3977 0.1918 1.40 0.4042 0.4067 0.3927 0.1755 1.50 0.3687 0.4019 0.3894 0.1605 1.60 0.3374 0.3933 0.3875 0.1467 1.80 0.2867 0.3698 0.3868 0.1227 2.00 0.2494 0.3426 0.3888 0.1028 2.50 0.1949 0.2769 0.3992 0.0645 3.00 0.1718 0.2234 0.4115 0.0454 3.50 0.1634 0.1810 0.4228 0.0311 4.50 0.1656 0.1290 0.4407 0.0158 6.00 0.1784 0.0779 0.4575 0.0065 8.00 0.1962 0.0490 0.4703 0.0026 10.00 0.2081 0.0262 0.4774 0.0012 The added-mass and damping coefficients (a11, b11) in surge and (a33, b33) in heave are evaluated by using this mesh. Their values divided by (ρR32π/3) for a11 and a33, and (ρR3ω2π/3) for b11 and b33 are given in tabulated form (Table 1) for various values of kR varying from 0.02 to 10. By comparing with Hulme's results which are also given in tabulated form, an the authoritative version for attribution. excellent precision of 2 to 4 exact figures is observed. The wave exciting forces (not given in Hulme, 1982) are also evaluated and presented in tabulated form (Table 2). The surge forces F 1 and heave F3 are divided respectively by (ρgR3ka02π/3) and (ρgπR 2a0) where a0 is the incident wave amplitude. As for the added-mass and damping, it is believed that an accuracy of 2 up to 4 decimal places is obtained. Farell's ellipsoid The second set of numerical results concerns wave resistance associated with the Neumann-Kelvin steady flow applied on an ellipsoid. An ellipsoid with its major axis parallel to the free surface and in the direction of its forward speed is considered in Farell (1973). The steady flow around the particular ellipsoid of revolution (with an identical length of two other small axes), called slender prolate spheroid by Farell, is analyzed by expanding the velocity potential in forms

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 74 Table 2: Wave loads on Hulme's hemisphere Table 3: Wave resistance coefficients on Farell's ellipsoid kR F CW(d/c)1 CW(d/c)2 CW(d/c)3 0.35 0.2907 0.0614 0.0049 −0.0006 −0.9751 0.02 1.4896 0.0000 0.40 1.2293 0.5620 0.1222 −0.0035 −0.9378 0.05 1.4732 0.0001 0.42 1.5862 0.8244 0.2167 −0.0121 −0.8777 0.10 1.4468 0.0010 0.45 2.0017 1.1799 0.3808 −0.0375 −0.7688 0.20 1.3959 0.0070 0.48 2.2685 1.4480 0.5396 −0.0661 −0.6751 0.30 1.3450 0.0201 0.50 2.3762 1.5739 0.6306 −0.0934 −0.5941 0.40 1.2898 0.0400 0.52 2.4386 1.6614 0.7059 −0.1174 −0.5234 0.50 1.2270 0.0643 0.54 2.4649 1.7163 0.7648 −0.1376 −0.4608 0.60 1.1550 0.0888 0.55 2.4673 1.7332 0.7883 −0.1541 −0.4051 0.70 1.0744 0.1091 0.56 2.4636 1.7442 0.8081 −0.1669 −0.3549 0.80 0.9880 0.1216 0.58 2.4410 1.7505 0.8372 −0.1763 −0.3095 0.90 0.8998 0.1244 0.60 2.4027 1.7401 0.8541 −0.1828 −0.2684 1.00 0.8136 0.1181 0.62 2.3534 1.7169 0.8607 −0.1867 −0.2309 1.10 0.7324 0.1043 0.65 2.2655 1.6655 0.8557 −0.1882 −0.1968 1.20 0.6578 0.0853 0.70 2.1017 1.5543 0.8201 −0.1877 −0.1657 1.30 0.5905 0.0635 0.75 1.9351 1.4318 0.7663 −0.1854 −0.1374 1.40 0.5302 0.0406 0.80 1.7766 1.3108 0.7059 −0.1815 −0.1117 1.50 0.4763 0.0180 − 0.0033 −0.1763 −0.0884 1.60 0.4280 − 0.0403 −0.1628 −0.0483 1.80 0.3455 − 0.0688 −0.1461 −0.0162 2.00 0.2775 − 0.1059 −0.0887 2.50 0.1505 0.0279 − 0.1076 −0.0550 3.00 0.0649 0.0597 − 0.0884 −0.0162 3.50 0.0092 0.0592 −0.0384 − 0.0360 4.50 0.0249 0.0298 −0.0202 −0.0111 6.00 0.0164 0.0178 −0.0097 −0.0070 8.00 0.0119 0.0064 − 0.0056 −0.0003 10.00 0.0011 0.0072 of source distribution of spheroidal harmonics. The wave resistances are then evaluated for a series of ellipsoids by varying the ratio of length between the major and small axes at three depths of submergence relative to the focal distance. A mesh composed of a total of 90 bi-quadratic patches representing the ellipsoid's hull is constructed and illustrated by a perspective view (Figure 3). The lengths of major and small axes are respectively 2a= 2.3 and 2b=0.4 so that the focal distance is The wave resistance applied on the ellipsoid is evaluated by the present numerical method. The results are presented in tabulated form (Table 3) for three relative depths of submergence as used by Farell. The Froude number and the wave-resistance coefficients in Table 3 are defined as Figure 3: Mesh used to represent Farell's ellipsoid respectively. C W is given at three relative depths of submergence (d/c). By reporting the values of C W given in Table 3 at (d/c)1=0.252 and (d/c)2=0.3266 to Fig. 1 and Fig. 2 in Farell (1973), any visual difference from the corresponding curves can be observed. Furthermore, the values of C W for a/b=5.75 at (d/c)3=0.5 given in Table 3 are well located in the middle between the curves for a/b=5.5 and a/b=6 on Fig. 3 in Farell (1973). These observations show that the present numerical method provides results that match Farell's. the authoritative version for attribution. Further to above satisfactory validations for two special cases at τ=0, namely Hulme's analytical solution for a hemisphere at zero forward speed (Fn=0) and Farell's semi-analytical solution for an immerged ellipsoid advancing beneath the free surface at zero frequency (ω=0), the next logical step is to consider the general case at τ ≠ 0. Wu's sphere The first benchmark for the general case is due to the recent work by Wu (1995). Wu used the multipole expansion extended from that developed for zero forward speed in Ursell (1949) to derive directly the velocity potential around a submerged sphere advancing in water waves. The linearization of the condition on the free surface leads to the decomposition of total velocity potential into a steady part and an unsteady part as (1). Wu assumed further that the steady flow is small on the free surface for a submerged body but not on the body surface, so that the exact interaction between steady and unsteady flows through the mj terms (8) derived from the Neumann-Kelvin solution is taken into account in

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 75 the evaluation of unsteady velocity potential. This original work yields accurate results (with 4 significant figures) for wave resistance and lift forces, added-mass and damping coefficients as well as wave loads. A mesh composed of 56 bi-quadratic patches representing the sphere of radius R =1 is constructed and illustrated by a perspective view (Figure 4). One forward speed corresponding to a Froude number and two depths of submergence (h/R) were studied. The case of h/R=2 is one presented in Wu (1995) for which the added-mass and damping coefficients are evaluated and presented in Figures 5–8 as well as wave loads in Figures 9–11, and wave resistance and lift forces in Figure 12. To note that Wu's method gives results in water of finite depth and to be close to those in deep water, a depth of 10 times the sphere radius is used in Wu's computations. Figure 4: Mesh used to represent Wu's sphere Figure 5: Added-mass and damping coefficients in surge motion (a11, b11) of Wu's sphere The abscissae in Figures 5–8 are the reduced wavenumber kR while the values of added-mass and damping coefficients are non-dimensionalized by respectively. The values of added-mass coefficients akj are represented by the solid line obtained from the present numerical method and by the circle symbols for Wu's semi-analytical solution. Those of damping coefficients bkj are represented by the dot-dash line and square symbols for the present numerical method and Wu's semi-analytical method respectively. It is observed that very good agreement between the two methods is obtained, despite the difference in the evaluation of mj terms and in waterdepth. The present method uses the double-body flow to derive the mj terms, while Wu's uses the Neumann-Kelvin flow. The difference may be considered as negligible for the sphere at this depth of submergence. Figure 6: Added-mass and damping coefficients in sway Figure 7: Added-mass and damping coefficients in heave motion (a22, b22) of Wu's sphere motion (a33, b33) of Wu's sphere The critical value τ=1/4 is located at kR ≈ 0.3906 where the effect of change of flow regime is remarkable in all figures except the added-mass and damping coefficients in sway motion (a22, b22) illustrated on Figure 6, which seem not to be affected. It is known from previous studies (Grue & Palm, 1983 and Liu & Yue, 1993) that hydrodynamic coefficients vary the authoritative version for attribution. sharply for τ crossing the critical value but not singular. The present results seem to confirm that a finite value does exist at τ=1/4. Due to the effect of forward speed, the symmetry

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 76 between the coefficients in surge and sway motions is lost (comparing Figures 5 and 6). Furthermore, the coupling coefficients in surge-heave motions are not zero and far from negligible as shown on Figure 8. Numerical results shows that the reverse flow relation given in Timman & Newman (1962) Figure 9: Wave loads F1 on Wu's sphere Figure 8: Added-mass and damping coefficients in surge- heave motion (a13, b31) of Wu's sphere is satisfied accurately to 3 decimal places, so that only (a13, b31) are illustrated on Figure 8. Solutions to the diffraction problem are represented on Figures 9–11 for wave loads in surge (F 1), sway (F2) and heave (F3), respectively. The heading of incoming waves is β=3π/4. The abscissae are the reduced wavenumber k0R with k0 the wavenumber of incoming waves. The values of forces are divided by where k is the wavenumber associated with the encounter frequency and a0 the amplitude of incoming waves. It may be noted that (F1, F 2, F3) in the present study are equivalent to the complex conjugate of (iFx, iFy, iFz) in Wu (1995), following the convention used in the present method and that used by Wu. Figure 10: Wave loads F2 on Wu's sphere Figure 11: Wave loads F3 on Wu's sphere In Figures 9–11, the real and imaginary parts of (F 1, F2, F3) are represented respectively by the solid and dot-dash lines for the present numerical method, and by the circles and squares for Wu's method. Except at small wavenumber where waterdepth may have noticeable effects, the results of both methods are in very good agreement. In the same way as the added-mass and damping coefficients, the change of flow regime at k0R ≈ 0.2937 corresponding to the critical value τ=1/4 yields rapid variation in surge forces F1 and heave forces F3. Finally, the wave-resistance and lift coefficients (CW, C L) on Wu's sphere at this depth (h/R=2) of submergence are evaluated and presented on Figure 12 by the solid and dot-dash lines respectively. The abscissae are the Froude number while the values of both coefficients here are non-dimensionalized as and the authoritative version for attribution. A precision of at least 2 exact figures is found for the results of the present method by comparing with Wu's represented by the circles and squares. The case of a smaller depth of submergence h/R=

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 77 1.1 (h is the distance of the sphere center from the free surface) is now considered. The same mesh (Figure 4) is used in the computation of added-mass and damping coefficients, wave loadings and wave-resistance and lift. Due to the small submergence, differences between the present numerical method involving mj terms derived from the double-body flow and Wu's semi-analytical method involving mj terms derived from the Neumann-Kelvin flow may be expected. Indeed, some discrepancy between two methods for added-mass and damping coefficients, and wave loadings, is found and considered to be reasonable because of differences in the two methods explained above. To confirm this, an interesting case, to compute added-mass and damping coefficients by setting zero values for m1,2,3 terms, is studied. The results presented in tabulated form (Tables 4 and 5) are in excellent agreement with Wu's (private communication) with an accuracy of 2 exact figures after the decimal point. Figure 12: Wave-resistance and lift coefficients on Wu's Table 4: Added-mass coefficients for Wu's sphere by setting sphere m1,2,3=0 kR a11 a22 a33 a13 a31 − 0.0485 0.05 0.6709 0.6111 0.8358 0.0489 − 0.1038 0.10 0.7090 0.6320 0.9068 0.1043 − 0.2047 0.15 0.7421 0.6582 0.9699 0.2052 − 0.3689 0.20 0.7300 0.6895 0.9554 0.3692 − 0.5273 0.25 0.6026 0.7245 0.7453 0.5274 − 0.5053 0.30 0.4021 0.7589 0.4196 0.5052 − 0.3236 0.35 0.2782 0.7785 0.2406 0.3235 − 0.0995 0.40 0.3025 0.7021 0.3481 0.0994 − 0.0762 0.45 0.3616 0.6096 0.4040 0.0761 − 0.0504 0.50 0.3903 0.5599 0.4173 0.0504 − 0.0075 0.60 0.4250 0.4997 0.4169 0.0075 −0.0262 0.70 0.4456 0.4619 0.4051 0.0261 −0.0526 0.80 0.4580 0.4351 0.3901 0.0526 −0.0732 0.90 0.4648 0.4153 0.3748 0.0731 −0.0890 1.00 0.4675 0.4002 0.3602 0.0889 −0.1007 1.10 0.4673 0.3886 0.3468 0.1006 −0.1151 1.30 0.4614 0.3726 0.3241 0.1149 −0.1207 1.50 0.4520 0.3633 0.3068 0.1205 Table 5: Damping coefficients for Wu's sphere by setting Table 6: Added-mass coefficients for Wu's sphere with double- m1,2,3=0 body mj terms at (h/R=1.1) kR b11 b22 b33 b 13 b31 kR a11 a22 a33 a13 a31 −0.1758 − 2.4652 0.05 0.0684 0.0024 0.1188 0.1760 0.05 8.3899 5.6621 10.1824 2.4868 −0.2832 − 2.0618 0.10 0.1576 0.0098 0.2732 0.2836 0.10 4.5933 3.2424 5.4949 2.0775 −0.3654 − 2.1359 0.15 0.3178 0.0276 0.5508 0.3662 0.15 3.2064 2.4703 3.6898 2.1494 −0.3236 − 2.3140 0.20 0.5776 0.0628 0.9992 0.3246 0.20 2.2659 2.1173 2.3215 2.3257 − 0.0202 − 2.2320 0.25 0.8342 0.1254 1.4356 0.0214 0.25 1.3489 1.9354 0.8796 2.2409 − 0.5502 −0.1786 − 1.6248 0.30 0.8206 0.2316 1.4052 0.5512 0.30 0.6394 1.8324 1.6298 − 0.8670 −0.4628 − 0.8836 0.35 0.5636 0.4078 0.9724 0.8676 0.35 0.3494 1.7358 0.8859 − 0.7646 − 0.2378 0.40 0.2122 0.6570 0.5066 0.7650 0.40 0.4361 1.3827 0.0009 0.2389 − 0.6480 − 0.1947 0.45 0.1698 0.6460 0.5516 0.6484 0.45 0.5678 1.0471 0.2002 0.1959 − 0.5996 − 0.1400 0.50 0.1440 0.6192 0.5588 0.6000 0.50 0.6108 0.8823 0.2606 0.1411 − 0.5398 − 0.0606 0.60 0.1226 0.5680 0.5608 0.5402 0.60 0.6411 0.7070 0.2999 0.0615 − 0.4922 − 0.0105 0.70 0.1216 0.5228 0.5562 0.4928 0.70 0.6459 0.6138 0.3093 0.0111 − 0.4480 −0.0211 0.80 0.1300 0.4822 0.5468 0.4484 0.80 0.6407 0.5568 0.3110 0.0216 − 0.4052 −0.0417 0.90 0.1422 0.4454 0.5338 0.4056 0.90 0.6308 0.5193 0.3107 0.0421 − 0.3640 −0.0547 1.00 0.1552 0.4116 0.5180 0.3644 1.00 0.6187 0.4935 0.3101 0.0550 − 0.3246 −0.0625 1.10 0.1674 0.3806 0.5000 0.3252 1.10 0.6057 0.4753 0.3098 0.0627 − 0.2532 −0.0684 1.30 0.1864 0.3256 0.4610 0.2536 1.30 0.5800 0.4527 0.3105 0.0686 − 0.1918 −0.0674 1.50 0.1974 0.2788 0.4202 0.1922 1.50 0.5568 0.4406 0.3126 0.0674 The results of added-mass and damping coefficients with mj terms derived from the double-body flow are presented on Tables 6 and 7 respectively. Compared with the results obtained by setting m1,2,3=0 (Tables 4 and 5), the importance of mj terms is evident, especially in the region of small wavenumber. the authoritative version for attribution. The wave loads are presented in Table 8. The forces (F1, F2, F 3) are non-dimensionalized by (ρgπR 3ka0) with k the wavenumber associated with encounter frequency and a0 the amplitude of incoming waves. Finally, the wave-resistance and lift forces (CW , CL) as well as the trim moment (CM) about the y-axis at the sphere center (0, 0, −1.1R) are given in Table 9. As in Figure 12, the Froude number is defined as while the wave-resistance and lift coefficients are defined by and

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 78 and the trim moment coefficient by An excellent agreement is observed between the results of (CW, CL) and those by Wu (private communication), as at least two exact figures after the decimal point are found in both sets of results. Compared with the values presented on Figure 12, the wave resistance and lift forces are augmented by a factor of 5 and 7 respectively. Table 7: Damping coefficients for Wu's sphere with double-body Table 8: Wave loads on Wu's sphere mj terms at (h/R =1.1) kR b11 b22 b33 b 13 b31 k0R F1 F2 F3 −3.4454 0.05 (1.2025, −0.1412) (−1.3135 −0.0076) (−0.3104, 1.6410) 0.05 2.7592 0.1186 6.3416 3.4246 0.10 (0.9897, −0.2357) (−1.2200 −0.0155) (−0.3887, 1.6639) −2.2844 0.10 2.6528 0.1334 5.7424 2.2746 0.15 (0.9398, −0.3721) (−1.1581 −0.0351) (−0.5668, 1.5282) −1.4544 0.15 2.9166 0.1804 6.0026 1.4518 −0.3190 0.20 (1.0862, −0.4564) (−1.1170 −0.0776) (−0.6814, 1.1516) 0.20 3.2782 0.2672 6.4542 0.3232 − 1.3126 0.25 (1.2295, −0.2677) (−1.0793 −0.1673) (−0.4387, 0.8159) 0.25 3.2540 0.4168 6.1498 1.3224 − 2.6232 0.30 (1.0870, −0.0292) (−0.9178 −0.3356) (−0.1865, 0.8901) 0.30 2.4422 0.6700 4.4356 2.6348 0.35 (0.9705, −0.0172) (−0.7326 −0.2838) (−0.2283, 0.8666) − 2.9276 0.35 1.3998 1.1018 2.4332 2.9372 − 2.2364 0.40 (0.8909, 0.0088) (−0.6350 −0.2383) (−0.2310, 0.8061) 0.40 0.4062 1.7212 0.9122 2.2420 − 1.7172 0.45 (0.8207, 0.0320) (−0.5648 −0.2036) (−0.2262, 0.7416) 0.45 0.2992 1.4996 0.9668 1.7216 − 1.4580 0.50 (0.7559, 0.0512) (−0.5089 −0.1763) (−0.2179, 0.6791) 0.50 0.2244 1.2914 0.8942 1.4616 0.60 (0.6390, 0.0774) (−0.4217 −0.1360) (−0.1964, 0.5661) − 1.1398 0.60 0.1484 0.9948 0.7474 1.1426 0.70 (0.5380, 0.0904) (−0.3546 −0.1075) (−0.1726, 0.4708) − 0.9338 0.70 0.1210 0.7968 0.6306 0.9360 − 0.7834 0.80 (0.4520, 0.0942) (−0.3006 −0.0864) (−0.1492, 0.3917) 0.80 0.1140 0.6550 0.5388 0.7852 − 0.6670 0.90 (0.3794, 0.0923) (−0.2561 −0.0703) (−0.1276, 0.3264) 0.90 0.1154 0.5482 0.4654 0.6686 − 0.5740 1.00 (0.3187, 0.0869) (−0.2189 −0.0577) (−0.1082, 0.2724) 1.00 0.1196 0.4650 0.4054 0.5756 1.10 (0.2679, 0.0798) (−0.1874 −0.0476) (−0.0913, 0.2277) − 0.4982 1.10 0.1242 0.3982 0.3558 0.4996 − 0.3832 1.30 (0.1901, 0.0642) (−0.1379 −0.0329) (−0.0641, 0.1595) 1.30 0.1300 0.2986 0.2782 0.3840 − 0.3014 1.50 (0.1356, 0.0497) (−0.1017 −0.0229) (−0.0444, 0.1117) 1.50 0.1304 0.2286 0.2208 0.3022 Journée's Wigley-hull The second benchmark study for the general case at τ ≠ 0 involves confronting the results of experimental measurements on Wigley hulls. Further to the classic experimental data on vertical motions, wave loads and added resistances of two first Wigley hullforms in Gerritsma (1988), new tests on two other Wigley hullforms with a midship section coefficient of 2/3 and length-breadth ratios of 10 and 5 have been made in Journée (1992). Although only the head sea is tested and the surge motion is restricted (i.e. the model was free to perform heave and pitch motions only), a large number of reliable data were achieved. For the sake of space, the results on the model called Wigley model III are used for the present comparative study. Its main dimensions (length/breadth/draught=L/B/T) and dynamic characteristics are Figure 13: Mesh representing Journée's Wigley-hull Table 9: Wave-resistance, lift and moment coefficients on Wu's sphere Fn CW CL CM 0.20 0.0001 0.0221 0.0001 0.30 0.0296 0.1204 0.0026 0.40 0.1497 0.2154 0.0085 0.50 0.2759 0.1806 0.0130 0.60 0.3270 0.0640 0.0154 −0.0601 0.70 0.3178 0.0160 −0.1654 0.80 0.2824 0.0157 −0.2517 0.90 0.2415 0.0151 −0.3248 1.00 0.2035 0.0145 −0.3905 1.10 0.1710 0.0140 −0.4526 1.20 0.1442 0.0135 −0.5137 1.30 0.1224 0.0130 −0.5756 1.40 0.1047 0.0127 −0.6392 1.50 0.0903 0.0123 −0.7053 1.60 0.0784 0.0120 −0.7742 1.70 0.0687 0.0118 −0.8464 1.80 0.0605 0.0116 −0.9219 1.90 0.0537 0.0114 −1.0010 2.00 0.0480 0.0112 the authoritative version for attribution. The hull shape is defined analytically by in a coordinate system with the origin at the amidship and on the waterplane. The Wigley-hull is panelized with 128 bi-quadratic patches and illustrated by a perspective view (Figure 13). This mesh is used, and the case of a forward speed corresponding to the Froude number (and at zero forward speed for motions) is considered in the present numerical method.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 79 The added-mass and damping coefficients in heave (a33, b33), in pitch (a55, b55) and in coupling heave-pitch (a35, b35) are presented in Figures 14, 15 and 16, respectively. The abscissae are the reduced circular frequency while the values of added-mass (a33, a35, a55) and damping coefficients (b33, b35, b55) are non-dimensionalized respectively by (ρ ∀ L0,1,2) and with ∀=0.0780 the displaced volume, and depicted by the solid and dot-dash lines. The results of experimental measurements are represented by the circles and downward-pointing triangles (corresponding to two sets of repeated model tests) for the added-mass coefficients, and the squares and upward-pointing triangles for the damping coefficients. The numerical results are in very good agreement with those of experimental measurements, except for those of damping coefficients b55 in pitch. However, the two sets of results (b55) have similar variation and close values. The critical value of τ=1/4 is located at where a rapid variation is expected due to the change of flow regime. Figure 14: Added-mass and damping coefficients (a33, b33) Figure 15: Added-mass and damping coefficients (a55, b55) for Journée's Wigley-hull for Journée's Wigley-hull The vertical wave exciting forces in heave (F3) and moments in pitch (F 5) are depicted on Figures 17 and 18. The abscissae are the ratio between the wavelength of incoming waves and hull length (λ/L). Wave loads (F 3) and (F5) are non- dimensionalized respectively by (C 33a0) and (C55k0a0) where k0 and a0 are the wavenumber and amplitude of incoming waves, C33=6119 and C55=2874 are the hydrostatic stiffness in heave and pitch motions. The numerical results are depicted by the solid lines and two sets of model tests by the circles and downward-pointing triangles. Again, the numerical and experimental results are very close to each other. Figure 16: Added-mass and damping coefficients (a35, b35) Figure 17: Wave loads on Journée's Wigley-hull Exciting forces in heave F3 for Journée's Wigley-hull Finally, response-amplitude operators (RAOs) of heave and pitch motions are represented in Figures 19 and 20 at forward speed (Fn=0.3) as well as at zero forward speed (F n=0). The abscissae in both figures are the ratio (λ/L), while the amplitude values of heave displacements and pitch angles are non-dimensionalized by the wave amplitude (a0) and (a02π/L) respectively. The numerical and experimental results at Fn=0.3 are depicted respectively by the solid lines and circles and those at zero forward speed by the dot-dash lines and squares. The important ef the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 80 fect of forward speed on heave and pitch motions is evident by comparing the solid lines (F n=0.3) and dot-dash lines (Fn=0) in both figures. The excellent agreement between numerical results (solid and dot-dash lines) and experimental measurements (circles and squares) in both cases (F n=0.3 and F n=0) shows that the present method can provide very good prediction of wave-induced ship motions and loads. Figure 18: Wave loads on Journée's Wigley-hull Exciting Figure 19: Heave motions of Journée's Wigley-hull moments in pitch F5 DISCUSSIONS AND CONCLUSIONS In this paper, we have summarized the theoretical background and numerical development of a new 3D method to predict the wave-induced ship motions and loads. The excellent level of agreement with semi-analytical results and experimental measurements is due to several important features of the method. First of all, the new formulation of ship-motion Green function and its gradients expressed by various components makes it possible to deal with different components separately and to develop appropriate algorithms in the accurate and efficient computation of influence coefficients associated with them. The higher-order description of ship hull geometry by bi-quadratic patches and continuous representation of the velocity potential lead to the solution of very good precision and the reduction of computing time. In all previous studies, the indirect method corresponding to the source formulation is surprisingly used only. The present study makes use of the direct method corresponding to the potential formulation which is directly derived from the Green's identity and preferable although two methods are thought to be equivalent. Furthermore, the Galerkin solution procedure is adopted instead of the procedure of point collocation. This procedure of domain collocation is well recognized to be superior in numerical stability and convergence. Finally, the mj terms involving in the boundary condition on the hull surface are evaluated by using the numerical method which permits to have accurate double derivatives of velocity potential without any numerical extrapolation or interpolation. It is understood that the mj terms are critically important in the computation of added-mass and damping coefficients especially for large Froude number. Figure 20: Pitch motions of Journée's Wigley-hull The present numerical method taking advantage of above-mentioned properties can be considered to be sound and robust as it is able further to provide very good results in a number of applications, although more validations are needed especially for hulls of more complex form as well as for computing the values of local pressures and wave patterns. It also offers a solid basis of development for a number of perspectives. Firstly, the suppression of irregular-frequency effects can be made by applying an additional condition of zero-potential over the surface of waterplane. The present method makes it possible since the influence coefficients between two patches located at the waterplane can now be computed correctly. Secondly, the linearization of free-surface boundary condition about a basic flow like the double-body potential (instead of a uniform flow) yields an extension of the integral equation by an additional zonal integral on the free surface close to the waterline. Finally, the present the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as NEW GREEN-FUNCTION METHOD TO PREDICT WAVE-INDUCED SHIP MOTIONS AND LOADS 81 study can also be very useful in the development of a method to couple with a near-field calculating method accounting for nonlinear or/and viscous effects. ACKNOWLEDGMENTS The first author is greatly indebted to Professor Wu Guo Xiong from University College London for providing comprehensive data of his semi-analytical results for a submerged sphere which have been critically useful in the development of computer codes. 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