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Ship Motions by a Three-Dimensional Rankine Pane! Method D. Nakos, P. Sclavounos (Massachusetts Institute of Technology, USA) ABSTRACT A Rankine Panel Method is presented for the solution of the complete three-dimensional steady and time-harmonic potential flows past ships advancing with a forward veloc- ity. A new free-surface condition is derived, based on lin- earization about the double-body flow and valid uniformly from low to high Froude numbers. Computations of the steady ship wave patterns reveal sig- nificant detail in the Kelvin wake a significant distance downstream of the ship, permitted by the cubic order and zero numerical damping of the panel method. The wave pattern appears to be sensitive to the selection of the free- surface condition only for full ship forms. The heave and pitch hydrodynamic coefficients exciting forces and motions of a Wigley and a Series-60 hull have been evaluated in head waves over a wide range of fre- quencies and speeds. A robust treatment is proposed of the m-terms which are found to be critical importance for the accurate solution of the problem. In all cases the agreement with experiments is very satisfactory indicating a significant improvement over strip theory, particularly in the cross-coupling and diagonal pitch damping coefficients. 1. INTRODUCTION Theoretical methods for the prediction of the seakeeping of ships have evolved in three phases over the Dast 40 Years. The first phase involved the development of strip theory, and was followed by a series of developments in slender- body theory which formulated rationally the ship motion problem and produced several refinements of striD theory. The advent of powerful computers in the early 80's al- lowed the transition into the third and current phase of seakeeping research which aims at the numerical solution of the three-dimensional problem. This paper presents our progress in that direction. The pioneering work of Korvin-Kroukovsky (19S5) stimu- lated a number of studies on the strip method which led to the theory of Salvesen, Tuck and Faltinsen (1970~. Its pop- ularity to date arises from its satisfactory performance in the prediction of the motions of conventional ships and its computational simplicity. Well documented are however its limitations in the prediction of the derived responses, 21 structural wave loads and in general the seakeeping char- acteristics of ships advancing at high Froude numbers te.g. O'Dea and Jones (19833~. The 60's and 70's witnessed several analytical studies aim- ing to extend the slender-body theory of aerodynamics to the seakeeping of Slender ships. The rational justification of strip theory, as a method valid at high frequencies and moderate Froude numbers, was presented by Ogilvie and Tuck (1969~. This theory was extended to the diffrac- tion problem by Faltinsen (1971) and was further refined by Maruo and Sasaki (1974~. The high-frequency restric- tion in earlier slender-ship theories was removed by the unified theory framework presented by Newman (1978~. Its extension to the diffraction problem was derived by Sclavounos (1984) and applied to the seakeeping of ships by Newman and Sclavounos (1980) and Sclavounos (1984~. Subsequent slender-ship studies by Kim and Yeung (1984) and Nestegard (1986), accounted directly for convective forward-speed wave effects near the ship hull and repre- sented the transition to numerical studies aiming at the solution of the three-dimensional ship-motion problem. By the mid-80's, the performance of slender-body theory for the seakeeping problem could only be validated from experimental measurements. Moreover, it had become ev- ident that end-effects at high Froude numbers cannot be modelled accurately by slender-body approximations and the need for a numerical solution of the complete three- dimensional had emerged. Early efforts towards this coal - by Chang (1977), Inglis and Price (1981) and Guevel and Bougis (1982) were not conclusive because the significant computational effort necessary for the evaluation of the time-harmonic forward-speed Green function limited the total number of panels used on the ship surface. More recently, King, Beck and Magee (1988) circumvented this difficulty by solving the same problem in the time domain, therefore making use of the zero-speed transient Green function which is easier to evaluate. The last decade witnessed the growing popularity of Rank- ine Panel Methods for the solution of the steady poten- tial flow past ships. The success of the early work of G add (1976) and Dawson (1977) motivated several anal- ogous studies which concentrated upon the prediction of the Kelvin wake and evaluation of the wave resistance. The principal advantages of the method are twofold - the Rank- ine singularity is simple to treat computationally and the distribution of panels over the free surface allows the en- forcement of more general free-surface conditions with vari

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able coefficients. A drawback of Rankin~panel methods is that they require about twice as many panels as methods based on the distribution of wave singularities over the ship surface alone. The resulting computational overhead is as- sociated with the solution of the resulting matrix equation, but may not be significant if an out-of-core iterative solu- tion method is available. This paper outlines the solution of the three-dimensional time-harmonic ship motion problem by a Rankine Panel Method. For the steady problem, the theory for the anal- ysis of the properties for such numerical schemes was in- troduced by Piers (1983) and generalized by Sclavounos and Nakos (1988~. The extension of this numerical anal- ysis to the time-harmonic problem is presented in Nakos and Sclavounos (1990~. In this reference the convergence properties of a new quadratic-spline scheme are derived, which has been found to be accurate and robust for the so- lution of both steady and time-harmonic free-surface flows in three dimensions. This scheme is applied in this paper to the solution of the time-harmonic radiation/diffraction potential flows around realistic ship hulls and the evalu- ation of the hydrodynamic forces and motions in regular head waves. A new three-dimensional free-surface condition is derived, using the double-body flow as the base disturbance due to the forward translation of the ship. This is shown to be valid uniformly from low to high Froude numbers and over the entire frequency range. Known low-Froude-number conditions for the steady problem, as well as the Neumann- Kelvin condition, are obtained as special cases. The ship- hull condition includes the m-terms which are evaluated from the solution of the three-dimensional double-body flow. An important property of the solution scheme is that the evaluation of the double gradients of the double-body flow is circumvented by an application of Stokes theorem. Computations are presented of the steady wave patterns trailing a fine Wigley model and a fuller Series-60 hull. The cubic order and zero numerical damping of the free- surface discretization allows the prediction of significant detail of the Kelvin wake at a large distance downstream of the ship. A comparison of the wave patterns obtained form the Neumman-Kelvin and the more general double body free-surface conditions reveals good agreement for the Wigley hull, while evident differences appear in the respective Series-60 wakes. Predictions of the heave and pitch added-mass and damp- ing coefficients and exciting forces are found to be in very good agreement with experimental measurements both for the Wigley and the Series-60 hull. The contribution of the complete m-terms is found to be important, partic- ularly in the cross-coupling coefficients. The validity of a more general set of Timman-Newman relations is observed and conjectured in connection with free-surface conditions based on the double-body flow. The heave and pitch motion amplitudes and phases pre- dicted by the present method are found in very good agree- ment with experiments and present an improvement over strip theory. 2. THE BOUNDARY VALUE PROBLEM Define a Cartesian coordinate system x = (x,y,z) fixed on the ship which translates with a constant speed U. The positive x-direction points upstream and the posi- tive z-axis upwards. The boundary-value problem will be expressed relative to this translating coordinate system, therefore the flow at infinity is a uniform stream and the ship hull velocity is due to its oscillatory displacement from its mean position. The fluid is assumed incompressible and inviscid and the flow irrotational, governed by a potential function ~(x-,t) which satisfies the Laplace equation in the fluid domain V2~(X,t) = 0 ~ (2.1) Over the wetted portion of the ship hull (B), the compm nent of the fluid velocity normal to (B) is equal to the corresponding component of the ship velocity VB, or [} (x,t) = (VB n)(x,t), where the unit vector n points out of the fluid domain. (2.2) The fluid domain is also bounded by the free surface, de- fined by its elevation x = `(x, y, t) and subject to the kine- matic boundary condition, (fit +V~I ~V) Liz-~(x,y,t)] = 0 on z = ~(x~y~t) (2.3) The vanishing of the pressure on the free surface combined with Bernoulli's equation, leads to the dynamic free surface condition `(x, y, t) = - - (~' + 2V~ V4i - -u2~1 2 J z= s (2.4) The elimination of ~ from (2.3) and (2.4) leads to (Ptt + 2V~ ~ V~t + 2 V\li ~ V(V~ - V~) +g~z = 0 on z = ~ . (2.5) If the fluid domain is otherwise unbounded, the additional condition must be imposed that at finite times the flow velocity at infinity tends to that of the undisturbed stream. Linearization of the free surface condition Physical intuition suggests that linearization of the pre- ceding boundary value problem is justified when the dis- t,~rbance of the uniform incoming stream due to the ship is in some sense small. Small disturbances may be justi- fied by geometrical slenderness, slow forward translation, or a combination of the above. Full-shaped ships typically advance at low speed and cause a small steady wave distur- bance. Fine-shaped ships, on the other hand, often advance at high Froude numbers. Yet the steady disturbances they generate, is small if their geometry is sufficiently thin or slender. Linearization may therefore be justified both at low and high Froude numbers F, as long as it is tied to the hull slenderness c. Linearization of the unsteady flow is also supported by the assumption of a small ambient wave amplitude. The linearized free surface condition derived next is uni- formly valid between these two limits, and it~ validity is 22

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heuristically justified if the parameter eF2 is sufficiently small. The details of the derivation outlined below are given in Nakos (1990~. The total flow field ~(x,t) is de- composed into a basis flow ~(x), assumed to be of O(1), the steady wave flow Add, and the unsteady wave flow (x,t) ~(x, t) = ~ + ~ + (x, t) . (2.6) The double-body flow is chosen as the basis flow, a selection primarily motivated by the body boundary condition as well as the simplifications it allows in the ensuing analysis. Thus, ~ is subject to the rigid wall condition: As = 0 , on z = 0 . (2.7) The wave disturbances , and `6 are superposed upon the double-body flow and are taken to be small relative to the A. Linearization of (2.4-5), correct to leading order in and , leads to the conditions: Vie VIVA V~+2V(V4) V~V++g~z - ~X2(V~ V) = - 2V(V~ Vie) V~- -2 (U2-Vie Vow, on z = 0 ~ (~:E Y) = -go (2Vq) Vie- 2u2 +Vqi- v~) ~ ' + 2Vq} Vet + V4) V (V4} ~ V) +-V(V~ V~) V+gjx -~ + Vie V) = 0, on z .. .. <(x,y,t)=-~('h+v~ v)z=o =0~ (2.9) for the steady and unsteady flows, respectively. For slender/thin ships with ~ small, and for Froude num- bers of O(1), the uniform incident stream -Up may be used as the basis flow. In this case, (2.8-9) reduce to the well- known Neumann-Kelvin conditions. In the opposite limit of blue ships with ~ of O(1) advancing at low Froude num- bers, (2.8-9) reduce to the conditions of slow-ship theory. The condition (2.8) contains all terms present in Dawson's (1977) condition, and it is closest to the one proposed by Eggers (1981~. This property may explain the fact that, even though Dawson's and Egger's conditions have been derived as low Foude number approximations, they have been found to perform satisfactorily over a wider range of forward speeds. Linearization of the body boundary condition The linearization of the ship hull boundary condition may also be derived from the decomposition (2.6). By defi- nition, the velocity potential of the double-body flow is subject to 04P = 0 , on (B) (2.10) Consequently, the steady wave flow also satisfies the homo- geneous condition = 0 , on (B) , (2.11) leaving the right-hand-side of (2.8) as the only forcing of the steady wave problem. The unsteady forcing due to the oscillatory motion of the vessel is accounted for by the unsteady wave flow . If a is the oscillatory displacement vector measured from the mean position of the vessel (B), it follows by substituting of (2.6) in (2.2) that B~ = ~ . n - V(~ + f) ~ n , on (B) (2 12) Assuming that the magnitude of the displacement vector a is small and comparable to the ambient wave amplitude, the boundary condition (2.12) may be linearized about the mean position of the hull surface iTimman and New- man(l962)], B~ Ba n - [(a V)V~+(V~ V)~ n ,on ( ) (2.13) The last term in (2.13) accounts for the interaction be- tween the steady and unsteady disturbances in a manner consistent with the assumptions underlying the derivation of the free-surface conditions (2.8). An alternative form of (2.13) may be derived in terms of the rigid-body global displacements (hi, (2, (3) and rotations (54, (5, (6), along the axes (x, y, z) respectively, An ~ ~ ,,` nj + gmj) , on (B), (2~14) where my, j = 1, ...6, denote the so-called m-terms tOgilvie and Tuck (1969)~. If the basis flow is approximated by the uniform stream the only non-zero m-terms are ma = Un3 and me = -Un2, which merely account for the 'angle of attack eEect' due to yaw and pitch. This approximation of the m-terms has been employed in most previous studies of the ship motion problem, consistently with the linearization steps leading to the Neumann-Kelvin free surface boundary condition. The performance of this linearization in practice will be the subject of numerical experiments presented in section 7. Frequency domain formulation of the unsteady problem The unsteady excitation is due to an incident monochro- matic wave train. The frequency of the incident wave, as viewed from the stationary frame is we, while in the trans- lating frame of reference x, the incident wave arrives at 23

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the frequency of encounter w. If ,6 is the angle between the phase velocity of the incident wave and the forward velocity of the ship, ~ is given by ~ = ~0 - U costs . (2.15) In the frame x, the velocity potential of the incident wave of unit amplitude, in deep water, is given by the real part of the complex potential ~0: ~o~x,t)=i 9 eg (~-i2C0~-iY~in') ire (2.16) The linearity of the Boundary Value Problem that gov- erns the physical system, along with the form of the body boundary condition (2.14), suggest the decomposition of the wave flow as follows, (x,t) = ~ {e Pablo + ('P7) + ~ (j~j~ } , (2.17) where A is the amplitude of the incoming wave train, ~7 is the complex diffraction potential, and As, j = 1,...6, are the complex radiation potentials due to the harmonic oscillation of the ship in each of the six rigid-body degrees of freedom, at frequency ~ and with unit amplitude. Upon substitution of the linear decomposition into (2.9), the free surface conditions for As, j = 1,...7, are derived. It is important to point out that the free surface condition for the diffraction problem is inhomogeneous, the forcing arising from the interaction of the incoming wave train with the double-body flow. In the limit of slender/thin ships, where the uniform stream may be taken as the basis flow, this inhomogeneity vanishes. 3. THE HYDRODYNAMIC FORCES Given the solution of the potential flow problem formu- lated in the preceding section, the hydrodynamic pressure follows from Bernoulli's equation. Of particular interest, in practice, is the pressure distribution on the ship wetted surface and resultant forces and moments necessary for the determination of the ship motions. The pressure on the hull is given by p= -p [at + EVA- V4i - ~u2 +9x] . (3.1) ~6 (B ) The unsteady portion of (3.1), correct to leading order in , may be expressed as follows: p = _ p (~ + Vim V~(B) -p [(a V)~2 ](B) (3.2) Under the assumption of small monochromatic motions at the frequency of encounter A, the components of the un- steady force F = (F~, F2, F3) and moment M = (F4, F5, F6) acting on the ship, accept the familiar decomposition Fitt) = ~ feint Taxi + God (~2aijiwbij-calm ~ , {3.4] where, Xi = -pal T/J. [iw($oO + $o7) + V4, V(`po + C7~] nid' (is) aij = _ O2 ~ ~ /| (imp + Vie Vail ni d8 ) (B) bij = P ~ T || (its + V`P V~j) ni do ) (B) (3.5) cij = p /~(a.V)(,,v4! V~+9Z) ni ds, J J (B) for i, j = 1, ..., 6. The exciting forces Xi and the added mass and damping co- efficients, aij and bit are therefore functions of the forward speed and the frequency of oscillation w. The restoring coefficients ci`, on the other hand, include the classical hy- drostatic contribution augmented by a dynamic term due to the gradients of the double-body flow. The latter con- tribution depends linearly upon the deflection of the ship surface from its mean position and quadratically on the ship speed. It is therefore expected to be substantial at high Froude numbers. The equations governing the time-harmonic responses of the ship follow from Newton's law. Using the definitions (3.5) of the forces acting on the hull, the familiar six-degree of freedom system of equations is obtained 6 ~ [-w2(mij +aij)+i~bis +cij] fj =Xi, i= 1, ,6' j=1 (3.6) where mij is the ship inertia matrix, fj the complex ampli- tudes of the oscillatory ship displacements, and the restor- ing coefficients cij are modified to include the moments in pitch and roll due to the corresponding displacement of the center of gravity. 4. THE: INTEGRAL FORMULATION Green's second identity is applied for the unknown poten- tials, A, ~ or As j = 1,...,7, using the Rankine source potential, G(x; x') = 2 ~ ~ -A . (4.1) as the Green function. The fluid domain is bounded by the hull surface (B), the free surface (FS) and a cylindri- cal 'control' surface (SOO). The resulting integral equation takes the form 24

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+(X) - || ant, ~G(X;x')dx'+ll (X') ~ i, Did (FS) (FS)U (B ) || +~( , ~ G(X; X')dx' , (B) x ~ (FS) U (B) (4.2) where ~ stands for any of the potentials it, +, As, j = 1,...,7, introduced in the preceding sections. The surface integrals over the control surface (SO) can be shown to vanish in the limit as (SOO ) is removed to infinity with kept finite. The derivatives of A, ' and Hi normal to the ship surface (B) are known. The corresponding vertical derivative on the free surface (FS) is replaced by the appropriate com- bination of the value and tangential convective derivatives, according to the corresponding free surface condition. Of particular interest is the treatment of the integral over the ship hull which accounts for the m-terms in the bound- ary condition (2.14). This is of the form: i~mj G(xjx') do' (B) , j=1, ,6 . (4.3) The evaluation of the m-terms in (4.3) requires the com- putation of second order derivatives of the double-body po- tential ~ on the ship hull. When it comes to the evaluation of gradients of the solution potential, low-order panel meth- ods are known to be sensitive to discretization error, unless their implementation and panel distribution is carefully se- lected. The evaluation of double gradients of the solution are known to introduce serious difficulties, as illustrated by Nestegard (1984) and Zhao and Faltinsen (1989). Here, an alternative expression for the evaluation of the in- tegral (4.3) is derived by an application of Stokes' theorem. Given that the basis flow ~ satisfies a zero flux condition on the ship hull and the x = 0 plane, it follows that, for j = 1,,6, i/ mj G(X; X ) do = - // IV~(X ) V~iG(X; X )] ni do (B) (B) (4.4) The right-hand side of (4.4) involves only first derivatives of ~ on the hull, consequently it is clearly superior from the computational standpoint. The integral equation (4.2) will not accept unique solutions unless a radiation condition is imposed enforcing no waves upstream. In practice the solution domain of (4.2) on the z-0 plane will be truncated at a rectangular boundary located at some distance from the ship where appropriate 'end conditions' will be imposed enforcing the radiation condition. Due to the convective nature of the flow, the condition at the upstream boundary is the most critical and takes the form (in - Ups ) ~ = (in - U,~8l ) ~ = o, (4.5) where ~ stands for either the steady or the unsteady wave disturbance. The origin and physical interpretation of these two upstream conditions are discussed in detail in Sclavouno and Nakos (1988) for a tw~dimensional steady flow, and are extended to time-harmonic flows in Nakos (1990). It is shown that both are necessary in order to ensure physically meaningful numerical solutions of the steady and unsteady problems. For ~ = wU/g > 1/4 no wave disturbance is present upstream of the ship and the conditions (4.5) can be shown to enforce this property of the flow. For ~ < 1/4 and with increasing E,roude numbers, the amplitude of the waves upstream of the ship decreases relative to that of the trailing wave pattern and conditions (4.5) perform well if the truncation boundary is sufficiently removed from the ship. No conditions are necessary on the transverse and downstream truncation boundaries. 5. THE NUMERICAL SOLUTION ALGORITHM The solution of integral equation (4.2) for the steady and unsteady flows is obtained using a Panel Method. The sys- tematic methodology for the study of the numerical proper- ties of Rankine Panel Methods for free surface flows devel- oped in Sclavounos and Nakos (1988) led to the design of a bi-quadratic spline-collocation scheme of cubic order, zero numerical dissipation and capable to enforce accurately the radiation condition (4.5~. The boundary domain - including the ship hull and the free surface solution domain - is discretized by a collection of plane quadrilateral panels See Figure 1~. The unknown velocity potential is approximated by the linear superposi- tion of bi-quadratic spline basis functions Bid, as follows ~ ~ ~ aj Bj(~, (5.1) where Bj is the basis function centered at the j'th panel and at is the corresponding spline coefficient. By collocat- ing the integral equation (4.2) at the panel centroids and enforcing the upstream condition (4.5), the discrete for- mutation follows in the form of a system of simultaneous linear equations for the coefficients as. The relation (5.1) provides a C1-continuous representation of the velocity pm tential and may be differentiated to give the velocity field on the domain boundaries. The free surface elevation and hydrodynamic pressure are evaluated using the relations (2.8-9) and (3.1-2), respectively. The error and stability analysis of the bi-quadratic spline scheme is presented in Nakos and Sclavounos (1990~. It is based on the introduction of a discrete dispersion relation governing the wave propagation over the discretized free surface. Comparison of the continuous and discrete dis- persion relations allows the rational definition of the con- sistency, order and stability properties of the numerical solution scheme. It is shown that the numerical dispersion is of O(h3) where h is the typical panel size and that no numerical dissipation is present. Both are valuable prop- erties for the computation of ship wave patterns which are not substantially distorted, damped or amplified by the numerical algorithm. Essential for the performance of the method is a stability condition restricting the choice of the grid Froude number Fh = U/~ relative the panel aspect ratio, c' = h=/hy, where hr~hy are the panel dimensions in the streamwise 25

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and transverse directions respectively. This condition, 6. STEADY AND UNSTEADY Stile WAVE derived and discussed in detail in Nakos and Sclavounos (1990), establishes 'stable' domains on the (Fh,~x) plane with boundaries dependent on the frequency of oscillation. For a given a Froude number, a stable discretization for the highest frequency of oscillation is stable for all lowest fre quencies. Therefore, no regridding of the ship hull and free surface is necessary for the solution of the time harmonic problem over a range of frequencies. The resulting complex linear system is solved by an accelerated block Gauss-Siedel iterative scheme which makes extensive use of out-of-core storage therefore permitting the use of discretizations with several thousand panels. Experimental verification of the convergence of the solu tion algorithm has been established by comparing com putations of 'elementary' flows around singularities and thin-struts with analytical solutions iNakos and Sclavounos (1990) and Nakos (1990~. The convergence of the hydro dynamic added-mass and damping coefficients is discussed in Section 7. PATTERNS The forward-speed ship wave problems formulated in Sec- tion 2 have been solved for two hull forms using the nu- merical algorithm outlined in the preceding section. This section presents converged computations of the steady and time harmonic wave patterns around a Wigley and a Series- 60 hull. The Wigley model has parabolic sections and waterlines, a length-to-beam ratio L/B = 10 and beam-to-draft ratio B/T = 1.6. The grid used for the solution of the steady problem consists of 40xlO panels on half the hull, providing adequate resolution of the geometry, while the panels on the free surface are aligned with those on the hull and have a typical aspect ratio is c' = h~/hy = i. The grid Froude number is Fh~6.3 F. allowing an adequate resolution of the steady wave flow for Froude numbers as low as F = 0.20 isee Nakos (1990~. The free surface domain is truncated at a distance cup = 0.2L upstream of the bow and one ship length downstream of the stern. The truncation in the transverse direction is selected at You' =0.75L, so that the entire wave sector is included in the computational domain. The total number of panels in the grid is 2020. Figure 1: Discretization of the free surface and the hull for a modified Wigley model, using 1110 panels on half the configuration. 26

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Figure 2 shows contour plots of the wave patterns resulting from the steady forward translation of the Wigley model at F = 0.25,0.35,0.40 . Predictions based on both the Neumann-Kelvin and the double-body linearizations are presented. Due to the slenderness of this Wigley model, the two wave fields agree well even at high speeds. Small differences are visible along the diverging portion of the wave system which originates from the stern, where the Neumann-Kelvin solution tends to generate steeper waves, particularly along the caustic. The opposite appears to be true in the 'bow wave system'. For all Froude numbers, the calculated wavelengths are not affected significantly by -n 's _~ 75 -1 the selected linearization. The second ship tested is the Series-60-Cb = 0.6 hull which is significantly fuller than the Wigley model, with length- to-beam and beam-to-draft ratios L/B = 7.5 and B/T = 2.5, respectively. The principal characteristics of the grid used for the computations are the same to those employed for the Wigley model. Figure 3 illustrates the wave patterns around the Series- 60 model for F = 0.20,0.25,0.35, respectively. At low speeds (F < 0.30) the amplitude of the generated waves are comparable - if not smaller - than the ones computed I\~] 1~1 1 ~ I ~ I ~ I I I I I 1 ~ I I y I Double~Bod, J - tleumann-Kel~rin ~/~ {/''~J I 1 1 /l I /l I I I ~ I I I .50 - 1.00 - 0.50 0.00 ~ I ~ I I I I ~ 1 1 1 1 1 1 1 ) I 1 _ _ ~_ F = 0.35 F = ~ 25 l~t~ble~Bod' -0 75 J -1 TO -1 .00 - 0.50 0.00 0.50 O.SO o.oo -0.25 -0. 75 -0 75 ~ = 0.4C (/~/~//~< ~ J Neumann-Kelvin ~ - 1.50 - 1.00 -0.50 0.00 O.SO I 0.00 -0.25 -0 75 Figure 2: Contour plots of the steady wave patterns due to the parabolic Wigley model advance ig at Froude numbers F = 0.25, 0.35, 0.40. 27

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around the Wigley model, despite the increase in the 'full- ness' of the hull shape. For the Wigley model the bow- and stern-wave systems are well formed while the correspond- ing wave pattern around the Series-60 hull appears to be more 'confused'. Differences between the steady wave pattern computations from the Neumann-Kelvin and double-body linearizations are here clearly noticeable. Again, significant discrepan- cies occur along the diverging portion of the stern-wave system, where the Neumann-Kelvin solution shows larger amplitudes and shorter wavelengths. Moreover, the caustic 0.00 ~ Do -0.25 -0.7s lines originating from the bow and stern appear at a larger angle in the solution based on the double-body lineariza- tion. The differences between the two solutions become more pronounced as the speed increases, resulting in quite different wave patterns at F=0.35 (see Figure 3c). Figure 4 is a snapshot of the time-harmonic wave pattern around a modified Wigley model translating at F = 0.2 and oscillating in heave at frequencies we = 3 and 7=5. The grid used for this flow field has the same density as that in Figure 1. Both frequencies are over- critical (r = wU/g > 0.25), thus two wave systems appear j I I I I 1 1 1 1/ 1 _ Double-Bod)r / ~ Neumann Kelvin ~ -1.50 - 1.00 - 0.50 0.00 0.50 ~ <~1\\t 1 \1 I\ I 141 1 1 1 1 1 1 1 ~ I ~ Double~Bod, / - _ 0-50 0.00 -0.25 -0.75 . )/,/, ], 1, J. ,1 , , , , , , ,), ~ , -1.50 - 1 .00 - 0.50 0.00 0.50 I I I I I I I ~ I I _ Double-Body ~7 _ F = 0.35 -0 75 -0.75 - - 1.50 - 1 too -0.50 0.so 0.00 Figure 3: Contour plots of the steady wave patterns due to the Series-60-cb =0.6 Yesse1 advancing at Froude numbers F = 0.20, 0.25, 0.35. 28

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downstream. At F = 0.3, the time-harmonic wave fields around the modified Wigley model are illustrated in Fig- ure 5 and are obtained from the same grid as for F = 0.2. For this larger Eroude number, the wavelengths appearing in Figure 5 are larger than their counterparts of Figure 4, although the general structure of the wave field is similar. Figure 6 illustrates the wave patterns around the Series- 6~C~ = 0.7 hull advancing at F = 0.2 and heaving at fre- quencies /=3 and =4. Relative to the cor- responding patterns generated by the Wigley hull, the di- verging wave system originating from the stern is more pro- nounced and is attributed to the more three-dimensional shape of the Series-60 geometry. In all cases the steady wave pattern has been removed. Certain common features of these thre~dimensional time harmonic wave patterns are worth emphasizing. The short- est wavelength scales are associated with the transverse wave system which appears downstream of the stern and propagates in the streamwise direction. Along the ship length, on the other hand, the wave field is dominated by relatively long divergent waves which propagate in the transverse direction and tend to be become more two di- mensional as the frequency increases. This character of the time harmonic wave pattern therefore appears to sum port the 8,,e~,,~,~C I-, ~-nder~h~rlY thPr?rv Near +lle ship hull the wave disturbance is convected primarily in the transverse direction and becomes more focused as the free quency increases. Its variation in the lengthwise direction is gradual since cancellation effects appear to significantly reduce the amplitude of the short transverse waves which are clearly visible downstream of the stern. Figure d: Snapshots of the time-harmonic wave patterns due to a modified Wigley model ad- vaDcing at F=0.20 while oscillating in heave at frequencies wp7ij=3.0,5.0. 29

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7. HYDRODYNAMIC FORCES AND MOTIONS IN HEAD WAVES The unsteady hydrodynamic pressure on the hull is eval- uated from expression (3.2~. The restoring component of the pressure which depends on the ship displacement and the gradients of the steady flow has been neglected since it been found to be small for the ship hulls and Froude numbers considered below. The gradients of the steady and time-harmonic potentials are obtained from the formal differentiation of the spline representation of the velocity potential (5.1~. Integration of the pressure over the hull according to expressions (3.5), allows the determination of the added-mass, damping coefficients and exciting forces from expressions (3.5), and Response Amplitude Opera- tors from the solution of the linear system (3.6~. Only the coupled heave and pitch modes of motion in head waves are considered in this paper. In order to establish the convergence of the solution algm rithm, a systematic study of the effect of grid density on the computations of the hydrodynamic coefficients was car- ried out for a modified Wigley model with L/B = 10 and B/T = 1.6. The tim~harmonic wave flow was solved at a E`roude number F = 0.3 for several frequencies of oscillation in the range of practical interest w~ ~ [2.5, 5.0] . The free surface domain was truncated at a distance 0.25L upstream of the bow, 0.5L downstream of the stern and L in the transverse direction. Four different grids were considered, resulting in a systematic increase of the dim cretization density on both the free surface and the hull. These grids use 20, 30, 40 and 50 panels along the length of the hull, respectively, while for all of them the aspect ratio of the free surface panels is equal to 1. Computations of the heave and pitch added-mass and damp ing coefficients obtained from these grids, are illustrated in Figure 7. The convergence rate is very satisfactory and Figure 5: Snapshots of the time-harmonic wave patterns due to a modified Wigley model ad- vsacing at F-0.30 while oscillating in heave at frequencies 7= 3.0, 5.0. 30

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appears not to depend strongly on the frequency. Having established the convergence of the numerical algo- rithm, the hydrodynamic coefficients and ship motions are next compared to experimental measurements and strip theory. A systematic set of experiments for a modified Wigley hull were recently conducted by Gerritsma(1986~. The diagonal heave and pitch added-mass and damping coefficients at F = 0.3 are illustrated in Figure 8. The experimental measurements are compared to strip theory and the present method. The solid line, hereafter denoting results from SWAN (ShipWaveANalysis), is based on the double-body free~surface condition (2.9) and the complete treatment of the m-terms. The Neumman-Kelvin curve is obtained from the solution of the linearized problem using the present Rankine panel method and is obtained by ap- proximating the steady flow by the uniform stream -Up both in the free-surface and body boundary conditions. The agreement between SWAN and experiments is quite satisfactory and represents an improvement over strip the- ory. For the diagonal coefficients, SWAN and the Neumman Kelvin problem are in good qualitative and quantitative agreement. Significant differences between the three theoretical pre- dictions occur in the heave and pitch cross coupling co- efficients illustrated in Figure 9. These coefficients are known to be sensitive to end-effects, therefore their ac- curate prediction requires the complete treatment of the m-terms which attain large values near the ship ends. This is confirmed by the very good agreement between SWAN and the experimental measurements. In spite of its three-dimensional character, the departure of the Neumman Kelvin solution from the experiments is mainly attributed to the incomplete treatment of the m-terms. Figure 6: Snapshots of the time-harmonic wave patterns due to the Serie~60 cb = 0.7 crewel sd~rancing at F=0.20 while oscillating in heave at frequencies w~/~7;-3.0,4.0. 31

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AID A TIDE O o o to I|D I~ D D - Cretan (A) x D - cretizabon (B) a ~ D~cretizatioI1 (C) \ ~ Discretization (D) ~;m MID D~cretmation (a) r. Discretization (B) Discretization (C) o D~cret~ation (D) i'_ Moo s. - ..oo i: 4|D O I~ ~ o ~ D O 5.00 ~ ~ o . . to . . 2.00 3.00 ~ 00 w47~; 5.00 6.00 Figure 7: Numerical convergence study for the heave and pitch hydrodynamic coefficients of a modified Wigley model advancing at F = 0.3. Of interest is also the observed symmetry of the experimen- tal measurements and the SWAN predictions of the cross coupling coefficients. The modified Wigley hull is sym- metric fore and aft and a generalization of the Timman- Newman symmetry relations appears to hold. The origi- nal Timman-Newman relations were shown to be exact for submerged vessels and the Neumman-Kelvin free-surface condition. It is here conjectured that they are also exactly valid for surface piercing vessels when the free-surface con- dition is based on the double-body flow. No proof has yet been attempted using the condition (2.9~. 32 Figure 10 compares experimental measurements with the strip-theory and SWAN and predictions for the heave and pitch exciting-force and motion modulus and phase. The pitch radius of gyration of the modified Wigley hull is k', = 0.25L, and the center of gravity is taken at x = y = z = 0. The agreement of SWAN with the experiments is in all cases very satisfactory. The stri~theory predictions have been obtained from the MIT 5-D Ship Motion program which is regarded a standard strip-theory code. The dim crepancy between the strip-theory and experimental heave and pitch resonant frequencies, is attributed to the poor prediction of the b55 and the cros~coupling coefficients by strip theory (Figures 8 and 9~.

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a 0 W ~' - N "ID I~ D lo 2.00 3.00 4.00 5.00 B.00 0 Experimenb _._ Strip Theory SWAN Neumann Kelvin _; lo 0 Experimen" -Strip Theory SWAT "~ N.`lmann-Kelvin it. - \, o ~ |D O no 0 . lo 0 . . 02.00 ~ Do ~ Do w\~7; - 5.00 6.00 Figure 8: Diagonal hydrodynamic coefficients in heave and pitch for a modified Wigley model advancing at Eroude number F=0.3. Figures 11 and 12 compare experiments with the strip the- ory and SWAN predictions of the heave and pitch added- mass and damping coefficients of the Series = 0.7 model, advancing at E`roude number F = 0.2. The experi- mental data are due to Gerritema, Beukelman and Gland dorp (1974~. The performance of SWAN is in all cases very satisfactory, offereing a significant improvement over strip theory. Due to the fore-aft asymmetry of the Series-60 model, the Timman-Newman relations for the cross coupling coeffi- cients do not hold. It is interesting, however, to notice that the curares corresponding to a35 and bs5 are very close 33 to being mirror images of the those corresponding to a63 and a53, respectively about a non-zero value. In strip the- ory, for example, it may be shown easily that a38-a53 and b35 - b53 are symmetric about the corresponding cm efficients at zero forward speed (F=O), but no such proof is yet available in three dimensions. The Series 60 heave and pitch motion amplitude and phase are shown in Figure 13. The agreement between theory and experiments is again satisfactory for both strip-theory and SWAN, with a slight detuning of the strip-theory predic- tions again attributed to its discrepancies with experiments in the cross coupling coefficients and b55.

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!|D |D f,>., b 0 Experiment _. Strip Theory - SWAN ~Neumann-KelYin . o ~ , ~ 10~ - I trio o 0 Experiments --Strip Theory SWAN Neumann Kelvin cat _ o '2.00 3.00 `.00 a _ s.oo 6.00 2.0 3.0 To w~77 5.0 6.0 Figure 9: Cross coupling hydrodynamic coefficients between heave and pitch for a modified Wigley model advancing at Froude number F=0.3. 8. CONCLUSIONS AND FUTURE WORK A new three-dimensional Rankine Panel Method method, referred to as SWAN, has been developed for the solution of the complete three-dimensional steady and time-harmonic ship-motion problem. Its principal attributes are: The use of a new free-surface condition based on the doubl~body flow and valid uniformly from low to high Froude numbers. The complete and accurate treatment of the m-terms. A high-order non-dissipative numerical algorithm for the enforcement of the free-surface and radiation conditions. 34

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.1 a l ~ o ID o Experiments -~-Strip Theory SWAN j/ ,s t.o ~ _ 1 I A/L 1: 0 Experiments _. Stup Theory SWAN 1? // 1.~ . '~ . 2.0 2.5 ab.s t.0 t.5 2.0 2.5 A/L Figure 10: Heave and pitch exciting forces and motions of a modified Wigley model advancing at Froude number F=0.3 through regular head waves. 35

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LID ~ - o o 1 lo ID a lo :|D o lo a\ Experiments Stup Theory SWAN .` 0 ~ o ~.oo 3.004.00 w\~7; .. 1 . ~. - . MID lo o o 1 ~1 ~ 1|D ~ 5 00 6 00 ~ 00 \ ~ Experiments _._ Stup Theory ~ . SWAN it. it. : of o HI - 1 - ~ \. ~\ O - .\ \ ~\a 3 00 4 00 fit s. go 6. go Figure 11: Diagonal hydrodynamic coefficients in heave and pitch for the Serie~6~c.=0.7 vessel advancing at Froude number F=0.2. Computations of steady and time-harmonic ship wave pat- terns illustrate the capability of the method to resolve con- siderable detail in the wave disturbance and at a significant downstream of the ship. Predictions of the heave and pitch added-mass, damping coefficients, exciting forces and motions of a Wigley and the Serie~60 hull are found to be in very good agreement with experiments and present a significant improvement over strip theory. A complete treatment of the m-terms has been developed and found to be essential for the accu- rate prediction of the cros~coupling coefficients and ship motions. In summary, all important features of the three-dimensional time-harmonic flow around the ship appear to be well pre- dicted by the present method. This will permit the accu- rate prediction of the hydrodynamic pressure distribution, wave loads, derived responses and added-resistance by di- rect use of the velocity potential and its gradients on the ship hull and the free surface. 36

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||D - o ~ |D 0 o o . - I _. . at\ '2.00 3.00 ~ ~ |D ~ .0 ID ~ 0 . 4.00 5.oo 6.00 'ho w~7i 0 Experiments Strip Theory SWAN e=; s.o no w~7; To 6.0 Figure 12: Cross-coupling hydrodynamic coefficients between heave and pitch for the Series-60- Cb=0.7 newel advancing at Froude number F=0.2. Future research towards the further development of the present rankine panel method in the steady problem, will concentrate upon the determination of the ship wave spec- trum from the available numerical data over the discretized portion of the free surface. This information is useful for the characterization of ships from their Kelvin wake and the accurate and robust evaluation of the wave resin lance. The proper implementation of the present numeri- cal scheme to hull forms with significant flare will also be studied in both the steady and time-harmonic problems. 37 The application is also planned of the same method to the prediction of the seakeeping properties of unconventional ship forms (e.g. SWATH ships and SES's) the hydrody- namic analysis of which is particularly amenable by the present three-dimensional panel method.

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too - to to 0 1, ~ to ~ o to i] o I',~ .: -I ~ Experiments -._ Strip Theory SWAN 11 / 11 1.~.-~= ., l In l ), . . o ~.5 1~ _. .0 1.5 A/L 2.0 2.5 Figure 13: Heave and pitch motions of the Series-60-cb =0.7 vessel advancing at Froude number F = 0~2 through regular head waves. 38

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9. ACKNOWLEDGEMENTS This research has been supported by the Applied Hydrome- chanics Research Program administered by the Office of Natural Research and the David Taylor Research Center (Con- tract: N00167-8~K-0010) and by A. S. Veritas Research of Norway. The majority of the computations reported in this paper were carried out on the National Science Founda- tion Pittsburgh YMP Cray under the Grant OCE880003P. This award is greatly appreciated. We are also indebted to the Computer Aided Design Laboratory of the Depart- ment of Ocean Engineering at MIT for their assistance in the preparation of the time-harmonic ship wave patterns on their IRIS Workstation. REFERENCES - Chang, M.-S., 1977, 'Computations of three-dimensional ship motions with forward speed', 2nd International Con- ference on Numerical Ship Hydrodynamics, USA. Dawson, C. W., 1977, 'A practical computer method for solving ship-wave problems', 2nd International Conference on Numerical Ship Hydrodynamics, USA. Eggers, K., 1981, 'Non-Kelvin Dispersive Waves around Non-Slender Ships', Schif3stechnik, Bd. 28. Faltinsen, O., 1971, 'Wave Forces on a Restrained Ship in Head-Sea Waves', Ph.D. Thesis, University of Michigan, USA. Gadd, G. E., 1976, ' A method of computing the flow and surface wave pattern around full forms', Mans. Roy. Asst. Nav. Archit., Vol. 113, pg. 207. Gerritsma, J., 1986, 'Measurments of Hydrodynamic Forces and Motions for a modified Wigley Model', (unpublished). Gerritsma, J., Beukelman, W., and Glansdorp, C. C., 1974, 'The effects of beam on the hydrodynamic characteristics of ship hulls', 10th Symposium on Naval Hydrodynamics, USA. Guevel, P., and Bougis, J., 1982, 'Ship Motions with For- ward Speed in Infinite Depth', International Shipbuilding Progress, No. 29, pp. 103-117. Inglis, R. B., and Price, W. G., 1981, 'A Three-Dimensional Ship Motion Theory - Comparison between Theoretical Predictions and Experimental Data of Hydrodynamic Co- efficients with Forward Speed', Transactions of the Royal Institution on Naval A Tchitects, Vol.124, pp. 141-157. King, B. K., Beck, R. F., and Magee, A. R., 1988, 'Seakeep- ing Calculations with Forward Speed Using Time-Domain Analysis', 17th Symposium on Naval Hydrodynamics, The Netherlands. Korvin-Kroukovsky, B. V., 1955, 'Investigation of ship mo- tions in regular waves', Soc. Nav. Archit. Mar. Eng., Trans. 6S, pp. 386-435. Maruo, H., and Sasaki, N., 1974, 'On the Wave Pressure Acting on the Surface of an Elongated Body Fixed in Head Seas', Journal of the Society of Naval Architects of Japan, Vol. 136, pp. 3~42. Nakos, D. E., 1990, 'Ship Wave Patterns and Motions by a Three-Dimensional Rankine Panel Method', Ph.D. Thesis, Mass. Inst. of Technology, USA. Nakos, D. E., and Sclavounos, P. D., 1990, 'Steady and Un- steady Ship Wave Patterns', Journal of Fluid Mechanics, Vol 215, pp. 265-288. Nestegard, A., 1984, 'End effects in the forward speed ra- diation problem for ships', Ph.D. Thesis, Mass. Inst. of Technology, USA. Newman, J. N., 1978, 'The theory of ship motions', Ad- vances in Applied Mechanics, Vol. 18, pp. 221-283. Newman, J. N., and Sclavounos, P. D., 1980, 'The Uni- fied Theory of Ship Motions', 13th Symposium on Naval Hydrodynamics, Japan. O'Dea, J. F., and Jones, H. D., 1983, 'Absolute and relative motion measurments on a model of a high-speed contain- ership', Proceedings of the 20th ATTC, USA. Ogilvie, T. F., and Tuck, E. O., 1969, 'A rational Strip Theory for Ship Motions- Part 1', Report No. 013, Dept. of Naval Architecture and Marine Engineering, Univ. of Michigan, USA. Piers, W. J., 1983, 'Discretization schemes for the mod- elling of water surface effects in first-order panel methods for hydrodynamic applications', NLR report TR-83-093L, The Netherlands. Salvesen, N., Tuck, E. O., and Faltinsen, O., 1970, 'Ship motions and wave loads', Soc. Nav. Archit. Mar. Eng., Trails 78, pp. 25~287. Sclavounos, P. D., 1984a, 'The Diffraction of Free-Surface Waves by a Slender Ship', Journal of Ship Research, Vol. 28, No. 1, pp. 29-47. Sclavounos, P. D., 1984b, 'The unified slender-body the- ory: Ship motions in waves' 15th Symposium on Naval Hydrodynamics, Germany. Sclavounos, P. D., and Nakos, D. E., 1988, 'Stability anal- ysis of panel methods for free surface flows with forward speed', 17th Symposium on Naval Hydrodynamics, The Netherlands. Timman, R., and Newman, J. N., 1962, 'The coupled damp- ing coefficients of symmetric ships', Journal of Sh* Re- search, Vol. 5, No. 4, pp. 34-55. Yeung, R. W., and Kim, S. H., 1984, 'A New Development in the Theory of Oscillating and Translating Slender Ships', 15th Symposium on Naval Hydrodynamics, Germany. Zhao, R., and Faltinsen, O., 1989, 'A discussion of the m-terms in the wave-current-body interaction problem', 3rd International Workshop on Water Waves and Float- ing Bodies, Norway. 39 1

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DISCUSSION William R. McCreight David Taylor Research Center, USA Your predictions of added mass and damping for the Series 60 hull are better than those for the Wigley hull, yet the motion predictions are not as good. Could you describe the accuracy on the Series 60 exciting-force computations, which are not shown. If this does not account for the discrepancy, what do you believe is the cause of this? AUTHORS' REPLY In response to Dr. McCreight's question we want to state that the calculation of the heave/pitch exciting forces typically compare very well with corresponding experimental data. Discrepancies between the numerical and experimental results for the motions of the Series-60 may be partly attributed to the speed dependent portion of the restoring force, which was not included in the presented calculations. Additional differences may also arise due to ambiguities about the appropriate values for the pitch moment of inertia and the vertical position of the center of gravity, as well as about the location of the point about which the heave/pitch motions are referenced. DISCUSSION Hoyte Raven Maritime Research Institute Netherlands, The Netherlands This paper is very interesting for me, in particular, as it addresses some points studied in my paper. I have a question on the steady wave resistance. You found differences in the remote wave pattern between the Kelvin and the show-ship condition. These may, however, be due to subtle changes in interference between wave components. Did you find any substantial difference in wave resistance? Secondly, as you noticed your free surface condition is intermediate in form between those of Dawson and Eggers, 1979. I have implemented your FSC in our code to make the same comparisons as in my paper, and found that the result was also intermediate for the Series 60 CB=0.60 model: the predicted Rw is 6-8% lower than with Dawson's condition, while Eggers is 20% lower. For a full hull form, again the resistance is lower than Dawson, but better behaved than Egger's condition. Ref. Raven, H.C., Adequacy of Free Surface Conditions for the Wave Resistance Problem,. this volume. AUTHORS' REPLY We would like to thank Dr. Raven for implementing and testing the free surface condition proposed in this paper. The differences of the wave patterns, as predicted by different free surface linearization models are indeed reflected on the corresponding wave resistance calculations. We strongly believe, however, that ~numerical" evaluation of the relative performance of different linearization models is still clouded due to the delicate nature of the underlying calculations. The robustness of each scheme ought to be established individually before comparison arguments can be stated. We are currently working towards this direction by employing the conservation of momentum as the self-consistency criterion ([1]). [1] Nakos, D.E., 1991, "Transverse Wave Cut Analysis by a Rankine Panel Method, 6th Int. Workshop on Water Waves and Floating Bodies, Woods Hole, MA, USA. 40