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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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 863 On Roll Hydrodynamics of Cylinders Fitted with Bilge Keels R.W.Yeung1, D.Roddier1, B.Alessandrini2*, L.Gentaz2, and S.-W.Liao1 (1University of California at Berkeley, USA, 2Ecole Centrale de Nantes, France) ABSTRACT Bilge keels are commonly used appendages that are effective in reducing roll motion. The complex fluid flow around such sharp-fin objects in the presence of hull geometry and a free surface has hardly been addressed. Two recently developed Navier-Stokes Solvers are utilized to examine the forced-motion hydrodynamic properties of rectangular cylinders fitted with bilge keels. One solver is the Free-Surface Random Vortex Method (FSRVM, Yeung & Vaidhyanathan, 1994) developed at the University of California at Berkeley, while the other is a Boundary-Fitted Finite- Difference Method (BFFDM, Alessandrini & Delhommeau, 1995) developed at the Ecole Centrale de Nantes. Cross- continents collaborations provided the opportunity for the authors to evaluate the consistency of these two different but equally successful methods. Where available, these theoretical calculations are validated against laboratory measurements taken from forced-oscillation experiments. The capabilities of such first-principle approaches to predict the added roll inertia and the roll damping are shown to be excellent. The shortcomings of traditional estimates of roll hydrodynamic coefficients based on potential-flow calculations are illustrated. The dependence of these coefficients on the parametric space of frequency, bilge-keel depths, and amplitude of roll is examined and discussed. KEY WORDS: Bilge keels, roll motion, viscous damping, vorticity, experimental hydrodynamics, vortex methods, Navier-Stokes equations. 1 INTRODUCTION Predictions of the roll motion of ships have traditionally relied on the empirical estimates of relevant hydrodynamic coefficients (Himeno, 1981) since viscosity effects cannot be easily incorporated analytically. The presence of stabilizing appendages such as bilge keels have made the motion prediction even more difficult (Miller et al., 1974, Cox and Lloyd, 1977). Martin (1958) and Ridjanovic (1962) were among the first to obtain experimental data for a flat plate in oscillating flows. Even though the mitigation effect of bilge keels on ship motion has been known from the time of William Froude, who proposed the usage of âbilge piecesâ in 1865 and later measured its resistance, very little advances, based on fluid- mechanics first principles have taken place. The modeling of unsteady viscous forces by sharp-edge fins have been a difficult one. Amid the long history of the subject on roll motion, we will mention a few recent references which is not an exhaustive list. Robinson and Stoddart (1987) and Standing et al. (1992) discussed the effects of damping on a barge's roll response. Using a single vortex method, Faltinsen and Sortland (1987) investigated the eddy-making damping in slow- drift motions and showed the importance of bilge-keel depth, especially at low KC (Keulegan-Carpenter) numbers. Cermelli (1995) observed a very distinctive asymmetric flow pattern associated with oscillating a flat plate near a free surface. He meticulously investigated the vortical structure of the flow around the tip of the plate using DPIV (Digital Particle Image Velocimetry) and compared them with the FSRVM (Free-surface Random Vortex Method) the authoritative version for attribution. *Visiting Scholar, University of California at Berkeley

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 864 of Yeung and Vaidhyanathan (1994). Sarpkaya and O'Keefe (1996) provided related, and similarly interesting features of the oscillating flows about bilge keels. Roll damping of ships is not strictly related only to flow separation around the bilges and keels. Of equal importance is that bilge keels, if present, also generate lift like a low-aspect wing because of the forward speed of the hull. It is primarily the first aspect that we address in this paper, even though the latter aspect could be effectively modeled by the first without a full-fledge three-dimensional solution. An extensive amount of literature exists in motion predictions by inviscid-fluid theories. It is generally assumed that inviscid-fluid ship-motion theory (Wehausen, 1971) can be largely improved by including viscous effects as ad-hoc hydrodynamic coefficients. On the other hand there are good reasons to believe that viscous effects could be fully coupled to the inviscid-fluid motion. The nature of this inviscid and viscous coupling was first examined by (Yeung and Wu, 1991) using a sophisticated Green function, followed by Yeung and Ananthakrishnan (1992), and Gentaz et al. (1997), using boundary-fitted coordinates methods. A complete understanding of this subject is yet to be achieved. Similar to the work of Fink and Soh (1974), Braathen (1987) used a vortex tracking method to predict the roll damping of a two- dimensional body. Korpus and Falzarano (1997) studied the viscous damping caused by bilge keels using a finite analytic technique to discretize the unsteady Reynolds-averaged Navier-Stokes equations. They used a -Îµ turbulence model, but did not consider the effect of the free-surface. Yeung et al. (1996) examined the flow about a rolling plate using FSRVM and validated their results against their DPIV experiments. Very recently, Yeung et al. (1998) applied the FSRVM to a rectangular ship-like section oscillating in roll motion. These results were substantiated with new experimental data, which revealed some discrepancies with the classical measurements reported by Vugts (1968), namely, in the diagonal terms of the added moment of inertia matrix. In this paper, bilge keels of various depths are fitted to the corners of a rectangular cylinder (See Fig. 1). The effects of viscosity, vortex shedding, and the associated fluid inertia and damping will be investigated systematically. Numerical results based on two rational methods of very different formulations: the FSRVM and a boundary-fitted finite-difference method (BFFDM), are compared with new experimental measurements. The validity of solving the roll hydrodynamics problem from first principles can thus be ascertained. 2 THEORETICAL MODELS 2.1 The Free Surface Random Vortex Method The Free-Surface Random-Vortex Method (FSRVM), as introduced first by Yeung and Vaidhyanathan (1994), has been successfully validated for the relatively simple geometry of a plate and for a rectangular cylinder (Yeung and Cermelli, 1998, Yeung et al., 1998). The theoretical formulation allows for arbitrary body shapes. It is therefore a relatively straight-forward extension to apply the same algorithm to the more complex problem of a rectangular cylinder fitted with keels around the bilge corners. As this method is relatively new, a brief exposition of the is given here, while more details are available from Vaidhyanathan (1993) and the three references cited above. Figure 2 shows a schematic of a floating rectangular cylinder undergoing prescribed roll motions. In indicial notations, the Cartesian coordinates are designated by (x=x1, y=x2) with the instantaneous angular position of the cylinder â Db denoted by Î±(t), measured positive counterclockwise. The velocity field is given by (u1, u2). This grid-free method of FSRVM solves the flow field by decomposing it into irrotational and vortical parts. The irrotational part is solved using a complex-valued boundary-integral method, utilizing Cauchy's integral theorem for a region bounded by the body, the free- surface, and the open boundary. The rotational part is described by the vorticity equation for Î¶3, where is the vorticity in the direction normal to the two-dimensional space, and Îµjkl is the alternating tensor. The vorticity equation is solved using a fractional step method. In two-dimensional flow, the governing equations for an incompressible, viscous fluid based on a vor the authoritative version for attribution.

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 865 ticity and stream-function formulation are given by: Figure 1: Geometry of cylinders with bilge keels. Figure 2: Computational domain and definitions for FSRVM. (1) (2) where Dt is the material derivative and v the kinematic viscosity. The velocity components of the flow are given in terms of Ï by (u1, u2)=(Ï,2,âÏ,1). In the context of a RVM, the vorticity field is assumed to be a collection of discrete vortices of finite core size (Chorin, 1973), whose motion is governed successively by a convection step and a diffusion step. In the first âhalf step', the convection equation of Î¶3 is obtained by dropping the right-hand side of Eqn. (2) (3) Since vorticity lines are material lines in a two-dimensional flow and the vorticity field is assumed to be an aggregation of vortices, Eqn. (3) implies that the blobs are convected with the flow velocity at their locations. Thus the location of the ith vortex blob is given by (4) In the second âhalf step', to simulate solution of the diffusion equation, (5) every blob is given a ârandom walkâ with a Gaussian distribution of zero mean and standard deviation. The diffusion step computations are simple whereas the convection-step computations are time consuming. This latter the authoritative version for attribution. difficulty is overcome by an O(N) algorithm (Yeung et al., 1996). The diffusion process is assumed to be sufficiently local so that effects of free surface is negligible. To compute the stream function, we let Ï be written as the sum of a vortical part and a homogeneous part, (6) where (7) Ïv is known if the position and strength of each vortex blob is known. For instance, for N blobs, each of circulation Îi,

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 866 (8) for any point xj outside of the core of the blob. It can be seen that only Ïh needs to be solved, which must consider all of the conditions on the fluid boundary â D. A complex potential formulation similar to Grosenbaugh and Yeung (1989) (see also Vinje and Brevig, 1980) is used. Let the complex potential be defined by where is the (conjugate) velocity potential associated with Ïh and z=x1+ix2. An outline of the solution method for Î²h follows. 1. At any given time t, Cauchy's integral theorem is applied so that either or Ïh is solved on the fluid boundary when its conjugate part is specified: (9) On the body boundary, the no-leak condition can be shown to yield: (10) where is the angular velocity of roll. On the free surface â DF, the kinematic boundary condition for the complex velocity w= u1âiu2 is used to advance the location of the free surface, while the dynamic condition is used to advance (11) (12) where indicates complex conjugate. The damping function vd in Eqns. (11) and (12) is zero except in the damping layers âL<x1â¤ xl and L>x1â¥xr on the left and right ends of the free surface, and zo is the initial location of the lead free-surface node of the layers at t=0 (Israeli and Orszag, 1981, Cointe et al. 1991, Liao and Roddier, 1998). Because of the use of damping layers, the no-disturbance condition can be used on the open boundary âDÎ£, (13) 2. After Î²h is solved, the âno-slipâ boundary condition on âDb needs to be satisfied since Î²h generates a relative tangential velocity. Vortices are generated to satisfy the no-slip boundary condition. On ith body panel, the strength of the vortex is (14) where is the relative velocity between the body and fluid, Ïi is the tangential unit vector and si is the panel length. Then, the vortex is convected and diffused into the fluid domain according to Eqns. (4) and (5). 3. In order to obtain the forces and moment on the body, we need to solve for Î²h,t, since is needed in Euler's integral to obtain the surface pressure. An integral equation similar to Eqn. (9) can be set up in parallel with are given by Eqns. (12) and (13). On âDb, Î²h,t. The boundary conditions on â Df and âDo of (15) 4. The new boundary conditions are now available for the next step. Step 1 to step 3 can be repeated till the end of simulation. Note that in the absence of Ïv, the flow is entirely irrotational. Thus, a fully nonlinear inviscid solution can be recovered by FSRVM by shutting off the vorticity generation process, with few changes in the solution procedure. 2.2 The Boundary-Fitted Finite-Difference Method Boundary-fitted coordinate methods for free surface flows have been investigated by Shanks and Thompson (1977) for a hydrofoil, by Yeung and Ananthakrishnan (1992) for heaving two-dimensional bodies with elaborate grid-control algorithms of Steinberg and Roache (1986) that were based on variational principles. This present method based on the RANS equations was developed by Alessandrini and Delhommeau (1995), which has been successfully validated in both two and three dimensional geometry. The convective form of Reynolds Averaged Navier-Stokes Equations are written through partial transformation from Cartesian space (x1, x2) to the authoritative version for attribution.

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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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 868 free surface can be manipulated into the following discrete form in time: (22) The use of the discrete normal dynamic condition can lead to the following final form of free surface boundary conditions or velocities: (23) where the Mik and f are quantities defined in Alessandrini and Delhommeau (1995). The âtildeâ notation indicates that interpolation to the grid points from cell center is necessary. The pressure equation, in matrix form and utilizing the pseudo-velocity, is solved by using the Rhie and Chow method (1983) so as to avoid checkerboard oscillations. (24) where Î pv, Î pp, and Î¦p result from the left-hand and right-hand sides of Eqns. (22) and (23), respectively. This full set of equations is solved at each iteration in a coupled manner using the BI-CGSTAB algorithm of Van Der Vorst (1992), conditioned with an incomplete LU decomposition. Figure 3 illustrates the Ãâtype mesh fitted around the cylinder at one instant of time. 2.3 Forces and moment The hydrodynamic forces and moment acting on the body can be calculated according to (25) (26) (27) (28) In the case of the FSRVM, the only stress on the body is the pressure which is given by Euler's integral: (29) In Yeung and Ananthakrishnan (1992), the shear stress was found to be of secondary importance in strongly separated flow, the pressure is the dominant dynamic stress contributing to the total fluid force and moment. However, BFFDM can provide the shear stress as part of the solution procedure, and both components of the stress are used to compute the force and moment integrals (Eqns. 25â27). The hydrodynamic coefficients will be directly obtained from these time histories as explained further in Section 3.2. 3 EXPERIMENTAL SETUP AND HYDRODYNAMIC COEFFICIENTS The experiments were conducted at the Richmond Ship-Model Testing Facility of the University of California at Berkeley. A 2.54Ã0.3Ã0.3 m rectangular acrylic cylinder as shown in Fig. 4 was hinged at the water level by the sides of the tank. Bilge keels of 1.25 and 1.90 cm were mounted on the bilge corner of the cylinder at a 45Â° angle. The model weighted 100 kg and displaced 0.11 m3. The rolling motion was induced by a hydraulic cylinder that can accept a random- motion input (Random Motion Mechanism, Hodges and Webster, 1986). The cylinder was oscillated from 2.5 to 10 Hz. Further details on the apparatus are explained in Yeung et al. (1998). In âdry conditionsâ, the model was oscillated to obtain its mass moment of inertia (natural period=1.424 secs). The center of gravity was measured by an inclining experiment. Force transducers placed at the end of the piston rod and just above the hinges provide an analog signal to a computer. The transducers were calibrated statically. Measured forces and moment were obtained after filtering instrument noises and vibrations, 3.1 Measured forces and moment Figure 5 shows a free-body diagram of the system that allows one to analyse the applied forces. Let W be the weight of the cylinder, m the mass, I0 the mass moment of inertia about O, G the center of gravity at (xG, yG) with |OG|=rG and | OA|=rA. Let also Xw, Yw and Mw be the forces and moment exerted by the fluid on the cylinder, and Xâ²A and Yâ²A the forces exerted by the horizontal driving rod on the cylinder, The subscript o refers to quantities the authoritative version for attribution.

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 869 measured at the roller bearing O, and the Xâ² and Yâ² refers to the forces in Axâ²yâ² coordinate system as shown in Fig. 5. Force transducers are adjusted to zero and calibrated when the cylinder is installed on the hinges and in the upright position. Their outputs are zeroed when no motion is present even though there may be static loads. When the cylinder is in motion, the transducer respoonse are proportional to the additional forces present. Therefore, the total force applied to the cylinder at a hinge can be expressed as Figure 4: Experimental apparatus and setup. Figure 5: Free body diagram of the cylinder. (30) where F can be either X or Y and the second term âm' is proportional to the output voltage of the transducer. By making use of Eqn. (30) and noting that the F0's are the static terms computable from the geometric properties of the cylinder, we can arrive at the following equations for the hydrodynamic forces and moment with buoyancy effects removed: (31) (32) (33) where Î¸ is the angle about O between the center of gravity G and the vertical direction when the cylinder is at rest, i.e. Î¸=|atan(xG/yG)|. Wrod is the weight of the driving rod, â the submerged section area, with the last term of (33) representing buoyancy effects. We note that even though the measured force carries the dynamic information of the fluid inertia term, it can be shadowed by the hydrostatics terms unless the body weights cancels it. This can be done by making the cylinder having a zero metacentric height. By Fourier decomposing the measured time series, hydrodynamic coefficients are obtained as shown in Section 3.2. We note in passing that the roll motion of amplitude Î±o is started smoothly by using a hyperbolic-tangent ramped function of the form (34) the authoritative version for attribution. This is also the case in the numerical methods. 3.2 Hydrodynamic coefficients In linear, potential-flow theory, the hydrodynamic forces and moment are usually written as the sum of inertia and linear damping terms (Wehausen, 1971): (35) (36) where Xw and Mw are the hydrodynamic sway forces and roll moment as obtained from both the numerical and experimental simulations. Since the forced roll motion is of the form Î±(t)=Î±0 sin(Ït), the linear hydrodynamic coefficients at t=t0 can be calculated by extracting the Fourier coefficient of the

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 870 primary frequency over a period T. Hydrodynamic coefficients of the sway force induced by roll motion can be shown to be: (37) (38) while the diagonal terms are: (39) (40) Earlier, we have completed some experiments for a model without bilge keels Yeung et al. (1998) and compared some with the reported results of Vugts (1968). Here, for reason of consistency and ease of interpretations, we define the same normalized coefficients as before: (41) (42) Because the flow is often governed by viscous separation, a nonlinear formulation for the damping could also be expressed as: (43) where Î»â²66 is the quadratic damping coefficient and is a traditional way of representing the overall flow effects. However, if the process is assumed to be sinusoidal, equivalent energy dissipation rate would imply the following formula for Î»â²66: (44) Thus, under the stated assumption, Equation (44) is related to Eqn. (40) by (45) The damping coefficient presented in Section 4 will follow Eqn. (40) and the nonlinear coefficient can be deduced accordingly. However, if both linear and nonlinear damping are present, the uniqueness of each of these coefficients can be determined only with more information. A formulation for this case can be found in Yeung et al. (1998). 4 RESULTS AND DISCUSSIONS For a given cylinder geometry and with the assumption of prescribed periodic motion, the problem is characterized by the roll amplitude Î±o and the following nondimensional parameters: (46) where is the percentage of bilge-keel depth relative to the beam and Re is a characteristic Reynolds number similar to Yeung et al. (1996). In the FSRVM, the Reynolds number is taken the same as the physical one in the laboratory, which ranges from 16,000 to 144,000. Typically, the number of vortex blobs used is of O(50,000) though a larger number can be accommodated. In the BFFDM, the grid size is adapted to the medium Re to avoid remeshing and consists of 200Ã200 nodes. 4.1 Inviscid-fluid results Viscous-flow solutions are emerging because of the increasing power of computers. Inviscid results are still very useful in providing not only a guideline but also a validation for new numerical methods. The problem of a rectangular section in roll motion has long been solved by a variety of methods. Yeung (1975) used a Hybrid Integral-Equation Method, which is very powerful for the case of arbitrary geometry. Boundary-element methods based on various Green functions is also a possible way for solution. An advantage of the FSRVM is that it yields the inviscid (potential-flow) solution when the vortical part of the stream function corresponding to Eqn. (8) is not âturned on.â In Figs. 6 to 9, the inviscid-fluid hydrodynamic coefficients are presented. The solution for a rectangular cross section (with a small bilge radius, see Fig. 1) is included for reference purposes. These results can be considered as linear results, being independent the authoritative version for attribution. of the roll amplitude. In the actual FSRVM computations, Î±o is taken as 2.85 degrees. Even though the potential-flow problems solved were nonlinear, the inviscid calculations were independent of the amplitude of roll for angles less than 10 degrees. The FSRVM inviscid-fluid results for the âNo Keelsâ case are obtained using a rounded corner similar to the experimental model where the corner radius was 2% of the beam. Note that the FSRVM for a square-cylinder yielded the established

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 the authoritative version for attribution. solution. Figure 7: Added mass of roll into sway, inviscid-fluid Figure 6: Added moment of inertia, inviscid-fluid solution. ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS solution. fluid solution. Figure 8: Damping coefficient of roll, inviscid-fluid Figure 9: Damping coefficient of roll into sway, inviscid- 871

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 872 solution of Yeung (1975) abd was given in Yeung et al. (1998). As expected, the effect of the keels is to increase the added moment of inertia and the radiation damping considerably. The location of the peak values, around for the added inertia and for the damping, remain the same as the cylinder without keels. 4.2 Theory versus experimentsâ a validation In Figs. 10 and 11, typical time histories of the hydrodynamic moment at the condition of are shown for both numerical methods described in Sec. 2 and the experimental measurements. The lower figure presents the average values of all three set of time series over three or more periods from t/T=0.5 onwards. The agreement is seen to be very good between FSRVM and BFFDM, both solutions overpredicting slightly the experimental results. Because the phase difference is minimal between all three curves, we expect the added mass and damping coefficient predicted to be similar, as will be seen later. A more detailed comparison of the two methods of solution is shown in Fig. 12. The variation of the local pressure on the body contour is shown, in the fourth period, as functions of the arc-length parameter s/b measured in a counter- clockwise direction and with the zero reference point taken at the centerline. Each subfigure corresponds to an instant of time successively T/6 apart, with the cylinder rotating clockwise. The tips of the left and right bilge keels show up as s/b approximately equal to â1 and +1, respectively. Results from both numerical methods agree remarkably consistently, particularly in view of the drastic difference in formulations. Further, since the boundary condition on the free surface are treated differently, most of the disagreement is expected to occur at values of s/b near the waterline. The pressure difference between both faces of the keel lead to a jump in the pressure from the side to the bottom of the cylinder. The presence of the keel is therefore a major contributor to the hydrodynamic roll moment. It should be noted that during the second half-period which is not shown in Fig. 12, information indentical to those presented in the first half is expected since there is quasi-symmetry of the solution about the T/2 point. The pressure distribution will have a mirror image about the point s/b=0 to the first half-period. The time series for the sway force and roll moment in this study are very similar to the ones presented in Yeung et al. (1998), except for an increase in amplitude as the keel depth or the amplitude of motion increases. It is more useful, from the practical standpoint, to present the Fourier averaged values of the force and moment in the form of hydrodynamic coefficients as defined in Eqns. (37) to (40). In Figs. 13 to 16, the roll added moment of inertia and roll damping, as obtained experimentally and numerically using FSRVM, are presented for bilge keels whose depths are 4% and 6% of the beam of the cylinder, and for two amplitudes of roll angle. Results for the rounded-corners geometry can be found in Yeung et al. (1998). The diagonal terms of the coefficients Figs. 13â16 are grouped separately from the coupling terms, Figs. 17â20. Numerical results obtained using BFFDM are added to Figs. 13 and 15, when and Î±o=2.85 for comparison purposes. It is gratifying to see these two very different and independent methods yield very predictions close to each other. These predictions also tend to agree well, for the most part, with the experimental results. Generally speaking, FSRVM and BFFDM do agree better in the diagonal terms than the coupling coefficients. In the case of comparisons with experiments, one should keep in mind that the reliability of the experimental measurements decreases with an increase in frequency because of vibration of the test apparatus. Detail examination of this extensive set of data will suggest the following trends. â¢ An increase in keel depth (4% to 6%) would increase both the added inertia and damping for the entire range of frequency. This is expected intuitively and from the computations. The inertia measurements indicate otherwise even though the change is not seen as substantial. â¢ With bilge-keels size fixed, experiments and real-fluid theory suggest that an increase in roll amplitude lead to a decrease in the inertia coefficients for both diagonal and off-diagonal terms, or at least up to the largest angle of 5.75Â° investigated here. Larger roll amplitude yields an appreciably larger damping coefficients. â¢ The agreement between theory and experi the authoritative version for attribution.

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 the authoritative version for attribution. history, Figure 10: Moment of roll hydrodynamic moments. Figure 11: Steady-state averages Mw(t) ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS body solutions of two methods. for numerical Figure 12: Local pressure on the 873

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 874 ments is better for the diagonal terms than for the coupling terms. Larger discrepancies between theories and experiments exist in the high-frequency regime, in which experimental measurements are not as reliable. â¢ Traditional computations based on inviscid-fluid theory significantly over-estimate the inertia coefficients, and under-estimate the damping effects. Similar observations in this regard were reported in Yeung et al. (1998) for the same cylinder without any keels. â¢ The inertia coefficients approach the inviscid-fluid theory at high frequency. This seems consistent with an analytical viscous-fluid theory by Yeung and Wu (1991), which modeled only diffusion effects. â¢ In the high frequency range, both experiments and theory are suggesting that the damping are not vanishingly small as normally expected from inviscid-fluid computations. This can have important design implications. 4.3 Vorticity and flow patterns Figure 21 details the flow patterns for a period of one oscillation after a steady state is reached. Strong vortices are created in the vicinity of the aft edge of the bilge keel and are later shed away as the keel moves in the opposite direction. During every half period, the vortices from the fore keel have a strong tendency to move away sideways1, whereas those from the aft keel would move downward. The phenomenon is not entirely mirror in image for the keel during each half of one full period. This slight lack of symmetry was explained by Yeung and Cermelli (1998) as attributable to memory effects of the starting swing. In Fig. 22, the vorticity fields obtained using both numerical models are displayed as color-contour plots. The vortex method predicts larger and more distinct vortices, as well as larger distances from the body once they become separated from the keel. A stronger dissipation appears to have taken place in the BFFDM. As in most finite-differencing schemes, numerical (artificial) damping is always present. The grid-free method of FSRVM does not contain such dissipation, thus preserving the details better. However, FSRVM does not model the boundary layer flow as well as BFFDM. 5 CONCLUSIONS Experimental and theoretical studies of the forced roll-motion hydrodynamics of a cylinder with bilge keels are conducted. The theoretical model includes the use of two free-surface Navier-Stokes Solvers (FSRVM & BFFDM). Overall speaking, the results obtained using the FSRVM method are well validated by the experimental results. In cases where both numerical solutions are available, very acceptable agreement is observed, at both global and detailed level. This lends much credence to the present methodologies of predictions. Added moment of inertia and (equivalently linearized) damping coefficients are presented as functions of frequency and bilge-keel depth. The computational and experimental results are compared also with those from classical methods based on an ideal fluid. The computations suggest that an increase in keel depth will increase both the added inertia and damping. Traditional computations based on potential-flow theory will significantly over-estimate the added inertia coefficients and under-estimate the damping effects. Real-fluid effects tend to âroundâ the shape of the body, thus reducing the effective inertia, whereas flow separation around the keels increases the damping. Similar observations were given by Yeung et al. (1998) for a cylinder without keels. Details of such flows are easily visualizable from the solutions presented. For a given bilge-keel size, an increase in roll amplitude reduces the nondimensional inertia coefficients slightly but increases the damping coefficients appreciably. The observation applies to both diagonal and off-diagonal terms. Being investigated is the amplitude dependence at angles larger than the maximum value of 5.75Â° studied here. The complete treatment of the free motion of a floating cylinder in waves, with the full effects of viscosity included, can be found in Roddier et al. (2000). Effects of turbulence on the present solutions are being examined. the authoritative version for attribution. 1The fore keel is deined as the more forward keel in the direction of roll motion. Every half period, the fore keel becomes the aft keel, and vice-versa.

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 the authoritative version for attribution. Figure 14: Added moment of inertia coefficient, Î±o=5.75Â°. Figure 13: Added moment of inertia coefficient, Î±o=2.85Â°. Figure 16: Equivalent linear damping coefficient of roll, Î±o=5.75Â°. Figure 15: Equivalent linear damping coefficient of roll, Î±o=2.85Â°. ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 875

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 the authoritative version for attribution. Figure 20: Roll-sway coupled damping coefficient, Î±o=5.75Â°. Figure 19: Roll-sway coupled damping coefficient, Î±o=2.85Â°. Figure 18: Roll-sway coupled added inertia coefficient, ao=5.75Â°. Figure 17: Roll-sway coupled added inertia coefficient, Î±o=2.85Â°. ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 876

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 the authoritative version for attribution. ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS Figure 21: Flow visualization: velocity vectors of the vortex-blob field by FSRVM. 877

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 the authoritative version for attribution. based on a non-dimensional vorticity defined by ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS Figure 22: Vorticity contours for a rolling cylinder with 4% bilge keels at one instant of time, Î±o=5.75Â°. The contour scale is 878

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 879 ACKNOWLEDGEMENT This research reported has been supported in part by the Office of Naval Research and a Shell Foundation Grant of the US authors and by the French Ministere de la Recherche and the French Ministere de la Defense (DGA) of the French collaborators. We are appreciative of the Ecole Centrale de Nantes and the University of California at Berkeley for promoting the collaborative environments. REFERENCES [1] B.Alessandrini, B. and Delhommeau, G. (1995). âA multigrid velocity-pressure-free surface elevation fully coupled solver for turbulent incompressible flow around a hull calculationsâ, Proc. 9th International Conference on Numerical Methods in Laminar and Turbulent Flows, Atlanta, pp. 1173â1184. [2] Braathen, A. (1987). âApplication of a vortex tracking method to the prediction of roll damping of a two-dimensional floating bodyâ, Dept. Marine Tech. Norwegian. Inst. Tech. Rept., UR-87â56. [3] Cermelli, C.A. (1995). âVortical flows generated by a plate rolling in a free surfaceâ, Ph.D. dissertation, Dept. of Naval Arch. and Offshore Engrg., Univ. of Calif., Berkeley. [4] Chorin, A.J. (1973). âNumerical study of slightly viscous flowâ, J. Fluid Mech., 57, pp. 785â796. [5] Cointe, R., Geyer, P., King, B., Molin, B., and Tramoni, M. (1991). âNonlinear and linear motions of a rectangular barge in a perfect fluidâ, Proc. 18th Symp. on Naval Hydrodyn., Ann Arbor, Michigan. [6] Cox, G.C. and Lloyd, A.R. (1977). âHydrodynamic design basis for navy ship roll motion stabilizationâ, SNAME Transactions, 85. [7] Faltinsen, O.M. and Sortland, B. (1987). âSlow drift eddy making damping of a shipâ, Applied Ocean Research, 9, 1, pp 37â46 [8] Fink, P.T. and Soh, W.K. (1974). âCalculation of vortex sheets in unsteady flow and applications in ship hydrodynamics,â Proc. 10th Symp. on Naval Hydrodyn., Cambridge, MA, pp. 463â491. [9] Froude, W. (1863). âRemarks on Mr. Scott-Russell's paper on rollingâ, The papers of William, Froude, published by INA, 1955. See also, Transactions, INA, 1863. [10] Gentaz L., Alessandrini B., and Delhommeau, G. (1997). âMotion Simulation of a two-dimensional body at the Surface of a Viscous Fluid by a fully coupled solverâ, Proc. Workshop on Water Waves and Floating Bodies, Marseille, pp. 85â90. [11] Grosenbaugh, M.A. and Yeung, R.W. (1989). âNonlinear free-surface flow at a two-dimensional bowâ, J. Fluid Mech., 209, pp. 57â75. [12] Himeno, Y. (1981). âPrediction of ship roll dampingâstate of the artâ Dept. Naval Arch. and Mar. Engrg., Univ. of Michigan, Rep. no 239. [13] Hodges, S.B. and Webster, W.C. (1986). âMeasurement of the forces on a slightly submerged cylinderâ, Proc. American Towing Tank Conference, New Orleans, Louisiana. [14] Israeli, M. and Orszag, S.A. (1981). âApproximation of Radiation Boundary Conditionsâ, J. Comp. Phys., 41, pp. 115â135. [15] Korpus, R.A. and Falsarano, J.M. (1997). âPrediction of viscous ship roll damping by unsteady Navier-Stokes techniquesâ, JOMAE, 119, pp. 108â 113. [16] Liao, S.-W., and Roddier, D. (1998). âGeneration of plunging breakersâlaboratory and numerical simulationâ, OEG-98.1, Ocean Engrg. Group, University of California, Berkeley. [17] Martin, M. (1958). âRolling Damping Due to Bilge Keelsâ, Report prepared by the Iowa University Institute of Hydrolic research for the Office of Naval research under contract No. 1611 (01). [18] Miller, E.R., Slager, J.J., and Webster, W.C. (1974), âDevelopment of a Technical practice for roll stabilization system selectionâ, NAVSEC Report 6136â74â280 [19] Rhie C.M., Chow, W.L. (1983). âA Numerical Study of the Turbulent Flow past an Isolated Airfoil with Trailing Edge Separationâ, AIAA Journal, 21, pp. 179â195 [20] Ridjanovic, M. (1962). âDrag coefficient of flat plates oscillating normally to their planesâ, Schiffstechnik, 9, Heft 45. [21] Robinson, R.W. and Stoddart, A.W. (1987), âAn engineering assessment of the role of non-linearities in transportation barge roll responseâ, The Naval Architect, July/Aug. p. 65. [22] Roddier, D., Liao, S.-W., and Yeung, R.W. (2000). âOn freely-floating cylinders fitted with bilge keelsâ, Proc. 10th Int'l Offshore and Polar Engrg. Conf., Seattle, Washington, 3, pp. 377â384. [23] Sarpkaya, T. and O'Keefe, J.L. (1996). âOscillating flow about two and three-dimensional bilge keelsâ, J. of Offshore Mech. and Arctic Engrg., bf 118, pp. 1â6. [24] Shanks, S.P., Thompson, J.F. (1977). âNumerical Solution of the Navier Stokes equations for the 2-D hydrofoils in or below a free surface,â, Proc. 2nd Int. Conf. Ship Hydrodyn. Berkeley, pp. 202â224. [25] Standing, R.G., Jackson, G.E. and Brook, A.K. (1992) âExperimental and Theoretical Investigation into the roll damping of a systematic series of two-dimensional barge sectionsâ, Proc. Behavior of Offshore Structure, 92, London, UK, p. 1097. the authoritative version for attribution.

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 880 [26] Steinberg, S. and Roache, P. (1986). âVariational Grid Generation, Numerical Methods in Partial Differential Equations, vol. 2, pp. 71â96. [27] Vaidhyanathan, M. (1993). âSeparated flows near a free surfaceâ, Ph.D. dissertation, Dept. of Naval Arch. and Offshore Engrg., Univ. of Calif., Berkeley. [28] Vinje, T. and Brevig, P. (1980). âNonlinear, two-dimensional ship motionsâ, Tech. Rept., The Norwegian Inst. of Tech., Trondheim. [29] Van Der Vorst, H.A. (1992), âBi CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systemsâ, J. Sci. Stat. Comp., 13. [30] Vugts, J.H. (1968). âThe hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surfaceâ, Rept. No. 194, Laboratorium voor Scheepsboukunde, Technische Hogeschool Delft, The Netherlands. [31] Wehausen, J.V. (1971). âThe motion of floating bodiesâ, Annual Review of Fluid Mechanics, 3, pp. 237â268. [32] Wilcox, D.C. (1988). âMultiscale model for turbulent flowsâ, AIAA Journal, Vol. 26, pp. 1211â1320, November 1988. [33] Wilcox, D.C. (1988) âReassessment of the scale-determining equation for advanced turbulence modelsâ, AIAA Journal, 26, pp. 1299â1310, November 1988. [34] Yeung, R.W. (1975). âA Hybrid Integral-Equation Method for Time-Harmonic Free-Surface Flowsâ, Proceedings, 1st Int'l Conf. on Numerical Ship Hydrodyn., Gaithersburg, Maryland, pp. 581â607. [35] Yeung, R.W. and Wu, C-F. (1991). âViscosity effects on the radiation hydrodynamics of horizontal cylindersâ, J. Offshore Mech. and Arctic Engrg., 113, pp. 334â343. [36] Yeung, R.W. and Ananthakrishnan, P. (1992). âOscillation of a floating body in a viscous fluidâ, J. Engrg. Math, 26, pp. 211â230. [37] Yeung, R.W. and Vaidhyanathan, M. (1994). âHighly separated flows near a free surfaceâ, Proc. Int'l. Conference on Hydrodynamics, Wuxi, China. [38] Yeung, R.W., Cermelli, C.A. and Liao S-W. (1996). âVorticity Fields Due to Rolling Bodies in a Free SurfaceâExperiment and Theoryâ, Proc. 21st ONR Symposium on Naval Hydrodynamics, Trondheim, Norway. [39] Yeung, R.W. and Cermelli, C.A. (1998), âVortical Flow Generated by A Plate Rolling in a Free Surfaceâ, Chapter 1, Free Surface Flow with Viscosity, Advances in Fluid Mechanics, vol. 16, ed. P.Tyvand, Computational Mechanics Publications, Southampton, England. [40] Yeung, R.W., Liao S.-W., and Roddier D. (1998). âHydrodynamic coefficients of rolling rectangular cylindersâ, Int'l J. Offshore and Polar Engrg., 8, No 4, pp. 241â250. the authoritative version for attribution.

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 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 ON ROLL HYDRODYNAMICS OF CYLINDERS FITTED WITH BILGE KEELS 881 DISCUSSION A.Clement Laboratoire MÃ¨canique des Fluides, Ecole Centrale de Lyon, France I may be wrong, but it seems to me, looking at moment curve results and at the animated video, that the global quantitive like forces and moments are periodically established long before the vortical flow around the keel is itself periodic. Moreover, one may wonder if the vortical flow ever reaches a periodical steady state? Could you comment about that? AUTHORS' REPLY The amount of time that a particular solution takes to attain a periodical steady-state behavior depends on the frequency of excitation. Lower frequency excitation generally takes more time to achieve steady state. Figure 10 of the paper indicates that it took about 4 to 5 periods to attain a steady behavior for the frequency in question. It is worthwhile to note that while the roll moment curve might not appear to vary significantly in amplitude during the âtransient stateâ, the phasing of the moment relative the to roll angle continue to adjust itself as more vortices are shed off the keels. This phasing naturally determines the damping and added moment of inertia. DISCUSSION M.Kashiwagi Kyushu University, Japan In the classical potential theory, the conservation principles can be used to check the validity of numerical results without resorting to comparison with experiments. I wonder if the same kind of principles can be used to validate the numerical results in this paper. What is the relation between the coupled coefficient of roll into sway and the coefficient of sway into roll? AUTHORS' REPLY Unlike classical potential-flow theory, the flows being considered in this paper are governed by the fully nonlinear Navier-Stokes equations. Thus, none of the normal conservation or reciprocity relations, such as the wave-amplitude to damping relation, symmetric of hydrodynamic coefficients, would be applicable. The FSRVM method contains no artificial or numerical viscosity in the fluid domain, aside from the sponge layer used to damp out reflected waves on the free surface. Its accuracy was earlier tested out by a number of experiments and numerical solutions reported in the references. The nonlinear nature of the solution, particularly the importance of convective effects in the flow, renders the sway-roll coupling coefficients to be non-symmetrical. About a decade ago, we developed a viscous-flow formulation (Yeung & Wu, Ref. [35]) under a small motion- amplitude assumption. The linearization process was applied to both the Navier-Stokes equations and the free-surface boundary conditions It is conceivable that some kind of conservation and symmetry conditions can be derived for such a linear model. The short-coming of such an approach is that if a substantial amount of flow separation takes place, as in the present bilge-keel problem, the convective effects in the field equations cannot be neglected. The linearized model has been pursued by a number of workers since our initial work. the authoritative version for attribution.