URANS Investigation of Ship Roll Motion Damping Using...

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URANS Investigation of Ship Roll Motion Damping Using Bilge Keels Prepared by: Joshua Counsil Kiari Goni Boulama Department of Mechanical Engineering, Royal Military College of Canada, 13 General Crerar Crescent, Kingston, ON K7K 7B4 DRDC-RMCC Project Arrangement: PA 13013 Technical Authority: Kevin McTaggart, Defence Scientist The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of the Department of National Defence of Canada. Contract Report DRDC-RDDC-2016-C134 April 2016

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URANS Investigation of Ship Roll Motion Damping Using Bilge Keels

Prepared by: Joshua Counsil Kiari Goni Boulama Department of Mechanical Engineering, Royal Military College of Canada, 13 General Crerar Crescent, Kingston, ON K7K 7B4

DRDC-RMCC Project Arrangement: PA 13013 Technical Authority: Kevin McTaggart, Defence Scientist

The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of the Department of National Defence of Canada.

Contract Report DRDC-RDDC-2016-C134 April 2016

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© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2016

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2016

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URANS Investigation of Ship Roll Motion Damping Using Bilge Keels Submitted by: Joshua Counsil

Research Assistant Royal Military College of Canada

Kiari Goni Boulama Associate Professor Royal Military College of Canada

Submitted to: Kevin McTaggart Warship Performance Section DRDC Atlantic Date: 27 April 2016

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Abstract It is widely accepted in ship hydrodynamics research that rolling is the least understood of the six degrees of freedom. Bilge keels are passive devices which are appended to the hull of a ship in order to increase the ship’s natural roll damping, thereby reducing the amplitude of roll motions. The optimum size of these bilge keels for effective roll damping and the influence of the roll amplitude on the performance of the bilge keels require careful research. Computational Fluid Dynamics (CFD) and particularly Unsteady Reynolds-Averaged Navier-Stokes methods (URANS) have some advantages over experimental methods for their cost, simplicity of parameterization, and data extraction, although their use in roll damping investigations is still in its infancy due to the inherent difficulty in modelling turbulence, free-surface interactions and moving meshes, among other considerations. Traditional empirically derived analytical methods are still in use today due to their ease of use, though their prediction abilities in the presence of strong viscous interactions are debatable. DRDC Atlantic has partnered with the Royal Military College of Canada as part of ongoing research into bilge keel roll damping. Using their in-house software, DRDC has obtained excellent results for many hydrodynamics and maneuvering cases though viscous roll damping still proves challenging. Hence, the commercial CFD code STAR-CCM+ is employed here to study a variety of bilge keel spans at different roll amplitudes. Two goals are presented: 1) test the value of URANS and semi-empirical methods in the face of complex viscous roll phenomena, and 2) provide insight into the selection of bilge keel span under varying roll amplitudes. First, two studies of submerged rolling cylinders are undertaken to verify and validate the selected computational setup with focus on grid and timestep selection. Next, a cylinder is appended with bilge keels representative of those on a Canadian vessel and rolled at varying amplitudes. Data extracted include the added mass and roll damping coefficients to evaluate the efficacy of the bilge keels, and vorticity flow-field visualizations to evaluate the interactions between the bilge keels and the hull. Comparisons are made with hydrodynamic coefficients calculated using semi-empirical methods and conclusions are drawn. A brief free-surface exercise is also presented as a precursor for future work. Extracted hydrodynamic coefficients indicate that in general, increasing displacement amplitude and bilge keel span yields increased added mass and increased damping. Drag coefficient is shown to increase with bilge keel span and decrease with increasing angular amplitude on account of its inverse relationship with rotational velocity. The trends, however, are nonlinear and not applicable to all cases. Hydrodynamic performance is subject to the interactions of the vortices with each other and the body. Damping is optimal for hull-bound vortices and lower for vortices that propagate from the hull. The semi-empirical methods used for comparison, particularly the popular Ikeda method, yield good predictions for simple vortex interactions but fail where viscous effects are strong, such as for low-amplitude roll cases. Hence, URANS methods are shown to be necessary for viscous-dominant flows while semi-empirical methods remain useful for initial design considerations. The largest bilge keels are shown to be superior damping agents and should be implemented when possible. The free-surface exercise indicated the damping trend with angular frequency is also strongly nonlinear. Results compared somewhat favourably with published literature, though more insight into the extraction of the coefficients is recommended for future work.

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Table of Contents URANS Investigation of Ship Roll Motion Damping Using Bilge Keels ........................................................ 1

Abstract ......................................................................................................................................................... 2

Table of Contents .......................................................................................................................................... 3

List of Figures ................................................................................................................................................ 5

List of Tables ................................................................................................................................................. 6

Nomenclature ................................................................................................................................................ 7

1. Introduction ............................................................................................................................................... 9

2. Literature Review .................................................................................................................................... 10

2.1 Geometry ........................................................................................................................................... 11

2.2 Roll Amplitude ................................................................................................................................... 12

2.3 Forward Speed .................................................................................................................................. 12

3. Approaches to Roll Damping Investigation ............................................................................................. 13

3.1 Experimental Approach ..................................................................................................................... 13

3.2 Analytical Approach ........................................................................................................................... 13

3.3 Numerical Approach .......................................................................................................................... 14

4. Numerical Method ................................................................................................................................... 16

4.1 Computational Setup ......................................................................................................................... 16

4.2 Equations of Motion ........................................................................................................................... 17

5. Validation and Verification....................................................................................................................... 19

5.1 Bare-Hull Square with Rounded Bilges ............................................................................................. 19

5.2 Circle with Bilge Keels ....................................................................................................................... 22

6. Results and Discussion ........................................................................................................................... 24

6.1 Moment History ................................................................................................................................. 25

6.2 Hydrodynamic Coefficients ................................................................................................................ 26

6.3 Flow-field Analysis ............................................................................................................................. 31

6.3.1 Typical Case ............................................................................................................................... 31

6.3.2 Smallest Angular Displacement .................................................................................................. 33

6.3.3 Smallest Bilge Keel ..................................................................................................................... 36

6.3.4 Largest Bilge Keel ....................................................................................................................... 37

7. Additional Study: 2D Partially Emerged Cylinder .................................................................................... 41

8. Conclusions ............................................................................................................................................. 44

8.1 Findings from Present Work .............................................................................................................. 44

8.2 Recommendations and Future Work ................................................................................................. 45

References .................................................................................................................................................. 46

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List of Figures Figure 1: Bilge keel orientation and mounting (Baniela, 2008). .................................................................. 10 Figure 2: Geometry and grid for the 2D submerged rectangular prism with round bilges. ......................... 19 Figure 3: Grid validation study for the 2D submerged rectangular prism with round bilges in comparison with the published data of Yu (2008). ......................................................................................................... 20 Figure 4: Timestep validation study for the 2D submerged rectangular prism with round bilges in comparison with the published data of Yu (2008). ...................................................................................... 22 Figure 5: Geometry and grid for the 2D submerged circular prism with bilge keels. .................................. 22 Figure 6: Bilge keel local grid validation study for the 2D submerged circle with bilge keels in comparison with the published data of Miller et al. (2002). ............................................................................................ 24 Figure 7: Velocity vector field (left) in comparison with Miller et al. (2002) published PIV data at t/T = 0.250. .......................................................................................................................................................... 24 Figure 8: Moment histories at various amplitudes. ..................................................................................... 26 Figure 9: Added mass coefficients (solid line) compared to flat-plate theory (dashed line). Legend labels correspond to bilge keel span. .................................................................................................................... 27 Figure 10: Drag coefficients compared to semi-empirical methods. ........................................................... 29 Figure 11: Drag coefficients for all cases. Legend labels correspond to roll amplitude. ............................ 30 Figure 12: Linear damping coefficients for all cases. .................................................................................. 31 Figure 13: Vorticity plots for the case at s = 0.8 m, θ = 8°. ......................................................................... 33 Figure 14: Vorticity plots for several bilge keel spans at θ = 2°. ................................................................. 35 Figure 15: Vorticity plots for the smallest bilge keel (s = 0.2 m) at θ = 14° and θ = 20°. ............................ 37 Figure 16: Vorticity plots for various cases at s = 1.0 m. ............................................................................ 39 Figure 17: Linear nondimensional added mass coefficient for the Vugts rectangle at rolling amplitude of 0.05 rad. ...................................................................................................................................................... 43 Figure 18: Linear nondimensional damping coefficient for the Vugts rectangle at rolling amplitude of 0.05 rad. .............................................................................................................................................................. 43

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List of Tables Table 1: Grid dimensions and y+ values for grid refinement studies........................................................... 20 Table 2: Grid convergence study of moment amplitude and its components for the 2D submerged rectangular prism with round bilges. ........................................................................................................... 20 Table 3: Timestep and CFLmin values for temporal discretization studies. ................................................. 21 Table 4: Timestep convergence study of moment amplitude and its components for the 2D submerged rectangular prism with round bilges. ........................................................................................................... 21 Table 5: Bilge keel region grid dimensions and convergence study of normal force amplitude for the 2D submerged circle with bilge keels. .............................................................................................................. 23 Table 6: x-directional grid convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 1.41. ....................................................................................... 41 Table 7: y-directional grid convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 1.41. ....................................................................................... 42 Table 8: Timestep convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 1.41. .......................................................................................................... 42 Table 9: Timestep convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 0.80. .......................................................................................................... 44

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Nomenclature ε % difference between simulation solutions ω angular frequency rad/s ω* nondimensional angular frequency ρ fluid density kg/m3

θ instantaneous roll displacement rad θ0 roll amplitude rad

underwater/submerged area m2

a44 linear added mass coefficient kg.m a44

* nondimensional linear added mass coefficient Ap added mass of flat plate kg/m B beam or breadth of ship hull m b44 linear damping coefficient kg.m/s b44

* nondimensional linear damping coefficient B44 roll quadratic damping coefficient kg/m2

C generalized damping coefficient kg.m/s CD drag coefficient f empirical increment factor F external excitation force kg.m/s2

Fr Froude number g gravitational acceleration m/s2

GB centre of buoyancy from centre of gravity m I mass moment of inertia or total roll inertia kg.m2

K stiffness or restoring moment kg/s2

M total moment kg/s2 Mω hydrodynamic moment kg/s2 M0 moment amplitude kg/s2 Mf0 moment amplitude due to friction kg/s2 Mp0 moment amplitude due to pressure kg/s2 Mr restoring moment kg/s2 Q constant in damping component kg.m R moment arm from ship centre of gravity to centroid of bilge keel m s span of bilge keel m t time s T period s-1

y+ dimensionless wall distance 2D two-dimensional 3D three-dimensional BEM boundary element method BK bilge keel CFD computational fluid dynamics CFL Courant–Friedrichs–Lewy number EFD experimental fluid dynamics DRDC Defence Research and Development Canada FPSO floating production storage and offloading

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FSRVM free-surface random-vortex method HMCS Her Majesty’s Canadian Ship K-C Keulegan-Carpenter PISO pressure implicit with splitting of operator PIV particle image velocimetry RANS Reynolds-averaged Navier-Stokes RMC Royal Military College of Canada RMS root mean square SST k-ω shear stress transport STAR-CCM+ commercial CFD software package from CD-adapco URANS unsteady Reynolds-averaged Navier-Stokes VLCC very large crude carrier VOF volume of fluid

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1. Introduction Ships experience six degrees of motion: surge, sway, heave (translation motions), yaw, pitch and roll (rotation motions) (Baniela, 2008). Surge, sway and yaw occur in the horizontal plane and are unrestored, meaning that they do not exhibit resonance. Heave, pitch and roll occur in the vertical plane, experience roll and pitch restoration forces in the form of simple harmonic motion, and exhibit resonance. The roll and pitch restoration forces arise from the misalignment between the centre of gravity of the ship and the centre of buoyancy when wave excitation forces unbalance the ship. While weakly restored roll could result in ship overturning in the most extreme cases, overly large roll and pitch restoration forces will lead to small oscillation periods and high motion accelerations, which in turn will increase risks of structural damage to the ship, difficult operations (helicopter landing for example), and discomfort to the passengers (Na et al., 2002). Furthermore, when the wave excitation force frequency is equal to the natural frequency of the ship, resonance i.e. large-amplitude motions occur. Resonance tends to be most problematic for roll motions due to low wave radiation damping, especially for rounded hull forms. The replacement of steam by sails and iron by wood resulted in significant design modifications that affected transversal ship stability, with large roll amplitudes even in moderate sea conditions (Baniela, 2008, Grace’s Guide, 2014). One of the most important first contributors to ship motion research, Froude, recorded the roll motions on the Great Eastern using a clever yet crude device appended to the hull and observed that the friction was a function of the velocity, and coming to the conclusion that bilge keels could be an efficient means of reducing roll. These consist of plates appended perpendicularly to the sides of the hull (see Figure 1). In fact, bilge keels represent simple, inexpensive and passive, anti-rolling devices, providing 30% to 60% roll attenuation (50 to 60% for naval vessels equipped with large keels) (Bassler et al., 2011). Bilge keels increase the energy dissipation through increased friction and pressure resistance, slowing down the roll motion and increasing the period. Other roll control devices include passive anti-roll tanks and active stability systems such as stabilizer fins (Yu, 2008). These will not be discussed in this report.

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Figure 1: Bilge keel orientation and mounting (Baniela, 2008). In this study, the contribution of bilge keels to ship damping is investigated via Unsteady Reynolds-Averaged Navier-Stokes (URANS) to assess the ability of Computational Fluid Dynamics (CFD) to capture viscous effects not accessible using potential and empirical methods. Two-dimensional (2D) bodies with prescribed sinusoidal roll motion are fully submerged to negate wave damping effects. First, two submerged cylinders are simulated with four different grid and timestepping schemes. The validated cylinder model is then appended with five bilge keels of varying span - 0.2, 0.4, 0.6, 0.8 and 1.0 m - and forced to roll with a period of 10.6 seconds at four angular displacement amplitudes - 2°, 8°, 14° and 20°. The bilge keel spans and period are scaled down for computational savings. Hydrodynamic coefficients are extracted and compared with empirical method results, while flow-fields are explored to explain the physical phenomena behind the observed trends. A preliminary free-surface study is also performed as a precursor to future work.

2. Literature Review Himeno (1981) presented an exhaustive summation of the available research on ship roll damping prediction and implementation up to the 1980s. At that time, empirical correlations on full ships were the most popular method of bilge keel damping prediction while advances in Japan were being made to split bilge keel damping into six components - lift, eddy, wave, bilge keel normal force, bilge-keel-hull pressure and bilge keel wave - with individual empirical correlations for each. These analyses provide great insight into the effect of various hydrodynamic parameters on ship roll damping. For example, bilge keel wave damping was determined to be usually negligible for standard bilge keel spans since viscous effects dominate at these scales. However, it was deemed important for large keels, like those on a warship, and limited research existed to that point on its effect in the presence of forward motion. For the other components of damping, with increased forward speed or roll angular frequency, it was shown that lift and

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wave damping increase, eddy and bilge keel damping decrease, and damping as a whole becomes almost linear in nature. Increasing roll angle amplitude yielded increased hull-keel, keel normal force and eddy damping, while little effect was observed on lift, hull friction and wave damping. Recent research on the contribution of geometry, angular amplitude and forward speed will be discussed in the following paragraphs, while the angular frequency parameter is ignored since it is linearly correlated with damping (Ikeda, 2004) and has largely the same effect as forward speed.

2.1 Geometry Geometry represents the most obvious parameter when studying roll damping. Beginning with the shape of the hull, Vugts (1968) was the first to investigate sharp-edged bodies under pure roll, observing for example that at certain beam-to-draft ratios and lower frequencies, a rectangular prism with round bilges did not generate waves and therefore did not damp roll motion. Yeung et al. (1998, 2000) and Sarkar and Vassalos (2000) studied similar shapes but with different bilge radii - 2% of beam for the former and 0.625% of beam for the latter - and obtained different results than Vugts, whose rectangular bilge radius measured 3.125% of beam. Korpus and Falzarano (1997) reexamined the Vugts roll cases and found a strong dependence of roll moment on shed vortices, confirming Himeno’s (1981) postulation. Martins et al. (2013) performed PIV measurements of the flow fields on four cylinders undergoing forced rolling motion: a round-bilge rectangular prism with and without keels of varying sizes, as well as a sharp-cornered rectangular prism without keels. The rectangular hull exhibited the most vortex presence, while the round-bilge prism without keels exhibited the least. In fact, rounder hulls intuitively yield less resistance and vortex generation, which in turn hinder damping. Furthermore, a perfectly round cylinder does not generate waves since it does not displace water (Wilson et al., 2006). Despite poor damping and lower payloads, however, rounder hull cross sections are gaining prominence since the decreased wetted area yields lower forward-motion resistance (Miyake and Ikeda, 2013). Many studies have demonstrated that with increasing bilge keel size, roll damping improves due to increased resistance (Pesman et al., 2001; Wilson et al., 2006; Esperança et al., 2008; Bangun et al., 2010; Martins et al., 2013). However, investigating a floating production storage and offloading (FPSO) unit, van’t Veer and Fathi (2011) found a critical bilge span below which the bilge keel damping increased linearly with bilge keel span, and above which the damping effectiveness was reduced as the generated vortices propagated away from the hull. The same conclusions were made by Thiagarajan and Braddock (2009) and Avalos et al. (2014). The latter study also indicated a minimum effective bilge keel span below which damping was actually reduced when compared to the case without appendages. Yu (2008) and Deng et al. (2013) concluded likewise for their parametric cylinder studies. No explanation was given by any of these authors for the existence of this minimum bilge keel span. It is suggested here that a very small bilge keel may act as a turbulence trip, providing rotational momentum from small vortices to help push and pull the hull in the direction of rotation. Ferrari and Ferreira (2002) studied both the span (across hull) and extent (along hull) of bilge keels on a large FPSO, concluding that the former had a far greater impact on damping than the latter. Souza et al. (1998) noted a substantial increase in damping for a bilge keel spanning 80% of the ship’s length as opposed to a bilge keel of identical span and length but segmented in three portions. Ikeda (2004) and Baniela (2008) observed that high-aspect-ratio bilge keels work best on slender ships, while the opposite is true for full ships. Na et al. (2002) stated that the end configuration of the bilge keel is as important as the span, insomuch that proper configuration allows for bilge keel span reduction without compromising damping. An additional consideration for the span of the bilge keel is the significant stress it causes to the structure (Pettersen et al., 2013), as well as its effect on forward motion resistance, ship docking clearance and biofouling (Baniela, 2008). Bilge keel span is

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therefore case-dependent and critical in design consideration. Another geometrical parameter to consider is the placement of the bilge keels. Na et al. (2002) and Seah and Yeung (2003) studied an FPSO hull, a rectangular shape similar to that of the aforementioned studies, concluding that horizontal bilge keels were more effective than vertical bilge keels as the generated vortices were closer to the free surface and therefore more energy was dissipated from the body. Bangun et al. (2010) studied six bilge keel orientations on a rolling rectangle and verified the superior damping effectiveness of horizontal bilge keels over vertical bilge keels, though the best damping results were obtained with bilge keels set at 10° with respect to the horizontal. Angled keels also proved more effective than vertical keels for Yu (2008) and Miyake and Ikeda (2013). With the cases investigated, it seems that bilge keels should be appended approximately in line with the radius of gyration to maximize bilge keel normal force.

2.2 Roll Amplitude In general, increasing angular amplitude has been shown to increase roll damping (Miller et al., 2002; Seah and Yeung, 2003 Fernandes and Oliveira, 2009; Irvine et al., 2013; Deng et al., 2013), though the relationship is nonlinear. Yeung et al. (1998) showed that the added-mass coefficient, a coefficient defining the added inertia on account of fluid resistance to object acceleration, was unaffected with changing roll amplitude at angles less than 5.75°. Souza et al. (1998) and Fernandes and Oliveira (2009) demonstrated strong nonlinearity between the hydrodynamic coefficients and angular displacement at the beginning and end stages of roll decay when angular displacement is large and small, respectively. Avalos et al. (2014) indicated a critical point beyond which further increases in roll amplitude did not increase damping. On the other hand, Bangun et al. (2010) found that at small roll amplitudes, bilge keel damping was less effective than that of the bare hull. An explanation would be that at high and low roll amplitudes, the vortices generated by the bilge keels interact destructively and in harmony, respectively, which affects damping effectiveness. Ikeda (1993) postulated that vortices enhancing the roll motion, i.e., the vortices in front of the bilge keel can overpower the vortices damping the roll motion, i.e., the vortices behind the bilge keel. At high roll amplitudes, the vortices behind the bilge keel may dissipate in the free surface.

2.3 Forward Speed With increasing forward speed, overall roll damping increases (Pesman et al., 2001; Wilson et al., 2006; Irvine et al., 2013; Miyake and Ikeda, 2013) on account of lift and wave damping increasing, while the bilge keel component of damping decreases on account of vortex dissipation (Miller et al., 2002; Aloisio and Di Felice, 2006; Irvine et al., 2013). Wilson et al. (2006) also verified that as their surface combatant gained forward momentum, the correlation with damping became increasingly linear. These results are consistent with those of Himeno (1981), though some other observations have been reported by other authors. For example, Wang et al. (2014) studied a ship in free motion using both the experimental and CFD approaches and reported the following interesting correlation between forward speed and roll angle: at low (respectively, high) roll angles, increased forward speed caused increased (respectively, decreased) stability; the decreased stability was said to be the result of less trim, i.e., wetted surface area and higher heave at the high roll angles. Sadat-Hosseini et al. (2010) found a similar stability correlation but pertaining to metacentric height values: at low metacentric height values corresponding to large roll periods, roll motion was sustained with increasing speed while parametric rolling in head waves was more likely at high metacentric height values. Similarly, Colbourne (1982) concluded that hull damping increased with forward speed at large roll periods and decreased with increasing forward speed at small

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roll periods. Esperança et al. (2008) showed that waves of similar frequency to that of the ship’s natural frequency caused increased loads and decreased stability on the boat, which was to be expected (Baniela, 2008). In all the aforementioned cases, forward motion was strongly correlated with ship stability and roll damping yet no general conclusion can be made for all ships and conditions.

3. Approaches to Roll Damping Investigation

3.1 Experimental Approach Obtaining acceptable experimental data from roll experiments is difficult. Vugts (1968) was uncertain about his measurements due to the unreliability of the mass moment of inertia and difficulty in separating the measured forces into added mass and damping components. Wilson et al. (2006) discussed errors specific to their case but overarching in many respects, including changing initial conditions, added energy loss on account of the friction from the system used to mount the ship, and geometrical asymmetries in model or setup. Avalos et al. (2014) found it especially difficult to obtain accurate data at small roll angles where experimental errors increase hyperbolically. In addition, empirical roll prediction methods cannot be applied to all cases and conditions, especially where viscous effects are prevalent such as large-scale roll amplitudes (Bassler et al., 2011), flat-bottom ships (Fernandes and Oliveira, 2009) and ships with small draft-beam ratios (Irvine et al., 2013). It is also difficult to segregate the individual damping components since, with the exception of hull friction and bilge keel wave damping, they are similar in magnitude (Himeno, 1981).

3.2 Analytical Approach One of the most common methods for analytically studying ship roll motion is to represent it in the form of a one-dimensional, mass-spring-damper equation of motion:

(1) where is the mass moment of inertia or total roll inertia, is the damping coefficient, represents the stiffness or restoring moment of the system, and F is the external excitation force. The mass moment of inertia includes the actual geometrical moment of inertia, as well as the added inertia resulting from the interaction between the hull and the water. The damping coefficient characterizes the energy dissipation resulting from the interaction between the ship body and the surrounding fluid domain. The total roll inertia is generally considered to be constant, while the restoring moment can be calculated from hydrostatic considerations and has generally been written in the form of an odd polynomial. The form of the damping coefficient is, however, rather uncertain (Chan et al., 1995). Following the steps of Froude more than two centuries ago, the three coefficients in Equation 1 have routinely been deduced from physical experiments on specific hull shapes and in specific sea conditions. Some authors use simple, linear analytical forms such as the one below by Thiagarajan and Braddock (2009), to predict the damping coefficient:

(2)

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where s is the bilge keel span and is the ship beam. Linear damping correlations are simple and quick, but due to the highly nonlinear nature of the flow, many authors represent damping in more complex nonlinear forms. By equating the energy that would be dissipated by a linear system to the actual energy dissipation measured on an experimental model, damping can be predicted via stochastic linearization as, for example, the sum of a linear and quadratic term (Souza et al., 1998; Taylan, 2000; Pesman et al., 2001; Ferrari and Ferreira, 2002; Zeraatgar et al., 2010; Perez and Blanke, 2012; Pettersen et al., 2013; de Oliveira et al., 2014):

(3) or as the sum of a linear, quadratic and cubic term (Taylan, 2000; Zeraatgar et al., 2010; Irvine et al., 2013):

(4) Restoring moment can also be represented in more complex forms, e.g., cubic, as was explored by Esperança et al. (2008):

(5) Many further variations of the above equations exist, including high-order restoring moment polynomials (Sadat-Hosseini et al., 2010) and piecewise linear models in which the damping coefficient changes over successive intervals of the roll angle (Bassler et al., 2011). Prediction success is case-dependent. In general, damping contributions due to lift and wave-making are considered linear while those of friction, eddy-making and bilge keels are quadratic (Himeno, 1981; Pesman et al., 2001; de Oliveira et al. 2014). The focus herein is on bilge keel damping and a quadratic correlation is employed.

3.3 Numerical Approach The first wave-body interaction studies began in 1955 with the development of strip methods (Yu, 2008) in which 2D solutions at each ship cross-section are integrated along the ship’s length. According to the Ikeda method, the damping coefficient in the equation of motion is the sum of four contributions: bilge keel, wave-making damping and exciting forces, eddy-making resistance on the hull, and skin friction (Himeno, 1981). A fifth term representing the lifting surface contribution to the damping and exciting forces should be added in the presence of finite forward motion. As the ship is assumed to maintain a straight, constant, forward speed, these methods are adequate for capturing forward waves but not high-frequency waves, lateral waves (Colbourne, 1982), viscous effects (Himeno, 1981; Nan et al., 2007), complex vortex interactions (Miyake and Ikeda, 2013), or blunt-body effects (McTaggart and Stredulinsky, 2004; Kawahara et al., 2012). During the 1970s, potential flow codes, including panel and boundary element (integral) method (BEM) solvers, gained prominence for inviscid, irrotational flows. These methods have been shown to predict some surface wave interactions for 2D (Na et al., 2002; Zhou et al., 2005; Avalos et al., 2014) and 3D (Souza et al., 1998; Ahmed and Guedes Soares, 2009) bodies but neglect important viscous contributions to roll (Handschel and Abdel-Maksoud, 2014), including vortex creation (Korpus and Falzarano, 1997), turbulence (Yu, 2005) and cross-flow velocities (Stern et al., 2012). While potential flow

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code improvements, e.g., via accounting for appendages (McTaggart, 2004), continue today with varying success, numerical predictions with potential flow models are limited to the range of geometry, frequency and operating parameters of the empirical data and suffer from scale effects (Wilson et al., 2006). Viscous flow computations for ships were first introduced in the 1960s with the simplified boundary layer approach, which failed in predicting flow at the stern and into the wake (Karim et al., 2010). Boundary layer finite difference methods have also been explored, resulting in improved 3D resolution while failing to capture thick separated flows (Karim et al., 2010). The free-surface random-vortex method (FSRVM) is a boundary-integral method developed by Yeung et al. (1998) and used in several studies (Seah and Yeung, 2003; Yeung et al., 2003). By merging inviscid flow modelling with limited grid-free Navier-Stokes resolution, the authors have developed an improved accuracy code with substantial cost savings and promising potential (van’t Veer et al., 2012). Agreement with experimental results tends to be satisfactory, though still less accurate than the resolution of the full Navier-Stokes equations. Advances in dynamic grids, local and adaptive grid refinement, free-surface tracking, turbulence modeling, high-performance computing and optimization have allowed for an enormous progress of RANS codes (Stern et al., 2012). One of the most popular CFD codes for ship motion simulations is the RANS/DES code developed at the University of Iowa for the US Department of Defense: CFDShip-Iowa. It has been used to qualitatively predict the flowfield around a navy destroyer (Kim, 2002), a tanker undergoing simple maneuver (Simonsen and Stern, 2005), a surface combatant in oscillation (Wilson et al., 2006), and a tumblehome ship in large-amplitude roll and forward speed (Sadat-Hosseini et al., 2010; Bassler et al., 2011). RANS has proved superior to existing theoretical models which tend to overestimate energy dissipation. Kinnas et al. (2003) for example used a RANS code to predict separation around bilge keels, obtaining substantially more accurate hydrodynamic coefficients than when they used a 2D Euler method to study an FPSO hull model in roll and heave. Bangun et al. (2010) confirmed the importance of viscous resolution on a rectangular cylinder with bilge keels by verifying large damping forces even in high wave frequencies in which one would suspect potential wave effects to dominate. Nevertheless, RANS methods still require important developments for roll motion problems, in addition to scrupulous modelling practices. RANS can be expensive for hydrodynamics problems and many authors employ different techniques to balance cost and accuracy. Korpus and Falzarano (1997) used a RANS code to investigate the flow around a fully submerged rectangular cylinder, extracting the individual viscous components of the damping for implementation in their potential code for future ship motion calculations. Nan et al. (2007) noted the importance of turbulence modelling for vortex damping on a 2D square cylinder with bilge keels but also emphasized that RANS should be used as a supplement to, rather than replacement of, strip-theory methods. Ahmed and Guedes Soares (2009) used both RANS and potential flow codes for a maneuvering very large crude carrier (VLCC) hull and obtained a reasonable agreement at high forward speeds, while RANS outperformed the potential code at lower speeds where viscous effects dominate. Noting that size differences between hull and bilge keel and minute timestep make RANS prohibitively expensive for detailed design applications, Piehl and el Moctar (2012) considered a computational domain in the immediate vicinity of the bilge keel, ignored hull curvature, and applied an oscillatory planar rather than rotational motion. It is interesting to note the similarities of this study with those of the flat-plate experiments (Kato et al., 1966; Ikeda et al., 1993) used to generate theoretical codes. RANS has been shown to be capable for simulations involving vortices, wave breakings, separations, resistance, and propulsion. Progress in ship roll simulations, however, remains slow (Querard et al., 2009) and roll remains a difficult phenomenon to capture. Special attention must be given to the domain,

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grid, timestep, numerical scheme, and interface capturing method. These components are discussed in the next section.

4. Numerical Method

4.1 Computational Setup STAR-CCM+ employs an algebraic, finite-volume-based multigrid method to solve the incompressible Navier-Stokes equations in their statistical Reynolds-averaged (URANS) formulation. A PISO correction procedure advances the solution: boundary conditions are used to solve the momentum equations, which are then used to calculate the pressure correction, and the momentum equations are revaluated with the newly found pressure value. Spatial and implicit temporal discretization schemes are second-order accurate in time. Turbulence is evaluated using the SST model, which has been shown to be effective for many hydrodynamics applications (Stern et al. 2012). All simulations are run as two-dimensional such that bilge keel span can be observed independently of other geometrical parameters. STAR-CCM+ being an intrinsically 3D code, a 3D mesh is first generated, then a 2D “slice” of that mesh is used for the computations. Domain sizes mimic the physical domains of the validation studies. The exterior boundaries surrounding the domain are set as free-slip walls. A prism mesh with five layers and a hybrid y+ treatment are set on the no-slip ship body, while a Cartesian-type “trimmed” mesh occupies the flowfield (Wang et al., 2014). Numerical errors are reduced via forced convergence of all normalized Navier-Stokes equation residuals to a minimum value of 10-4. Avalos et al. (2014) noted that roll damping is dominated by wave radiation and viscous effects, though many authors ignored the former as 1) wave radiation is easily predicted with potential codes, and 2) the latter is dominant, especially for hulls with keels. DRDC has successfully predicted wave propagation for models under similar conditions (McTaggart and Stredulinsky, 2004). The focus of this study is therefore on fully submerged conditions. The roll motion is simulated via field functions controlling a linearly interpolated overset mesh, which is advantageous for parametric design studies with large ranges of motion (Stern et al., 2012). Each body herein is subjected to forced harmonic oscillation about its center of gravity with the intent of removing free decay memory effects (Handschel and Abdel-Maksoud, 2014) and reducing restoring moment. The roll motion takes place over several cycles. When the absolute peak values of moment differ by less than 5% between cycles, the motion is simulated over six cycles and the results are averaged over the latter four cycles. For all other simulations, ten cycles are simulated and the results are averaged over the final five cycles. The roll angular motion is given by:

(6) where is instantaneous angular displacement, is average angular amplitude, is average angular frequency ( where is period) and t is time. As STAR-CCM+ motion field functions are velocity-based, Equation 6 is derived with respect to time to obtain:

(7)

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4.2 Equations of Motion The damping forces and moments here are presented based on the work of Schmitke (1978) as presented by McTaggart (2004). The added mass moment of inertia is proportional to the rolling radial acceleration. Since the focus here is on bilge keel damping for fully submerged bodies, radiation and diffraction forces can be ignored and the damping/drag force is assumed to be proportional to the roll radial velocity in quadratic form (Himeno, 1981; Pesman et al., 2001; de Oliveira et al., 2014):

(8) where is the roll added mass coefficient, is the roll quadratic damping coefficient, and is the hydrodynamic roll, which can be isolated from total roll via:

(9) where is the restoring moment given by:

(10) where is the density of the fluid, is the gravitational acceleration, is the displaced area of fluid (the total area of the object in the fully submerged case), and is the centre of buoyancy measured downward-positive from the centre of gravity. Hence, the restoring moment scales linearly with the roll amplitude. Since the bodies are submerged and rotate about their centres of gravity, the restoring force contribution is due only to bilge keel appendages and is therefore very small. Returning to Equation 8, the damping term can be written as a function of the drag coefficient as follows:

(11) where is a constant defined by:

(12) where is the bilge keel span and is the cubic moment arm from the ship centre of gravity to the bilge keel centroid (in the case of triangular bilge keels, /3 from the base of the bilge keel). Equation 8 represents the nonlinear model for predicting bilge keel roll added mass and drag. By applying the model to the moment history curve of a bilge keel, regression analysis via least squares can be used to derive the approximate nonlinear added mass and drag coefficients. Simpler, linear, empirically derived approximations exist and are used herein for comparison. The added mass of a 2D flat plate can be approximated by:

(13) The drag caused by bilge keels tends to be much higher than that of a flat plate in uniform flow due to the oscillating nature of the flow. Hence, several authors published alternative empirically derived drag coefficient equations. One popular method has been proposed by Ikeda (1976):

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(14) McTaggart (2004) developed the following equation from Kato method (1966) and seakeeping trial data for HMCS Nipigon (McTaggart and Stredulinsky, 2004):

(15) The latter correlation is used here for comparison as the current cases are scaled according to the dimensions of the HMCS Nipigon. It is also more relevant to this study than the Kato’s original correlation since it uses the roll amplitude rather than the roll velocity amplitude, which is constant in this study. One final empirically derived method for determining bilge keel drag was also proposed by Ikeda et al. (1976) based on the free-plate (rather than wall-bounded-plate) data of Keulegan and Carpenter (1958) as described by van’t Veer et al. (2012):

(16) where is an empirical increment factor given by:

(17) where denotes the area coefficient of the ship hull section (ratio of area to product of beam and draft). This increment factor also appears in the Ikeda’s bilge keel damping correlation and represents a modification to local bilge keel velocity on account of hull geometry. In the case of a circular, submerged cylinder as studied here, = 1. In order to compare the computed results in this study to published linear added mass and damping terms, Equation 8 can be revisited according to the linear potential theory (Vugts, 1968):

(18) where and are the linear added mass and damping terms, respectively. The following linear expressions are obtained by extracting the Fourier coefficient of the primary frequency over a period :

(19)

(20) Equations 19 and 20 are nondimensionalized as follows:

(21)

(22)

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5. Validation and Verification Numerical validation and verification are extremely important for roll hydrodynamics given its modelling difficulty and large disparities over wide ranges of amplitudes and frequencies (McTaggart and Stredulinsky, 2004); yet many studies neglect, or perform minimal effort in, validating and verifying their results. This may be attributable to the lack of published high-fidelity experimental results. The verification here is accomplished through a systematic parameter refinement by varying the kth input parameter (in the verification process here, grid and timestep refinement) while holding all other parameters constant (Wilson et al., 2006) for the purpose of eliminating numerical errors (Stern et al., 2012). The validation is accomplished through comparison of simulation results with available experimental.

5.1 Bare-Hull Square with Rounded Bilges The first case of this validation exercise consists of a bare-hull rectangular prism of breadth 0.4 m with bilge radii 0.00625B (Yeung et al., 1998; Yu, 2008). The prism undergoes forced harmonic oscillations about its centre of gravity with a displacement amplitude of 5.75° and nondimensional angular frequency ( ) of 0.4. Figure 2 shows an example of the adopted grid topology. Four grids are tested as shown in Table 1. The Overset Cell Size indicates the size of the cells immediately surrounding the square. These cells double in size every ten cells for approximately eight layers. Grid verification calculations for the moment amplitudes are shown in Table 2. Since convergence is shown to be oscillatory, Richardson extrapolation is not possible (Wilson et al., 2006) and grid convergence is stipulated by the magnitude of the difference in the solution from grid to grid. In this case, the finest grid (G1) shows only 0.04% difference from the next coarsest grid (G2) which shows 2.07% difference from the next coarsest grid. Hence, grid G2 is considered acceptable for this solution. The moment due to shear stress shows negligible difference between grids G3 through G1, indicating the boundary layer is well resolved, noting that many studies in the literature neglect to resolve the viscous sublayer. Figure 3 shows the moment history for each grid study in comparison to the 2D URANS data of Yu (2008). The agreement is satisfactory, and generally improves as the Star-CCM+ plus mesh becomes finer.

Figure 2: Geometry and grid for the 2D submerged rectangular prism with round bilges.

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Table 1: Grid dimensions and y+ values for grid refinement studies.

Grid Overset Cell Size Total Cells y+avg

G1 3.125x10-3B 183233 0.26

G2 6.250x10-3B 94611 0.51

G3 1.250x10-2B 50962 1.03

G4 2.500x10-2B 27526 2.15

Table 2: Grid convergence study of moment amplitude and its components for the 2D submerged rectangular prism with round bilges.

Grid G4 G3 G2 G1

M0 0.9306 1.0156 0.9945 0.9950

Ε 9.14% -2.07% 0.04%

Mp0 0.9027 0.9860 0.9651 0.9654

Ε 9.22% -2.12% 0.03%

Mf0 0.0293 0.0296 0.0297 0.0298

Ε 1.21% 0.30% 0.36%

Figure 3: Grid validation study for the 2D submerged rectangular prism with round bilges in comparison with the published data of Yu (2008).

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Table 3 shows the four timesteps considered for the temporal discretization. The solution history maximum Courant–Friedrichs–Lewy (CFL) number based on minimum cell size and maximum velocity in the overset region is reported. While irrelevant to implicit timestepping schemes, this nondimensional parameter provides a gauge of numerical robustness. The convergence results obtained with the four timesteps are displayed in Table 4. Less than 1% variability in moment amplitude is exhibited between the simulations with timestep size reduction factors of two. In addition, Figure 4 also shows that the agreement with the 2D results by Yu (2008) is not significantly altered when using either of these timestep sizes. All simulations are therefore run with a variable timestep set such that CFLmax does not exceed 0.50 in the overset region, i.e., not coarser than T2. An adaptive timestepping scheme allows the timesteps to 1) automatically adjust to different runs, 2) fluctuate within runs to minimize solution runtime and 3) retain accuracy and residual convergence. Adaptive timestepping has been shown to be effective for similar simulations in previous studies (Quérard et al., 2009). Table 3: Timestep and CFLmax values for temporal discretization studies.

Timestep t/T CFLmax

T1 1.31x10-4 0.25

T2 2.68x10-4 0.49

T3 5.35x10-4 0.98

T4 1.07x10-3 1.97

Table 4: Timestep convergence study of moment amplitude and its components for the 2D submerged rectangular prism with round bilges.

Timestep T4 T3 T2 T1

M0 0.9865 0.9947 0.9984 0.9950

Ε 0.83% 0.37% -0.34%

Mp0 0.9571 0.9653 0.9690 0.9654

Ε 0.86% 0.38% -0.37%

Mf0 0.0297 0.0297 0.0297 0.0298

Ε 0.01% 0.04% 0.39%

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Figure 4: Timestep validation study for the 2D submerged rectangular prism with round bilges in comparison with the published data of Yu (2008).

5.2 Circle with Bilge Keels The second case of the validation exercise is that of Miller et al. (2002) for which experimental and numerical data are available. This is a cylinder of diameter 0.897 m with bilge keels of span 0.051 m spaced 90° apart, rolled at a frequency of 0.32 Hz and with amplitude of 15°. This geometry is reproduced in this study, with a progressively refined grid in the vicinity of the bilge keel (Figure 5) in order to capture local flow features. The grid size in Table 5 is defined with respect to the bilge keel span for scaling later. The results indicate that BK2 is sufficient for capturing the desired effects.

Figure 5: Geometry and grid for the 2D submerged circular prism with bilge keels.

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Table 5: Bilge keel region grid dimensions and convergence study of normal force amplitude for the 2D submerged circle with bilge keels.

Grid BK3 BK2 BK1

BK Region Cell Size 0.100B 0.050B 0.025B

FN0 Starboard 11.3810 11.0583 10.9720

Ε -2.84% -0.78%

FN0 Port 11.7199 11.1209 11.0246

Ε -5.11% -0.87%

Moment histories are shown in Figure 6. Only the starboard bilge keel force measurements of Miller et al. (2002) are shown for clarity. The noise of the experimental data was reported to result from fluctuations in angular velocity because of friction, asymmetrical force loads and control system limitations, which yielded 20.8% RMS error; these effects are compounded for slower angular velocities and small-amplitude oscillations (Vugts, 1968; Miller et al., 2002). Miller et al. also reported on the difficulties in obtaining accurate ensemble averaging of PIV data, particularly because of the turbulent cylinder motion (a distinct advantage for CFD). Furthermore, the experimental results varied significantly from cycle to cycle. Finally, regarding the bilge keel geometry, only the span was published in the experimental study, while the bilge keel in the present study is triangular, with a sharp vertex and a base size of 0.20B. In spite of these setup differences and assumptions, the comparison is quite satisfactory. The amplitude of the bilge keel normal force is well captured, as is the phase. Furthermore, the flow-field is well predicted as shown in Figure 7, which depicts the velocity vectors on the keel at the point of maximum angular displacement and zero angular velocity. The model captures the counter-rotating vortices on either side of the keel, as well as some additional weaker vortices not displayed in the published experimental and numerical images. As for the bare-hull case in the previous section, it is assumed that the simulations in the present study are better resolved than those of Miller et al. (2002).

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Figure 6: Bilge keel local grid validation study for the 2D submerged circle with bilge keels in comparison with the published data of Miller et al. (2002).

Figure 7: Velocity vector field (left) in comparison with Miller et al. (2002) published PIV data at t/T = 0.250. Note that left and right images use different relationships between arrow length and flow velocity.

6. Results and Discussion The validated submerged cylinder geometry of Miller et al. (2002) is used herein. The desired results are for a ship with beam 12.766 m, period 10.6 s, and bilge keels measuring 0.2, 0.4, 0.6, 0.8 and 1.0 m in

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span, mimicking the HMCS Nipigon (McTaggart and Stredulinsky, 2004). The bilge keels are scaled down by a factor 0.07 such that the simulated beam equals 0.897 m, as the Miller et al. cylinder. Note that the bilge keel spans herein are still referred to by their full-scale spans, e.g., 0.2 m. Froude scaling ( ) is applied to obtain a period of 2.81 s. Scaling in order to reduce computational effort is common in the literature (Vugts, 1968; Korpus and Falzarano, 1997; Bulian et al., 2010).

6.1 Moment History Flow-field behaviour and subsequent solution convergence differs substantially between simulations of varying amplitude and bilge keel span (see Figures 10-13). The moment history plots indicate that with increasing bilge keel span and especially angular amplitude, the moment history becomes increasingly noisy. Phase lag, in general, seems to increase with bilge keel span and angular amplitude. Studying the same cylinder, Miller et al. (2002) noted strong force history fluctuations at similar conditions while Yu (2008) witnessed sinusoidal disparities for displacement angles larger than 10° on a rotating rectangular cylinder. These observations will be further explored in the following sections.

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a) θ = 2° b) θ = 8°

c) θ = 14° d) θ = 20° Figure 8: Moment histories at various amplitudes.

6.2 Hydrodynamic Coefficients Figure 9 displays the added mass coefficients extracted via nonlinear regression, together with a comparison with flat-plate theory. In general, the added mass increases with bilge keel size and angular amplitude on account of increased acceleration of fluid particles from the bilge keels. This has been observed throughout various published literature. Thiagarajan and Braddock (2009), for example, noted a linear increase in added mass with bilge keel span for a rectangle rolling at 5°, 10° and 15°. The

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relationship seen here is less predictable. At small roll amplitudes, the added mass tends to increase linearly with bilge keel span. At higher roll amplitudes, the relationship between added mass and bilge keel span is more complex. This is the result of complex vortex interactions, which increase as the differential between body and fluid velocities increases. These complex viscous phenomena also cause difficulties for flat-plate calculations, which tend to over-predict added mass for viscous-dominated flows (Avalos et al., 2014) except for the smallest bilge keels. The relationship between the added mass coefficient and the bilge keel size and angular amplitude of motion was also reported to be dependent on frequency (Vugts, 1968; Yeung et al., 1998), though the latter authors observed no increase in added mass with increasing amplitude for angles less than 5.75° for periods ranging from approximately 1 to 4 seconds. While this is not the case here, the sensitivity of the added mass coefficient to the bilge keel size is certainly less pronounced for the 2° cases than for those of higher roll amplitudes.

Figure 9: Added mass coefficients (solid line) compared to flat-plate theory (dashed line). Legend labels correspond to bilge keel span. Figure 10 displays the drag coefficients extracted via nonlinear regression of moment histories in comparison with those predicted using the methods of Ikeda (1976) (referred to as “Ikeda”), Kato (1966) simplified by McTaggart and Stredulinsky (2004) for the HMCS Nipigon (referred to as “Kato”), and Ikeda based on Keulegan-Carpenter number (abbreviated as “K-C”). In general, drag coefficient decreases with increasing roll amplitude on account of exponentially higher velocities at higher amplitudes and increases with bilge keel span since drag increases while angular velocity remains relatively constant. The effect is especially pronounced for the smaller bilge keels (Figure 10a and 10b) as angular displacement increases from 2° to 8°. For the largest keel (Figure 10e), drag coefficient increases as angular displacement increases from 5° to 8°. These observations are the result of a change in vortex behaviour as will be discussed in Section 6.3. Regarding empirical methods, that of K-C tends to underpredict drag coefficient while that of Kato tends to overpredict drag coefficient. As these formulae are based on flat plates (simpler: low damping) and hull forms (complex: higher damping), this is unsurprising. Ikeda’s method performs well for larger amplitudes

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and bilge keels where pressure forces dominate (Kawahara et al., 2012), but tends to overpredict drag for flows where viscous effects dominate. As observed by van’t Veer et al. (2012), Ikeda’s formulation, which is based on twice the bilge keel span, agrees well with wall-bounded plate drag coefficients which were not known at the time Ikeda fitted his expression to the free-plate data. Conversely, the formulation based on Keulegan-Carpenter number tends to underpredict drag. The Ikeda method is therefore still recommended as an inexpensive first guess for initial design stages.

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a) = 0.2 m case. b) = 0.4 m case.

c) = 0.6 m case. d) = 0.8 m case.

e) = 1.0 m case. Figure 10: Drag coefficients compared to semi-empirical methods.

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Figure 11 shows the drag coefficients consolidated as one plot with respect to bilge keel span as the abscissa. In general, drag coefficient increases with increasing bilge keel span. At 2°, increasing bilge keel size first decreases, then increases drag coefficient, though the overall effect is almost negligible. At 8°, the effect is pronounced and highly nonlinear. As amplitude increases past 8°, the drag coefficient increases more linearly with bilge keel size, agreeing with the observations of Thiagarajan and Braddock (2009) at similar angles for a rectangular hull. These results are dissected via flow-field observations in Section 6.3.

Figure 11: Drag coefficients for all cases. Legend labels correspond to roll amplitude. Figure 12 shows the linear damping coefficients. Unlike drag coefficient, linear damping coefficient is not inversely proportional to angular velocity squared and therefore does not decrease with increasing roll amplitude, which helps to clarify trends. Note that the nonlinear terms are more representative of the actual drag values and Figure 12 only exists to help clarify trends and compare results to other authors who presented damping in this form. For the bilge keel of span 1.0 m, additional runs at 5° and 11° are presented. With increasing roll amplitude and bilge keel span, hydrodynamic moment increases rather linearly while shifts in phase (Figure 8) cause dramatic changes in the linear damping coefficient (Yu, 2008). At 14° and 20°, damping trends towards levelling out with further increases in bilge keel size, which Thiagarajan and Braddock (2009) also witnessed. The linear and nonlinear dependencies of damping on bilge keel span and roll amplitude have been suggested by Yeung et al. (1998), Fernandes and Oliveira (2009) and van’t Veer and Fathi (2011) to be dependent on the interaction of vortices with the hull. This will be explored in the next section.

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Figure 12: Linear damping coefficients for all cases. From a design perspective and for the angles investigated, the bilge keels larger than 0.4 m appear to be the best choices for damping. The smaller bilge keels perform well for small roll amplitudes but drag quickly tapers off as amplitude increases. The largest bilge keel yields the best damping, especially at larger amplitudes. Baniela (2008) notes that low-aspect-ratio bilge keels tend to be more effective on bulky ships.

6.3 Flow-field Analysis

6.3.1 Typical Case Most of the hydrodynamic coefficients follow predictable trends as discussed in the previous section. A full rolling cycle for a typical case (s = 0.8 m, θ = 8°) exhibiting those trends is presented in Figure 13. At t/T = -0.25, the bilge keel is at its maximum port displacement amplitude and is stationary. The bilge keel rolls counter-clockwise and draws low-momentum, high-turbulence fluid along an arc, thinning and thickening the hull boundary layer on the left and right sides, respectively, of the bilge keel. A low-momentum, counter-rotating vortex (“vortex 1”) is visible at mid-bilge while fluid begins separating from the bilge keel tip. At t/T = 0, angular velocity is maximised and the separated bilge keel tip flow develops into a large, clockwise-rotating vortex (“vortex 2”). As the bilge keel traverses towards the starboard (right) side, the centre of vortex 2 remains relatively stagnant while the vortex itself grows and stretches along the bilge keel arc, vortex 1 impinges on the bilge keel and flows off the bilge keel tip, and vortices 1 and 2 couple as a counter-rotating vortex pair. This vortex pair formation generates large viscous drag (Irvine et al., 2013) and, in some cases, fluctuating forces as shown in the moment history plots. Once the bilge keel tip slows to a stop at t/T = 0.25, inertia causes the fluid to overrun the bilge keel and spill around the tip in what becomes the inception of vortex 3 (Irvine et al., 2013). The vortex packet of vortices 1 and 2 rolls up into a large vortex rotating in place at mid-bilge. As the bilge keel begins its return to its neutral position at t/T = 0.50, it stretches vortex 3 along its arc path. At the end of the cycle, the bilge keel collides with vortex 2 and vortex 3 entrains itself in the blob, again creating a stagnant, low-momentum vortex pair. These results agree very well with those of Miller et al. (2002) who also studied a circular cylinder undergoing forced roll with similar parameters (period of 3.1 s and 15° amplitude), Aloisio and Di Felice (2009) who studied a 3D frigate hull with large bilge radius undergoing free roll decay with similar

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parameters (period of 3.0 s and initial displacement of 10°), Avalos et al. (2014) who studied a rectangular cylinder undergoing free roll decay at initial displacement of 12°, and Broglia et al. (2009), who studied a 3D vessel undergoing free roll decay with forward motion at various speeds and initial displacements of 5°, 10° and 15°.

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a) t/T = -0.25 b) t/T = 0

c) t/T = 0.50 d) t/T = 0.75 Figure 13: Vorticity plots for the case at s = 0.8 m, θ = 8°. In the surface combatant study of Irvine et al. (2013), vortex pairs appear to propagate from the bilge keel tip to the far-field. The flow-field in the study by Aloisio and Di Felice (2009) appears to follow suit for initial roll when amplitudes are high (10°), but as free-roll amplitude decays, the vortices seem to remain hull-bound. In addition, some disparate trends appear in the hydrodynamic coefficient plots herein. Flow-fields for additional cases are therefore examined.

6.3.2 Smallest Angular Displacement The smallest angular displacement yields strongly nonlinear trends for the hydrodynamic coefficients at various parameters. Figure 14 displays each of the bilge keels investigated rotating with 2° peak

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displacement at two instants of time. For the smallest bilge keel, shed vortex packets are shown to remain hull-bound. As bilge keel size increases to 0.6 m (as well as 0.4 m, which is not shown as the plots are similar), the added momentum of the fluid on account of travelling a longer distance on the bilge keel before separating generates higher momentum vortices that are “thrown” from the tip, propagating downward in a counter-rotating vortex packet and interacting very little with the hull. This is marked by a decrease in drag coefficient (Figure 11). As the bilge keel span increases to 1.0 m (as well as 0.8 m - again, not shown due to similarity), vortices continue to propagate outwards, though the vortices are larger, rotate slower and linger around the bilge keel tip. This tip interaction causes a subsequent increase in drag coefficient and exponential increases in added mass and linear damping coefficient. In the terminology of van’t Veer and Fathi (2011), bilge keel damping varies linearly with its span as long as the generated vortex interacts “well” with the hull. Avalos et al. (2014) concluded likewise, witnessing a critical bilge keel span at which further extensions actually reduced damping due to a reduction of vortex interaction with hull. Hence, both the normal drag force on the bilge keel as well as the interaction of the fluid with the hull on account of bilge keel are important for damping prediction (Himeno, 1981). The Ikeda method and other traditional empirical methods are therefore acceptable for simple cases but not complex vortex interactions (Miyake and Ikeda, 2013), which is verified by the poor predictions for 2° cases in Figure 10.

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a) s = 0.2 m, t/T = -0.25 b) s = 0.2 m, t/T = 0

c) s = 0.6 m, t/T = -0.25 d) s = 0.6 m, t/T = 0

e) s = 1.0 m, t/T = -0.25 f) s = 1.0 m, t/T = 0 Figure 14: Vorticity plots for several bilge keel spans at θ = 2°.

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6.3.3 Smallest Bilge Keel The moment history of the smallest bilge keel rotating at 20° (Figure 8d) shows multiple inflection points during one quarter cycle. Two instantaneous flow-field plots - one set each for the cases at 14° and 20° - are shown in Figure 15 to investigate why this case is unique. At 20° to port, the bilge keel is at rest and a trail of generated vortices can be seen in its wake along the hull for both cases. As the bilge keel remains at rest and the vortices remain rotating in place, a variation in pressure between the fore and aft sections of the bilge keel occurs and causes the inflections witnessed in the moment history plots. This is not unique to the case at 20°; all the simulations for the 0.2 m bilge keel generate multiple vortices during rotation. What is unique is that the vortices generated at 20° are large enough such that the drag on the bilge keel is momentarily reduced as the keel begins accelerating towards neutral. This dip in drag quickly changes once the keel gains sufficient momentum to overcome the vortices. Miller et al. (2002) attributed the force fluctuations in their cylinder simulations under similar conditions to radial velocity fluctuations, which do not occur here for ideal CFD motion. In real conditions, the generated vortices would affect rotational acceleration and hence force.

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a) θ = 14°, t/T = -0.25 b) θ = 14°, t/T = 0

c) θ = 20°, t/T = -0.25 d) θ = 20°, t/T = 0 Figure 15: Vorticity plots for the smallest bilge keel (s = 0.2 m) at θ = 14° and θ = 20°.

6.3.4 Largest Bilge Keel The largest bilge keel (1.0 m) generates interesting hydrodynamic trends as amplitude increases. As shown in Figures 9 and 10, added mass increases significantly. Drag coefficient decreases as roll amplitude increases, with the exception of a small increase in drag coefficient as roll amplitude increases from 5 degrees to 8 degrees. Figure 16 shows two instants of roll motion for roll amplitudes of 5°, 8° and 11°. For the 5° case at t/T = -0.25 (Figure 16a), the vortex generated from the roll towards port has shed from the keel tip and propels from the hull downward, not interacting closely with the newly generated tip vortex. The small amplitude and large bilge keel here propel the vortices from the hull rather than along the arc travelled by the bilge keel as was the case for the bilge keel with span 0.6 m rotating at 2° (Figures 14c and 14d). Conversely, for the case at 8° roll amplitude (Figures 16c and 16d), the vortex pair generated by the bilge keel rotation interacts strongly with the hull. The large vortex shed during the clockwise rotation cycle

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remains relatively stagnant and collides with the bilge keel during the second instant of time, becoming entrained with the newly formed tip vortex. This strong interaction of the vortex with the body is associated with increases in added mass, drag coefficient, phase lag and, subsequently, linear damping. As amplitude increases to 11° (Figures 16e and 16f), the effect is similar though the added angular momentum of the bilge keel helps to dissipate and propel hull-bound vortices tangentially from the tip rather than perpendicularly as witnessed at 5°. As peak displacement angle increases to 14° and 20° (not shown due to similarity), this behaviour continues with larger vortices with subsequently linear increases in added mass and decreases in drag coefficient.

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a) θ = 5°, t/T = -0.25 b) θ = 5°, t/T = 0

c) θ = 8°, t/T = -0.25 d) θ = 8°, t/T = 0

e) θ = 11°, t/T = -0.25 f) θ = 11°, t/T = 0 Figure 16: Vorticity plots for various cases at s = 1.0 m.

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Another interesting observation from the hydrodynamic coefficients is made for the 8° cases when the bilge keel span increases from 0.8 to 1.0 m: drag coefficient decreases slightly, phase lag increases significantly and subsequently, linear damping coefficient increases significantly. As shown in Figure 13, the vortex generated by the 0.8-m bilge keel maintains its centre of rotation along the arc the bilge keel travels and the vortex interacts strongly with the cylinder body. As the bilge keel span is increased to 1.0 m (Figure 16c and 16d), the fluid traversing the bilge keel is forced to travel further in the same amount of time, generating higher velocity fluid towards the keel tip. This added momentum propels the centre of the vortex away from the arc of the bilge keel. The vortices generated by the 0.8-m and 1.0-m keels are small with high angular velocity and large with low angular velocity, respectively, and the larger vortex interacts less with the hull. Once again, bilge-keel-hull vortex interactions cause nonlinear trends in the hydrodynamic coefficients and poorer predictions via traditional empirical methods (Figure 10d and 10e at 8°). 6.4 Error Discussion Numerical errors were limited as much as possible via verification. Regarding the computational setup, only 2D simulations were considered and the isotropic SST model was implemented. In reality, turbulence is highly 3D and anisotropic. Reynolds stress turbulence models were investigated early on but convergence could not be obtained. In addition, the cost associated with the models was deemed too high given the number of parameters to be investigated. When conclusions herein pertaining to vortex dynamics are made, consideration to the isotropic modelling limitations should be considered. The motion was simulated via forced sinusoidal roll with constant amplitudes and period. This is acceptable given the objective of the study, though free roll is more physically sound given memory effects. The model here was scaled down to reduce computational expense. This is very common in the literature but full-scale models should be examined given the thinner boundary layer and higher vortex damping experienced at larger scales (Hochkirch and Mallol, 2013). An adaptive timestepping scheme was implemented here to optimize accuracy-cost ratio. Quérard et al. (2009) used an adaptive timestepping scheme to find the lowest timestep required for their parametric study, then noted improved accuracy upon implementing that timestep as a constant throughout. Overset meshes cause viscous effects at interface, effectively acting as a numerical sink. Therefore, overset meshes are really only appropriate for simulations where convection significantly dominates viscosity, e.g., propeller. In the cases examined here, the only convection was that of the vortices propagating from the body. The overset interface was chosen at sufficient length from the body to avoid interacting with vortices as much as possible, but it still may be causing significant convective dispersion at interfaces, even if cell transitions are good. Extracting hydrodynamic coefficients remains a debated topic in marine simulation. Nonlinear regression of the histories allows for more accurate coefficients to be obtained in comparison to linear simplifications, but even with a quadratic model for the damping coefficient, regressed quadratic equations here were only able to capture 80 - 95% of the measured moment history values. The agreement became worse as complex vortex interactions increased fluctuations in the moment histories. Additional polynomials should be considered in future studies.

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7. Additional Study: 2D Partially Emerged Cylinder While wave damping can be predicted with potential codes and bilge keels dominate damping (Avalos et al., 2014), roll damping for partially emerged hulls constitutes a rather different endeavour. Studying the same circular cylinder herein, Miller et al. (2002) obtained good results with the fully submerged model but less accurate findings when the cylinder was partially emerged due to difficulties in capturing the effects as the bilge keel approaches the surface. Large-scale roll amplitudes present the same problem (Bassler et al., 2011). The rolling rectangular cylinder with round bilges investigated by Vugts (1968) has been used in many free-surface validation studies and is presented here. It is very similar to the initial validation exercise of Yu (2008) from the beginning of this study but with different bilge radii. In order to reduce the diffusive, blurred interface often encountered in volume of fluid (VOF) simulations (Stern et al., 2012), a high-resolution, interface-capturing (HRIC) scheme is enforced at the free surface. As 2D VOF simulation capability is not a built-in function for STAR-CCM+, field functions are employed to enforce a 50:50 air:water domain as an initial condition. Boundaries are chosen to be seven wavelengths from the rotating rectangular cylinder in either streamwise (x) direction to avoid wave reflections. Domain size therefore varies with frequency and becomes quite large with lower frequencies, increasing computational cost substantially. Roll amplitude is set as 0.05 rad and nondimensional frequency varies from 0.30 to 1.41. 7.1 Verification at ω* = 1.41 With free-surface activated, grid refinement in both the x (horizontal) and y (vertical) planes is analyzed independently. First, the case at ω* = 1.41 is investigated. This was the largest frequency investigated by Quérard et al. (2009). Table 6 displays the cycle-averaged total moment amplitudes for grids of varying x-refinement. All grids exhibit less than 1% change between cycles for this frequency. It is recommended that each wave be resolved with at least 20 cells (Yen, 2014). The wavelength at this frequency is approximately 0.63 m. Hence, grid FSX3 is considered verified. Table 6: x-directional grid convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 1.41.

Grid FSX4 FSX3 FSX2 FSX1

x-refinement 0.12500B 0.06250B 0.03125B 0.01562B

M0 3.7136 3.7192 3.7136 3.7140

Ε -0.23% -0.23% -0.02%

Table 7 shows a grid convergence study for the y-directional refinement using grid FSX3. Again, doubling grid resolution yields only small changes in moment amplitude. Grid FSY2 is considered verified.

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Table 7: y-directional grid convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 1.41.

Grid FSY3 FSY2 FSY1

y-refinement 0.01250B 0.00625 B 0.00312B

M0 3.7185 3.7170 3.7153

Ε 0.04% 0.04%

Table 8 shows a timestep convergence study. Once again, the timestep chosen such that maximum CFL is less than 0.50 is chosen. This is important for free-surface simulations using the second-order implicit time integration scheme, which require values from the previous two timesteps such that mesh movement at the air:water interface cannot exceed half the length of the smallest cell. The adaptive timestepping scheme with this constraint is again employed for the emerged simulations. Table 8: Timestep convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 1.41.

Timestep FST3 FST2 FST1

CFLmax 1.00 0.50 0.25

M0 3.7385 3.7153 3.7153

Ε 0.62% -0.05%

7.2 Application to Alternative Frequencies The chosen model is applied for the remaining frequencies at a roll amplitude of 0.05 rad and the results are shown in Figures 17 and 18 as compared to the experimental work of Vugts (1968), the URANS simulations of Quérard et al. (2009) and Sarkar and Vassalos (2000), and the potential flow results obtained by Bishop et al. (1979, 1980) using Lewis and seven-parameter conformal transformation representations. The linear added mass and damping coefficients show strong nonlinear correlations as frequency increases from low values to mid-range values, then increasingly linear dependency at higher frequencies, agreeing with the published studies. Added mass values are higher than those calculated by Vugts (1968), who noted the added mass values were underestimated, but match very well with the modern URANS study of Quérard et al. (2009). Roll damping, conversely, is typically underestimated, which indicates insufficient viscous resolution (Avalos et al., 2009). The results at ω* = 1.41 agree quite well with those of Quérard et al., but a second verification study is undertaken to improve results at other frequencies.

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Figure 17: Linear nondimensional added mass coefficient for the Vugts rectangle at rolling amplitude of 0.05 rad.

Figure 18: Linear nondimensional damping coefficient for the Vugts rectangle at rolling amplitude of 0.05 rad.

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7.3 Attempted Verification at ω* = 0.80 The timestep is revaluated here to improve predictive capabilities. Table 9 shows additional timesteps considered and the corresponding changes in moment. Interestingly, further timestep refinement beyond a maximum CFL value of 0.5 does change differences between solutions rather substantially. In this case, the timestep corresponding to a maximum CFL value of 0.06250 is verified. Table 9: Timestep convergence study of moment amplitude for the 2D partially emerged rectangular prism with round bilges at ω* = 0.80.

Timestep FST2 FST4 FST5 FST6 FST7

CFLmax 0.50000 0.25000 0.12500 0.06250 0.03125

M0 2.5139 2.5027 2.3873 2.3475 2.3392

Ε 0.45% 4.61% 1.67% 0.36%

The mesh is also reinvestigated for this frequency. Since it is uncertain whether increased resolution is required to resolve waves or viscous effects, the entire mesh is doubled in refinement and run at ω* = 0.80 to attempt better damping prediction. The average moment with this new mesh is predicted as 2.4121 kg.m2/s2, a slight decrease from that predicted with the mesh in the previous section (2.5139 kg.m2/s2). Due to project time constraints, additional meshes and timesteps are not investigated. At the time these results were obtained, a training seminar (Yen, 2014) was attended and project goals were revaluated to focus on submerged cases. Hence, the free-surface study is incomplete and requires further investigation.

8. Conclusions

8.1 Findings from Present Work As part of DRDC’s ongoing research into viscous roll prediction, a study was undertaken by the Royal Military College of Canada to simulate via URANS viscous bilge keel roll damping for bodies undergoing forced oscillation and compare those results with traditional empirical methods, such as that of Ikeda. Two-dimensional fully submerged bodies undergoing forced sinusoidal oscillation were simulated to focus on the viscous rather than wave effects of bilge keel damping. Two cases were studied for grid and timestep verification and validation. First, a bare-hull square cylinder was investigated and the global grid and timestep were resolved in comparison to published URANS moment histories. Next, the chosen grid and timestep were applied for a circular cylinder with bilge keels. The grid surrounding the bilge keels was refined and altered until verified and validated against published experimental moment histories and flow-field. The timestepping scheme was then altered to be adaptive for cost savings while still remaining within the bounds set by the verification process. Given the focus on bilge keel viscous damping, the circular cylinder was adopted as the hull model. Bilge keels of span 0.2, 0.4, 0.6, 0.8 and 1.0 m were chosen with angular amplitudes of 2°, 8°, 14° and 20° and a period of 10.6 s. The keels and period were scaled down based on the dimensions of the HMCS

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Nipigon to reduce computational cost. Simulations were run for at least six cycles of oscillation with data averaging over the latter four and, for more complex flows, 10 cycles with data averaging over the latter five. Hydrodynamic coefficients were extracted via nonlinear regression and assuming damping as a quadratic term. Drag coefficient results were compared with empirical methods. Added mass values were compared with flat plate theory. For most cases, with increasing angular amplitude, added mass increased, damping increased, drag coefficient decreased and vortices increased in size while decreasing in angular momentum. Likewise, with increasing bilge keel span, added mass increased, damping increased, drag coefficient increased and vortices increased in size while decreasing in angular momentum. (The increase in drag coefficient with bilge keel span but not amplitude was the result of significantly increased angular velocity, which drag coefficient is inversely proportional to, with amplitude while angular velocity only increased slightly with bilge keel span.) Some exceptions to these trends were observed, notably for the smallest angular displacement as bilge keel span increased, and the smallest and largest bilge keels as angular displacement increased. It was found as in other published studies that damping is strongly dependent upon the interaction of the vortex generated by the bilge keel with the rotating body. Higher damping was obtained when the vortex remained hull-bound. Increasingly complex vortex interactions also caused increasingly noisy moment histories. It is probable that these trend nonlinearities exist for other bilge keels, though relatively few amplitudes were investigated in this study and the nonlinearities may have been bypassed. The Ikeda method was shown to perform relatively well, though not in the presence of complex vortex interactions (e.g., s = 1.0 m at low angles) or viscous-dominant flows (e.g., all cases at 2°). Even still, the traditional empirical methods are excellent for initial design considerations when URANS or experimental methods prove too expensive.

8.2 Recommendations and Future Work The equation to describe angular displacement herein is based on the sine rather than cosine trigonometric function such that its derivative, the angular velocity, takes the cosine form. Hence, at start of simulation, the bilge keel is in its neutral position where angular velocity is at a maximum. It may be prudent for numerical stability for future simulations to use the cosine trigonometric function to define angular displacement such that at start of simulation, fluid is stagnant and velocity ramps up over a quarter period. The Ikeda method performed well for most cases, which involved simple submerged cylinders. Future work should include both free-surface and proper ship hull. Modern versions of the semi-empirical methods that account for these factors should also be tested. Forward-speed cases present an added level of complexity as vortex structures tend to dissipate and bilge keel damping reduces while lift damping increases. Future studies should incorporate 3D shape and forward speed. Alternative motion schemes to overset meshes should be attempted. A rotating domain is ideal, if possible, though the domain size can be quite large once free surface is introduced. Based on the information obtained here and in the Star-CCM+ Virtual Tow Tank course (Yen, 2014), the following recommendations are made for free-surface simulations:

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Grid refinement is a function of rolling frequency. Wave amplitudes and wavelengths should each be resolved with a minimum of 20 cells. Therefore, custom grids for each frequency are necessary.

CFL upper and lower limits for the HRIC numerical scheme should be set to be much larger than the maximum desired CFL value such that the scheme is always enforced at the free surface.

Waves should be artificially damped via mesh coarsening as they approach the streamwise boundaries.

References Ahmed, Y., Guedes Soares, C., 2009, "Simulation of Free Surface Flow around a VLCC Hull using Viscous and Potential Flow Methods", Ocean Engineering, 36(9-10), pp. 691-696. Aloisio, G., Di Felice, F., 2006, "PIV Analysis around the Bilge Keel of a Ship Model in Free Roll Decay", Proceedings XIV ConvegnoNazionale A.I.VE.LA, Rome, Italy. Avalos, G.O.G., Wanderley, J.B.V., Fernandes, A.C., Oliveira, A.C., 2014, “Roll Damping Decay of a FPSO with Bilge Keel”, Ocean Engineering, 87, pp. 111-120. Bangun, E.P., Wang, C.M., Utsunomiya, T., 2010, "Hydrodynamic Forces on a Rolling Barge with Bilge Keels", Applied Ocean Research, 32, pp. 219-232. Baniela, S.I., 2008, "Roll Motion of a Ship and the Roll Stabilising Effect of Bilge Keels", Journal Navigation, 61(4), pp. 667-686. Bassler, C.C., Miller, R., Reed, A.M., 2011, "Considerations for Bilge Keel Force Models in Potential Flow Simulations of Ship Maneuvering in Waves", Proceedings 12th International Ship Stability Workshop, Washington, D.C. Bassler, C.C., Reed, A., Brown, A.J., 2010, "A Method to Model Large Amplitude Ship Roll Damping", Proceedings 11th International Ship Stability Workshop, MARIN, Wageningen, The Netherlands. Bassler, C.C., Reed, A.M., Brown, A.J., 2011, “A Piecewise Model for Prediction of Large Amplitude Total Ship Roll Damping”, Proceedings of the ASME 30th International Conference on Offshore Mechanics and Arctic Engineering, Paper OMAE2011-49494, Rotterdam, The Netherlands. Broglia, R., Bouscasse, B., Di Mascio, A., Lugni, C., 2009, “Experimental and Numerical Analysis of the Roll Decay Motion for a Patrol Boat,” ISOPE-I-09-412, Osaka, Japan. Bulian, G., Francescutto, A., Fucile, F., 2010, "An Experimental Investigation in the Framework of the Alternative Assessment for the IMO Weather Criterion", Proceedings HYDRALAB III Joint User Meeting, Hannover, Germany. Chan, H.S.Y., Xu, Z., Huang, W.L, 1995, "Estimation of Nonlinear Damping Coefficients from Large-Amplitude Ship Rolling Motions", Applied Ocean Research, 17, pp. 217-224.

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