Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Viscous Roll Predictions of a Circular Cylinder with Bilge Keels Ronald W. Miller, Joseph J. Gorski, David J. Fly (Naval Surface Warfare Center, Carderock Division) Abstract The roll motion of a ship is largely influenced by viscous effects. This includes the drag on the hull form as it rolls and the flow separation from the bilge and keel where subsequent vortex formation accounts for a large amount of the roll damping. Bilge keels will significantly increase the damping of roll motions as well as generate a lift force if any forward motion of the ship is present. Predicting roll effects analytically has been problematic because of the significant viscous effects. Roll motions are an ideal area to pursue viscous calculations methods including the Navier- Stokes equations. Yeung et al (2000) showed results recently in this regard, though limited to two- dimensions. The current effort demonstrates some RANS calculations to simulate roll motions of a These are three- dimensional calculations, which will provide an assessment of the accuracy that can be obtained with RANS codes for predicting viscous roll damping effects. This assessment is performed by comparing RANS solution data with measurement data obtained from the experiment recently carried out in the Circulating Water Channel at the Naval Surface Warfare Center, Carderock Division. cylinder with bilge keels. Introduction The roll response of a ship is an important consideration in its design. Roll motion limits ship operability, affects crew performance and ship habitability, and affects dynamic stability and ship capsize. Viscous related effects have a large influence on the roll motion of a ship. These include the drag on the hull form as it rolls and the flow separation from the bilge and keel where subsequent vortex formation accounts for a large amount of the roll damping. Bilge keels will significantly increase the damping of roll motions as well as generate a lift force if any forward motion of the ship is present. The prediction of ship roll motions has been difficult because of its nonlinear nature as well as the strong dependence on forward speed. Current ship motion prediction methods rely primarily on strip-theory potential-flow based methods. Roll effects have largely been included in such predictions based on empirical results, flat plate lift and other simple theories. Such techniques can provide aspects of roll motion, but are not adequate for correctly predicting roll effects because of the viscous effects, even with bilge keels present, and dependence on specific hull geometry. As demonstrated by Sarpkaya and O'Keefe (1996) bilge keel damping is affected by the vortices shedding from the edge of the bilge keel and the use of damping coefficients from flat plate tests in a free stream are not necessarily accurate for wall bounded bilge keels. Consequently, these methods must be supplemented with empirical information and much effort has been directed at developing coefficient- based approaches for roll prediction. Himeno (1981) summarized much of the development of these coefficient-based methods and it is not clear that they have changed substantially to the present. These approaches have been used successfully when applied to hull forms for which they were developed. However, these methods required new data when applied to new hull forms as demonstrated by Blok and Aalbers (1991) for a high-speed displacement hull form. Additionally Liut, et al (2001), needed to tune the viscous model used in the motion prediction program LAMP to match roll-damping characteristics of CG47 based on experimental roll data. To remedy such deficiencies modifications to the basic model may be needed as well as new test data requiring numerous forced roll or roll extinction tests at model- scale to define coefficients that describe the roll motion. Additionally, model-scale coefficients may not relate well to full-scale behavior due to the differences in Reynolds number. As revolutionary new naval combatants are considered, where existing databases do not provide the necessary design

information in general (Rood, 2000), there is a strong desire to have better computational capability for ship roll motions as the Navy progresses to a more computational based design paradigm in general. Viscous flow computations can provide roll motion information. However, due to the need to evaluate ship motions in a variety of sea states under a large number of headings viscous flow calculations will not replace traditional strip-theory in the foreseeable future. However, viscous flow calculations, in particular Reynolds Averaged Navier-Stokes (RANS) codes, do have a role to play as considerable advancements have been made in the prediction of ship flows with them (Gorski a&b, 20011. RANS predictions have already been demonstrated for a destroyer model with propeller shafts, struts, rudders, and propellers for straight ahead and a restricted maneuver in the horizontal plane (Kim, 20014. This prediction demonstration, provided the complete flow field around the hull showing the ship motion predicted as a result of the propeller induced thrust and turning rudder. Such calculations can be run for a few conditions to help understand the flow field that may be causing a particular force distribution on the hull and appendages during a maneuver or other ship motion. Another area where RANS codes can contribute is in supplementing conventional strip-theory approaches, which rely on empirical databases for the damping coefficients. As already mentioned when calculating nonconventional hull forms, traditional roll damping coefficients may be inadequate requiring new model tests to replace them. Rather than model tests RANS predictions may be able to fulfill the same role. Viscous computations of roll motions have been reported in the past. Yeung, et al. (1998) presented two-dimensional computational results for a rectangular cylinder. Later, Yeung et al (2000) added bilge keels and demonstrated that viscous flow calculations could adequately be used for obtaining coefficients. These methods consisted of a Free- Surface Random Vortex Method and a Boundary- Fitted Finite Difference Method, but were only applied in two-dimensions. Korpus and Falzarano (1997) applied a finite analytic RANS calculation to a rotating square, demonstrating that the RANS code could be used to obtain roll motion coefficients, but again only in two-dimensions. Full three- dimensional RANS results have also been demonstrated for ship hull forms undergoing roll (Kim, 2001), but without data comparisons or the free-surface present. These calculations demonstrate that RANS may fulfill the role of traditional forced roll tests if adequate accuracy can be demonstrated. The current effort demonstrates some RANS calculations to simulate roll motions of a 3-D cylinder with bilge keels. Measurement data from experiments performed at Naval Surface Warfare Center, Carderock Division (NSWCCD) is used to validate the RANS solutions. The experiment provides data for a large set of test conditions available for numerical simulations, including the actual roll motion of the body. These conditions include a range of frequencies and amplitudes of forced roll motion, model sizes, forward speeds, and ballast conditions. Case-by-case comparisons of calculations with measurements for a set of these conditions are shown to be good. Additionally, the use of the RANS solver with the body undergoing an ideal sinusoidal roll motion demonstrates the effects of forward motion and scale on the bilge keel force. Experiment Experiments were recently carried out in the Circulating Water Channel (CWC) at NSWCCD. This recirculating water channel has a 9.1 x 6.7 x 2.7 m. (30 x 22 x 9 ft) test section with a free surface. Figure 1 shows the test section with the largest cylinder in place. Flow velocities into the CWC test section ranged from O to 2.0 m/s (4 knots). The cylinder mount allowed immersed and various emerged elevations. Under conditions of zero flow at partial submergence, keel generated waves reflect back from the test section sides. Wave absorbers and multiple short duration experiments prevented contamination from reflected waves. Figure 1 Test section with the largest cylinder Four different cylinder configurations are tested. These consist of a smaller, 0.48 m (19 in) diameter cylinder was tested with 0.025 m (1 in) and 2

OS9 ~ IS rr `34.9 rl. 1 l20.~ i: - q~ ~ - it Figure 2 Roll Cylinder Dimensions (Large Cylinder) 0.051 m (2 in) wide bilge keels and a large 0.897 m (35.3 in) diameter cylinder with 0.051 m (2 in) and 0.102 m (4 in) wide bilge keels. The basic cylinder dimensions appear in Figure 2 for the larger cylinder. The twin bilge keels (located 90 degrees apart on the circumference) run along the cylinder's constant diameter section. The model is forced to roll about its longitudinal axis at various frequencies (0.08 - 0.64 Hz) and amplitudes (15 - 60 degrees). The actual motion varies from an ideal sinusoidal variation of roll angle because of friction, asymmetrical force loads, and control system limitations. Roll position is measured by resolver sensors that are sampled at 100 Hz. The position at any time is quite close to a sinusoidal variation (Figure 31. However the angular velocity is far from ideal for the slowest roll oscillations (Figure 4~. There is also significant cycle to cycle variation. At higher amplitudes and frequencies, the angular velocity is closer to the sinusoidal ideal with variations consistent cycle to cycle(Figure 5~. RMS Error / Amnlitr~rt~ = n Oo/^ to in 1 Oscillation Cycles (12.5 see period) Figure 3 Slow Roll Displacement Force measurements are taken on a central 0.61 m (2 ft.) section of both the port and starboard bilge keels (Figure 2~. They are instrumented with strain gauge flexures which are sampled at 100 Hz.. O04,] Do. _ , j5 2 in __. i_,_ 0.2 r 0.15 ~ 0.1 Oh C 0.05 Ct o ~5 4, ~ -0.05 00 0 4) 1 -0.15 -1 RMS Error / Amplitude = 20.8% L2EO-A3F1~132 A, Measured , Sinusoid Van ', I, I · 1 1.2 1.4 1.6 1.8 2 22 2.4 2 6 2.8 3 Oscillation Cycles (12.5 see period Figure 4 Slow Roll rate RMS Error/Amplitude = 6.1% Oscillation Cycles (3.12 see periods Figure 5 Faster Roll Rate The two dimensional, flow field near the port bilge keel is measured using Particle Image Velocimetry (PIV). Camera and light sheet generating components are attached to the exterior of the rolling cylinder (Figure 6~. The support geometry and fairings are designed to have minimal impact on the sampled flow field and port keel force measurement. A fiber optic cable brings light from two stationary Dye Lasers out to the cylinder. Electrical cables connect the camera to power supplies and an image acquisition computer. 3

Autocorrelated images (2048 x 2048 pixels) are recorded at a rate of 4 Hz. Approximately 300 are collected for each cylinder/roll condition. The PIV images (Figure 7) also provide free surface deformation and air ingestion information for conditions when the keel interaction with the free surface is strong. CYLINDER ROLLING AT FREE SURFACE Figure 6 Particle Image Velocimetry (PIV) Camera and Light Sheet Components The small cylinder was tested under 55 different conditions. This included variations in cylinder elevation. freestream current. keel width motion amplitude and frequency. The large cylinder was tested under 70 different conditions. Figure 7 PIV Image showing bilge keel interaction with free surface Governing Equations and Numerical Implementation Math Mode/ and UNCLE To compute the viscous flow field the incompressible Reynolds Averaged Navier-Stokes equations are solved using the Mississippi State University code UNCLE, (Taylor et al 1991, 1995~. The UNCLE code is one of two RANS codes used for the ONR Surface Combatant Accelerated Hydro S&T Initiative to provide documented computational solutions for innovative propulsor/hull concepts of interest for DD-2 1 and beyond, e.g. Gorski et. al.~2002~. The equations are solved using the pseudo-compressibility approach of Chorin (1967) where an artificial time term is added to the continuity equation and all of the equations are marched in this artificial time to convergence. The UNCLE version used is 1St order accurate in time. Subiterations are necessary for convergence of the continuity equation at each time step. For the present calculations a third-order upwind biased discretization, based on the MUSCL approach of Van Leer et al. (1987), is used for the convective terms. The equations are solved implicitly using a discretized Newton-relaxation method (Whitfield and Taylor, 1991) with multigrid techniques implemented for faster convergence due to Sheng et al (1995~. The turbulence model used for the present calculations is a k-c model. The RANS equations are solved in an absolute frame of reference with a rotating body and deforming volume grid. An important factor in being able to compute and evaluate the operating conditions of interest is the implementation of a parallel version of the UNCLE code. The code uses MPI for message passing due to its portability. To run in parallel the computational grid is decomposed into various blocks, which are sent to different processors. Load balancing is obtained by making the blocks as equally sized as possible. More details of the solver can be found in the various references provided. Boundary conditions - surfaces Surfaces bounding the computational fluid volume require boundary conditions. For the RAN S calculations, the CWC entrance, exit, and side walls are replaced by farfield surfaces at least one body length forward and at least two to the sides and aft. On these surfaces characteristic boundary conditions are applied. Shown in Figure 8 is the port side of the extent of the immersed body configuration, but the grid wraps around the starboard side for the 3D 4

calculation. For the emerged cases, the fluid only extends to the mean free-surface level, where either a zero Froude number or linearized free surface boundary condition has been used. The last surface bounding the domain is the test model, where the noslip boundary condition is applied. This surface rotates at a specified angular velocity. The angular velocity, d'0 is specified from actual experimental dt data or from the ideal sinusoidal function Amcos(mt). Figures 4 and 5 show the actual experimental angular velocity versus the smooth analytic velocity. At each time step the body is ,1as rotated the amount ^0B = "v ~ about its dt longitudinal axis, where At is the computational time step. The calculated wall velocity required by the noslip boundary condition is then imposed. The cylindrical emerged body has been extended downstream to the outflow boundary because of difficulties with the free surface boundary conditions for the actual flat back of the cylinder. Flow region - volume grids Structured grids are used for the present calculations. Figure 8 shows the point clustering at body walls and at the bilge keels. Because the angular velocity of the test body is time dependent, the discretization of the computational fluid volume changes with time. However the basic grid topology for each configuration, in an immersed or emerged ballast condition, will remain the same for all time. The initial grids were created using GRIDGEN. The flow region is decomposed into equal sized blocks, i.e. an equal number of cell volumes per block, for load balancing on a parallel processor. For the immersed body cases the fluid domain is decomposed into 84 blocks each with dimensions 33x33x33. This region consists of about 3 million grid points. There are 193 points in the axial direction and 65 in the radial direction. Circumferentially there are 193 points. At every time step all grid points rotate about the longitudinal axis an amount AB = ABB SO that the entire region rotates as a solid body. In this region all volumes remain constant for all time although their location in the absolute frame is changing. Figure 8 Grid and extent of computational domain for immersed body For the emerged configuration the domain is decomposed into 56 blocks each with dimensions 33x33x33. This region contains about 2 million grid points. Circumferentially there are 129 points. The tree surface must always be located on the computational parameter j = jmax surface. Since the constant j-surface at the free surface does not rotate, computational blocks containing this surface must deform in time and must be modified differently. For each of these blocks a dummy block has been created whose j-varying lines extend above the free surface level for any amount of specified rotation. These dummy blocks, shown as blue grids in Figure 9, are rotated an amount, /~0 = /~eB, at every time step as before and the cell volumes remain constant. The computational blocks are then created from these blocks. The j-varying curves in the dummy blocks are cut at the free surface. The points remaining below the free surface makeup the curves used to interpolate a new set of points with a hyperbolic tangent distribution with specified arclengths at the ends. The coefficients for the distribution are calculated according to the algorithm found in Thompson (1985). In this region, /~0 charges in space resulting in expanding and contracting cell volumes. Figure 9 shows this process. The upper figures show the computational grid and the dummy grid at some time. The lower two figures show the resulting rotated grids at a later time. The rotation amount used here has been exaggerated for purposes of illustration. Blocks not bounded by the free surface are modified as in the immersed case. At every time step, UNCLE calculates grid velocities and updates all volume quantities. s

if Test Cases (Actual Roll Motion) For each experimental configuration, measured time histories of both roll motion (angle and angular velocity) and flow quantities are recorded. This data is time averaged to eliminate some of the noise over small intervals and then ensemble averaged to create one period of data. The one period of measured angular velocity,—, is dt used for input to the RANS solver for all time and the Figure 9 Grid rotation for the emerged body same one period of measured flow results is used for validating the RANS solver solutions. Force comparisons are made of the measured normal force on the individual bilge keels with the RANS force for one period of roll. The RANS force is found by integrating the calculated pressures and viscous stresses along the corresponding section of the port bilge keel computational surface. For some experimental cases data for only one bilge keel is available. For all cases, differences (sometimes large) between the port and starboard measurements occur. For the small amplitude submerged cases, one would expect the forces on the port and starboard bilge keels to be the 6

same. For the larger amplitude cases, where interference between the keels could be present, the force magnitudes should only be 180° out of phase. Similarly for the emerged cases, the force differences between the two keels, should only be evident in the phase. Flow field comparisons are also made of the RANS velocity vectors with experimental velocities created from PIV data. The calculated velocities are interpolated to match the time and position of the measured velocities. Overlaid or side-by-side animations or still pictures at constant times can be created from this data. The large cylinder (D = 0.897 m), small bilge keel (w = 0.051 m) configuration is used for all of the calculations. From the large set of experimental configurations a subset is chosen to simulate using the RANS code. Calculations are done for the immersed and emerged body configurations with zero and forward speeds. Two amplitude-frequency combinations are chosen from the available data for comparison purposes. Both low and high-speed amplitude roll velocity cases with good experimental roll motion and force measurements are desired as well as a combination that spans the range of ballast conditions and forward speeds tested. The lowest frequency tested in all of the configurations for comparisons in the calculations is .32 Hz. Experiments with this frequency and the relatively small 15° amplitude provide a good set of low speed data for all speeds and ballast conditions. Experiments for a immersed body with amplitude 40° and frequency 40 Hz. are used for higher speed compansons. Computational time steps are chosen so that 360 time steps per period results. Six subiterations are used to get a converged solution at each time step. Using 30 subiterations does not change the calculated force results. The roll motion starts abruptly with its maximum velocity, at the position of zero roll angle. At this time the port side bilge keel is moving upward. At a quarter of a period, the bilge keel is at its maximum roll angle where the velocity is zero. The roll accelerates down past the zero roll angle position at a half of a period. The motion continues downward to its lowest point at three quarters of period and then returns to its starting position at the end of the period. For forward motion cases, the forward speed solution is converged, to set up the boundary layers on the cylinder, before the roll motion started. The force generally becomes periodic after about 5 or 6 periods. The calculation takes approximately 24 hours for 10 periods of roll on an IBM SP3. immersed Body Comparisons Figures 10-13 show comparisons for the immersed body. Each figure shows the calculated and measured normal force on the bilge keels versus time for one period of roll motion. The different angular roll velocities at which the model is forced to roll is also shown in the figures. Figures 10 and 12 show the results for zero forward speed, basically a 2D flow, and Figures 11 and 13 show the results for 1.0 m/s (2 kts.) forward speed. The figures show that the RAN S calculations predict both the magnitude and phase of the measured data accurately and the significantly larger forces of the high amplitude roll as compared to the low amplitude. Also predicted are the highly oscillating variations in the force data. The rapid acceleration and deceleration of the actual roll motion causes the sharp peaks in the force data. Comparing the motion of these two cases shows that the higher speed roll case has smaller oscillations. Because of the sharp peaks, case to case comparisons are difficult, but given accurate roll motion data to use in the RANS solver, very good validation comparisons can be made. 20 10 He Go' o IL -10 _ _ 1 ~ Experiment - Port _ RXPNesment-Staboald —~ ~ r 20 1 1 1 1 -1 0 0.25 0.5 0.75 1 AT Figure 10 f = .32 Hz., A = 15°, U = 0 m/s., Immersed 7 o

~1 10 At - o o -10 - Experiment Starboard | RANS | deeds I _~: it\ _~ As- ~~W -0.S =~ ~= -1 0.75 1 0.25 0.5 AT 0.5 US In O ~ Figure 11 f = .32 Hz., A = 15° U = 1.0 m/s (2kts), Immersed 7OI 35 - z - ~ 0 o u" -35 -70 0 Experiment - Polt it- Add'- ~ _ / ~e 0.75 1 0.25 t).5 AT 2 - u, - u Figure 12 f = .40 Hz., A = 40°, U = 0 kts., Immersed 7OI 35 z o o IL -35 _ _ 4 == Expenme~t-Starboard RANS Lab ~ 0.: !5 0 5 0.75 . VT 7 O Figure 13 f = .40 Hz., A = 40°, U = 1.0 m/s (2 kts), Immersed Figure 14 shows a comparison of the calculated velocity vectors with the velocities obtained from PIV data, looking aft at the port bilge keel, for the zero forward speed case where the roll motion amplitude is approximately 15° and its frequency is .32 Hz. The calculated velocity was interpolated to the PIV point locations. The left column shows the experimental data and the right column shows the RANS solution data. The time sequence begins at (a) t/T = -0.125 where the body is accelerating in the counterclockwise direction. At this time, we see large flow velocity passing over and around the keel, from top to bottom, creating the large separated zone behind it. The next row shows the results at (b) t/T = 0.0 where the maximum counterclockwise velocity of the body occurs. At this time the flow at the bilge keel tip has reduced and the vortex center has moved down as the vortex is elongating. At (c) t/T = 0.125 the body is slowing down. Here the flow on the top and the tip of the bilge keel continues to reduce, but the trailing vortex center position remains approximately the same. As the counterclockwise motion stops ~ (d) t/T = 0.250) we see that the cross flow impinging on the bilge keel starts to reverse. This creates an upflow on the bilge keel. Although the large separated flow above the bilge keel is still present, the upflow around the keel creates a counter rotating vortex on the topside of the keel. Overall the predicted and measured flowfields have very similar behaviors with similar magnitudes for the velocities. The calculations predict the location of the trailing vortex fairly well, but the predicted location of the new counterrotating vortex appears to be somewhat off as shown Figure 14d. Some of the differences are because of the difficulty in obtaining accurate ensemble averaging of the PIV data at the flowhield locations in time, particularly because of the unsmooth motion. Because of this it is unrealistic to expect that the experiment and the computations should compare one-to-one. However, this demonstrates significant progress in using unsteady PIV data in a time dependent manor for comparing with RANS solutions. 8

Experiment _ _ _ _ _ ~~ItIJIt~ fIll ~ J ~ I ~ ~ I I J I dlJItlIIIJI l I I I I ~ I ~ ~ ~ I I I 1 1 1 1 1 1 1 ~ 1 1 1 ~ ~ ~ t ~ I ~ I I I I I ~ \~\ t I ~ l l I l l I ~ ~~~\ _-~\ ~1' -;;~.~; , ''''',/1 7 '''/~y 1 'l,/~/~t 'IJI/lI33~! . I l l lull I I l I ~ I 1 1 1 1 l 1 t I t I ~ 1 I l ~ I ~ 1 I ~ l I l 1 till t1 I l I l l 1 I l , I ~ 1 l,,,lt l ~ , I l 1 l l , I ~ 1 l ~ , I 1 1 ~ ~ l I 1 1 !!'~i!l!i l ~ ~ !!! !1 ~: tt\~\ "'1~11 tjlll\~ t\~) · ~ ~ ~ \\s ~\ RANS '.3iiii,,iTiT~ `'-ii,,i,iii5 ' ~ " " 5 Itit iJ~ {,,,~ ~ (a)t/T= 0.125 (b) t/T = 0.0 9 ;. .1!!! , , , , , , , ~ ~ 1 i, l ., . . I ~ ~ I t I ~ t t t I ~ ~ 1 it l ~ ~ ~ t t`~il 1 `~1 r ~ ~ ~\~I ~,\( 1 _ vY , 1 ~ ~~~ 1 ~ ~ ~ ~ - ~ ~ \~3,1 ,,' `~t ~1 -~/2 'f] ~ t ~ ~ ~ `,

I ~ 1 `~! \~` \~! 1. \~\ ~ ~ ~ `~\ (c) t/T = ~ 125 (d) t/T = 0.250 1~,: Figure 14 Velocity vectors: Actual roll motion, PIV measurements vs. RANS solution (experiment on left) Emerged Body Comparisons Figures 15-17 show the results for the emerged body moving forward at 0, 1.0 m/s (2 kts), and 1.5 rn/s (3 kts.) The amplitude of the roll motion is 15° and the frequency is 0.32 Hz. The water level is approximately 0.108 m (4.25 in.) below the longitudinal axis of the cylinder. The linearized free surface boundary condition is used in these calculations. For the O forward speed case there is a significant difference between the measured port and starboard bilge keel forces. The starboard force data is suspect in Figure 15, since it should be more symmetric. Physically they should be the same, albeit out of phase with each other. The computation predicts the port force very well, but this is a significantly lower then the measured starboard force. For the nonzero forward speed cases only starboard 10

data is available for comparison and at 1.0 m/s (2.0 kts.) the prediction is less than the measured starboard force, but at 1.5 m/s (3.0 kts.) it compares well. In these forward speed cases the starboard force data appears more symmetric as physically expected. There does appear to be a drop in the force from 0 to 1.5 m/s (3 Its.), but with the port-starboard difference in the data as well as the oscillating variation, it is hard to draw definitive conclusions. Figure 18 shows the computed free surface elevations at the time t/T = 0.0 for the 1.5 m/s (3kts) case. Only very small port-starboard differences in the wave elevations are seen over a roll period at this amplitude. High amplitudes could not be run with this RANS code due to difficulties associated with the bilge keels nearing the free surface. 10 ,,, s c Figure 15 f = 32 Hz., A = 15° U = 0 kts. Emerged on 1n __ _ l 1 _~_: ZocL:N -20 , 0 0.25 0.5 0.75 1 AT Figure 16 f = 32 Hz., A = 15°, U = 1.0 m/s (2 kts), Emerged 20 ~ Expelunenl - Starboard RANS dew 1 ~` ~ ~ ._. f O ~-10 ~ ~: ~ ~ -20 1 1 1 , 0 0.2fi 0.5 0.75 1 AT no V no Figure 17 f = 32 Hz., A = 15° U = 1.5 m/s (3kts), Emerged ~ I / Hi '''I'd il.// 1,, ~ `; J ~ ~ Hi Figure 18 FS contours, f = .32 Hz., A = 15° U = 1.5 m/s (3kts), Test Cases (Ideal Roll Motion) Once convinced of the good performance of the RANS code for predicting bilge keel forces for forced roll body tests, the solutions are used to compare differences in force data due to varying roll test parameters. These tests use an ideal sinusoidal roll motion, resulting in smooth force results. Figure 19 shows the comparison of the normal force on the port bilge keel for an immersed body rolling at .32 Hz. and amplitude 15° for a range of forward speeds. The results show the maximum force decreasing as the forward speed increases. The shape is also changing. At low speeds the force curves show narrower and steeper peaks. 11

10 5 ~ _ l U = 0.0 misdo 0} - - - U = 1.0 miss dial U-~.5~s(3~5.\ ~ZD~ ~ ~ 0.: 25 0 5 0. 5 AT indicated by very small mean value, aO, in the decomposition. 0.5 - O - 3 _0.5 Figure 19 RUNS Solution: f = .32 Hz., A = 15°, Fwd. Speed Comparison (Smooth Roll) Traditionally the bilge keel force is decomposed into an inertial term proportional to the acceleration (IS' sine harmonic) and into drag terms proportional to the rotational speed and speed squared (dominated by 1St and 3r~ cosine harmonics). The Fourier decomposition of the force histories shown in Figure 20, shows that the first harmonic sin and cos terms have about equal contributions at a particular speed, but reduce as the forward speed increases. At zero forward speed, a significant contribution from the third harmonic cos term is seen. As the forward speed is increased to 2.0 m/s (4 kts.) this harmonic disappears as the force curve approaches a purer sine wave with lower amplitude. 1 , i i I , i , ' . . ~ ~ · Is a,, cosco'~b~nlo' a 0 ~~ bh dill conlrib~nion5 ! ~ ! . ~ 4 Is ~ Is corm anion ! ~ a 4 - As b - ' oot~lbutia~s ~ ~ ~ I—- —t ~ ~ ~ , i _ _ I ~ ~ . ~ L __ ~ . ail_., .. - i. ~~ .. ......... ...... .. . ~ , . ~ . o 1 k 5 Figure 20 Fourier coefficients 10 Figure 19 also shows that magnitude of the force does not depend on the direction of rotation, which indicates that for these conditions there is little or no influence on the port bilge keel force due to the starboard bilge keel and vice versa. This is also Figures 21 and 22 show the calculated vorticity for the O and 2.0 m/s (4 kts.) forward speed cases at different times in a partial period of counterclockwise rotation. The vorticity being created at the bilge keel tip rotates in a clockwise direction (indicated by red contours), opposite to the rotation of the bilge keel. This vortex is also shown in Figure 14. The counterclockwise vorticity (indicated by blue contours) is the disturbance remaining from the previous clockwise motion of the bilge keel. Figures 21a and 22a show the bilge keel at the tlT = -0.125 (t/T = 0.875~. Since the body is accelerating at this time, the inertial force (first sin harmonic) adds to the drag force (first cos harmonic) resulting in a peak in the total bilge keel damping force. These dominating contributions produce a maximum force somewhere in the period between the maximum acceleration and maximum velocity. Additional drag occurs in the zero forward speed case where the left over "interference" vorticity (third cos harmonic) is present. This vorticity could possibly be contributing to the large velocity vectors at the bilge keel tip as shown in Figure 14a. This vorticity (drag contribution) is not seen in the 2.0 m/s (4 kts.) case, most likely because the forward speed has swept it downstream out of the way by the time the bilge keel rotates back. 12

'1 (a) t/T = -0.125 it' (c) t/T = 0.125 ~ : (b) t/T = 0.0 / (d) t/T = 0.250 Figure 21 RANS Vorticity: Ideal Roll Motion f = .32 Hz., A = 15°, U = 0.0 m/s (a) t/T = -0.125 fit ( c) t/T = 0.125 hi.... <{I .. ( ,__ I.. At' (b) t/T = 0.0 hi" (d) t/T = 0.250 Figure 22 RANS Vorticity: Ideal Roll Motion f = .32 Hz., A = 15°, U = 2.0 m/s (4 Its.) 13

Figures 21b and 22b show the bilge keel at t/T = 0.0 (tlT = 1.0). At this time, the acceleration is zero and the velocity has obtained its maximum value. The inertial forces are zero here so that the bilge keel force is entirely due to the drag terms. Figures 21c and 22c show the bilge keel at t/T = 0.125. At this time the body is slowing down so that the inertial force subtracts from the drag forces resulting in a small total bilge keel force. The last figures, 21d and 22d show the bilge keel at t/T = 0.25 where the roll velocity is zero and the acceleration is maximum. Here the total bilge keel force is mostly due to the inertial contribution. Also seen in these figures is the counterclockwise vorticity created by fluid catching up with and passing the bilge keel tip, also demonstrated in Figure 14d. Figures 23 and 24 show the calculated axial velocity for two different times of a period roll motion. The roll velocity is at its maximum in the first and zero in the second. Both figures show the thickening of the boundary layer and the growth of the vorticity in the downstream direction. Also seen is the depression of the boundary layer in the region behind the bilge keels due to the vertical motion over them. When the roll is at its maximum velocity, Figure 23, we see the vortex roll up over the length of the keel much like a wing tip vortex. At zero roll velocity, Figure 24, the vortex behavior is much different with no tip vortex like flow present. Axial Velocity 0.90 0.80 Q.70 0.60 0.50 0.40 0.30 0.20 o1n Figure 23 RANS: f = .32 Hz., A = 15°, U = 2.0 m/s (4 Its.), t/T = 0.0 Figure 25 shows the comparison of the normal force on the port bilge keel for the immersed body rolling at .32 Hz and amplitude 60° for zero and 2.0 m/s (4 kts.) forward speed. At this amplitude the bilge keel rotates 15° beyond the centerline of the body into the wake of the opposite bilge keel. The dashed curves are the forces plotted 180° out of phase and opposite sign. At zero forward speed this illustrates that the force magnitude is different when the body is rolling counterclockwise then when rolling clockwise. axial Velocity 1 o.9o Q.8O i o.7o ; 0.60 0.50 OAO 0.30 0.20 ~ O1D Figure 24 RANS: f = .32 Hz., A = 15°, U = 2.0 m/s (4 kts.), t/T = 0.25 This indicates an influence on the port bilge keel force due the starboard bilge keel and vice versa. The interference is not seen at 2.0 m/s (4 kts.), again because the forward speed has swept the vortices, created by each bilge keel, downstream out of the way by the time the opposite keel sweeps past the centerline of the body. 14

70 35 o o -35 /\ /f' \ ~ ~ \ -- ~ - - 0.' 25 0 5 VT U · 0.0 ants (A fits.) - Pot BK . _ — — — U · G.0 m/s tO Ids.) - Sta,rbo, rdi BK U ~ 2.0 m/5 (4 ~as.) - Pod BK — - — U ~ ~0 ~S (4 - .) - ~i~i7i~ BK dell I/ . _ \\~~ _ 0.75 1 u, O '- _ .' Figure 25 f = .32 Hz,., A = 60°, Fwd. Speed Comparison (Smooth Roll) The Fourier decomposition of the port bilge force histories is shown in Figure 26. As in the 15° case the magnitude of the force decreases and the 3r~ cosine harmonic disappears as the forward speed increases. Also shown is the nonzero mean value in the decomposition for the zero forward speed case, indicating the interference of the opposite bilge keel. 9 t 8 3 r O : -6 ~- == I.~-~......... _ . ~ . =_ = ~.._..~._... =1 =1 _, = o | O - US. is - e phi - him a 0 H~ be shy co~un~ns ~ 4 54s a,, ads Henry, tio,-?s 0 4 - 54s b,, Tiff U - - liens ~-1 1- ·f~l-~-.L~ l-~--l j . _.~ __ I.... .~ EMU ! in ' ~ I -t---'i~--'-'-'!~---'t"~ '--"-- 1 t i . t ~ | I . 1 . T I t , , t ~ I t ' . Fi-~ ~.. ,~Ui . it t t I I ! ! , 1 1 s k Figure 26 Fourier Coefficients 10 One advantage of having reliable computational capabilities is the ability to quantify scaling effects when moving from small to large geometries. These effects can be used to justify the use of a smaller model versus a larger model or extrapolating small model data to full-scale data. Calculations have been performed using the same computational geometry, with smooth motion as before but with half of the Reynolds number. No noticeable effects were seen in the force data. Conclusions Comparisons of RANS calculated force data with measured data and calculated velocity vectors with PIV data have been made using the actual roll motion of the cylinder for a range of test conditions. Overall, these demonstrate that the RANS solver can be used to accurately simulate forced roll motion. Particularly, the solver provides excellent bilge keel force data for the submerged, forward speed cases. Emerged case calculations also compared well to experimental data, but are limited due to numerical stablility problems when the bilge keels approach the free surface. Comparisons with PIV data also demonstrate that the calculations are predicting the correct flow physics. Calculations performed using an ideal sinusoidal roll motion allowed trends to be investigated by enforcing the same roll motion for a number of forward speeds and simplified (somewhat) the harmonic content of the total bilge keel force. The results showed that the magnitude of the total bilge keel force decreased as the forward speed was increased. Acknowledgements This effort was sponsored by NAVSEA PMS 500. The authors would like to thank Mr. James Webster for his direction and encouragement during this effort. Partial support for this effort from ONR under Dr. _ Pat Purtell is gratefully acknowledged. The authors would also like to acknowledge Mr. Martin Donnelly for providing the processed experimental bilge keel force data. The computations were performed using facilities at Maui and ARSC under DOD HPCMO. 15

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