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Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

Chapter: Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway

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Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 459
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 460
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 461
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 462
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 463
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 465
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 466
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 467
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 468
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 469
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 470
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 471
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Page 472
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
×
Page 473
Suggested Citation:"Validation and Application of Chimera RANS Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Validation and Application of Chimera RAN S Method for Ship-Ship Interactions in Shallow Water and Restricted Waterway Hamn-Ching Chenl, Woei-Min Lin2, and Wei-Yuan Hwang3 (~Texas A&M University, 2Science Applications :International Corporation, 3Us Merchant Marine Academy, USA) ABSTRACT A Reynolds-Averaged Navier-S tokes (RAN S) method has been employed in conjunction with a chimera domain decomposition approach to compute the effects of unsteady ship-ship interaction in shallow water and restricted navigation channels. For the simulation of ship-ship interactions, it is convenient to construct body-fitted numerical grids for each ship and the navigation channel separately. The numerical grids around the passing ships and the own ship are allowed to move relative to each other and relative to the grid of the navigation channel. The computation results were first validated by comparing directly with the time record of the experimental measurements by Dand ( 1981 ) for ship-ship head-on encounter and overtaking cases. The method was then applied to navigation channel design problem of computing the effects of moving ships on a ship moored next to a pier. More than 40 time-domain simulations were carried out parametrically for different ship types, wharf line distances, ship speeds, and wind directions that are manifested in ship crabbing angles. The results of these computations were systematically organized and compared to investigate the ship speed effect, wind direction effect, wharf line distance effect, ship type effect, ship sheltering effect, and bottom clearance effect. Although a great deal of further study is required, the results in this paper clearly demonstrate the potential of using the chimera RANS method for ship-ship interaction and navigation channel design problems. INTRODUCTION water effects as well as ships are operating near obstacles such as other ships and piers. Traditionally, for various ship maneuvering simulation purposes, the semi-empirical formula are used to report and represent the towing tank test based ship-ship interactions in deep as well as shallow water when these vessels meet or overtake each other on parallel course. In ship maneuvering simulator community, these formulae are typically used to calculate the ship- ship interactions during real-time simulation. There are simulators that calculate the ship-ship interactions by solving potential flow in real time with the assumptions of open water, rigid free surface, parallel course, and single passing ship. Whether the simulated interactions on parallel course are calculated empirically or solved numerically, further adjustments are applied to account for crabbing angle and various bank effects. When multiple passing ships are in the proximity of own ship in a simulation scenario, the passing ship effects are typically superimposed. This approach certainly has a great deal of room for improvements in order to provide better quality of simulator research and training. In order to more accurately capture the complex phenomena of ship-ship interactions with the presence of variable bottom in a navigation channels, it is desirable to use more sophisticated numerical methods that are capable of dealing with three- dimensional turbulent flows induced by arbitrary vessel motions in confined shallow water channels. In the past several years, Chen and Chen (1998) and Chen et al. (2000) developed a chimera RANS method for the computation of ship-fender coupling during berthing operations. The method was generalized recently by Chen et al. (2001~ 2002) for time-domain The ship maneuvering and ship-ship interaction in confined water have been important problems in channel design and ship operation in harbors. The problems are very complicated because of the shallow 1 ~ . , simulation or large amplitude ship roll motions including capsizing. In the present study, the chimera RAN S method was further generalized to compute the interaction of two moving ships and the effects of

moving ships on a ship moored next to a pier. To demonstrate the validity of the current approach, the numerical results were first compared with experimental measurements by Dand (1981) for ship-ship head-on encounter and overtaking cases. The method was then applied to compute the effect of moving ships on a ship moored next to a pier. A total of 44 runs were carried out parametrically for various combinations of ship speeds, wind directions, ship types and wharf line distances. The results of these computations were systematically analyzed to determine the ship speed effect, wind direction effect, wharf line distance effect, ship type effect, ship sheltering effect, and bottom clearance effect. NUMERICAL METHOD In the present study, calculations have been performed using the free-surface chimera RANS method of Chen and Chen (1998) and Chen et al. (2000, 2001, 2002) to determine the multiple-ship interactions in a shallow- water navigation channel. The method solves the non- dimensional RAN S equations for incompressible flow in general curvilinear coordinates (5~;, i): u'!i =o (1) au + u jUt + U'U] j + g j p j ——g U,jk O (2) where ~ and ui represent the mean and fluctuating velocity components, and go is the conjugate metric tensor. t is time p is pressure, and Re = ULIv is the Reynolds number based on a characteristic length L, a reference velocity U. and the kinematic viscosity v. Equations (1) and (2) represent the continuity and mean momentum equations, respectively. The equations are written in tensor notation with the subscripts, ,j and ,jk, represent the covariant derivatives. In the present study, the two-layer turbulence model of Chen and Patel (1988) is employed to provide closure for the Reynolds stress tensor uiuj The RAINS equations have been employed in conjunction with a chimera domain decomposition technique for accurate and efficient resolution of turbulent boundary layer and wake flows around the moving and moored ships. The method solves the mean flow and turbulence quantities on embedded, overlapped, or matched multiblock grids including relative motions. Within each computational block, the finite-analytic method of Chen, Patel and Ju (1990) is employed to solve the RAN S equations in a general, curvilinear, body-fitted coordinate system. The overall numerical solution is completed by the hybrid PISO/SIMPLER pressure solver of Chen and Korpus (1993) that satisfies the equation of continuity at each time step. The present method was used in conjunction with the PEGSUS program of Subs and Tramel (1991) that provides interpolation information between different grid blocks. The free surface boundary conditions for viscous flow consist of one kinematic condition and three dynamic conditions. The kinematic condition ensures that the free surface fluid particles always stay on the free surface: 77,+U0x+v~y-w=o on z = ~ (3) where ~ is the wave elevation and (U,V,W) are the mean velocity components on the free surface. The dynamic conditions represent the continuity of stresses on the free surface. When the surface tension and free surface turbulence are neglected, the dynamic boundary conditions reduce to zero velocity gradient and constant total pressure on the free surface. A more detailed description of the chimera RANS/free-surface method was given in Chen and Chen (1998) and Chen et al. (2000, 2001, 2002~. VALIDATION OF SHIP-SHIP INTERACTION Although the chimera RANS method has been validated and successfully applied to many different hydrodynamic and body fluid interaction problems, it is necessary to validate its capabilities in predicting the ship-ship interaction in shallow navigating channel due to the complexity of the problem involving multiple moving ships, stationary ship, variable bottom topography, and channel banks. The whole procedure of dealing with multiple moving vessels needs to be checked to make sure the results are valid. The capabilities of chimera RANS method, including its supporting pre-processing and post-processing software modules, to handle the ship-ship interactions are thus validated against the model test data. A favorable comparison would provide the confidence to apply the code to full-scale predictions. Ship Model and Experimental Setup The validation data used here are the towing tank test results reported by Dand (1981). That experiment study was designed to provide insights into the hydrodynamic interactions between ships moving on parallel courses in shallow water for: (1) overtaking encounter, both ships moving; (2) head-on encounter, both ships moving; and (3) a stationary vessel when passed by another. 2

Two 48.2:1 scaled models were used. As shown in Figure 1 and Figure 2, both models were chosen as representative of"averaged" vessels, model 5232 being of the single-screw cargo-liner type while model 5233 represented a tanker. A fully instrumented model 5233 (own ship) was attached to the towing carriage with its track on the tank centerline. Model 5233 has a Lpp of 3.962m, a beam of 0.506m, a draft of 0.208m at FP and 0.218m at AP. Model 5233 was restrained in surge, sway, and yaw; but allowed to pitch and heave freely, and roll to a limited extent. Model 5232 (passing ship) was running on a track and carried no measuring instrumentation. Model 5232 has a Lpp of 3.323m, a beam of 0.473m, a draft of 0.162m at FP and 0.170m at AP. Both models were fitted with propeller and rudder. tI\\\~ TO I I I I 1 1 11 \ \ \ \ \ ~ Figure 1. Body Plan of Model 5232 11\ \ \ \ i\ \ \ \ \ A\\ \ \ \ \\\\-\ \ \ \\\\\\~\\\ ~ \ \\\ \ \ \ \ \ \ \ \ \ \ \ \ ~~— 54\ \ \ N \ \ \ \. \ \ \\ \ Aft\ \\ \ \ \,9. \ \ \ \ i\ \e X\ \\ \ ' 1 \\.~'\\ \\ 5~\ ~ fib\ \ ~ \ \ \ ~ 1 i; 1 / / / / / I / / / / / 1 7 /_/ - l it I / / / /7 [l 11 11 / / / 7 1~ r, _ I -4/ / / 7 1 11 / l I I rI 1 7 I I 11 . I I ~ I I I 111~7 1 7 ITS ' 47/ ~ PI Figure 2. Body Plan of Mode 5233 The model tests were conducted in a towing tank of 90m long, 6.1m wide, and a depth that can be adjusted between O and 0.56m. Note that the combined width of the tanker and the cargo liner is about 16% of the tank width. It was decided to model the tank walls in the numerical simulation as solid wall boundary conditions. A low-pass filter set at 10 Hertz was used to eliminate the noises originated from the vibration of towing carriage. In addition, the model test data underwent a screening process to reject 'wild' data values and also a curve-fitting procedure to fit the data into a modified sine function format. The curve fitting strategy was probably influenced by the observations of typical calculations of interactions based on potential flow and rigid free surface flow assumptions. The original measured data was not available in the original report. Comparison between Numerical Results and Experimental Measurements To provide a critical assessment on the capability of using the chimera RANG method for time-domain simulation of ship-ship interactions in shallow and confined water, computations were performed for two selected cases and compared with the experimental data. These two cases are a head-on encounter case (case 1) and an overtaking case (case 2~. In order to facilitate a direct comparison with the experimental data, the governing equations were normalized using the water depth h, a characteristic velocity A= ,/~, and a characteristic time of T = h I U = it. The Froude numbers are defined by Fnhp = UplU for the passing ship with speed Up and Fnho = UOIU for the own ship with speed UO. In case 1 (the head-on encounter case), the Froude numbers for the passing and own ships were chosen to be 0.421 and 0.250, respectively, in the time- domain simulation. The lateral separation distance YO between the centerlines of the passing and own ships is 1.6 Bo' where Bo = 0.506m is the beam of the own ship. The water-depth-to-draft ratio was chosen to be hlTo = 1.19, where To = 0.213m is the mean draft ofthe own ship. The present chimera grid system consists of 7 computational blocks (3 blocks for each ship and one block for the channel) with 811,587 volume grid points and 5 free-surface blocks with 29,895 free surface grid points. A close up bird-eye view of the free surface grid is shown in Figure 3 at two different time instants during the head-on encounter. In case 2 (the overtaking case), the Froude number of the passing ship was chosen to be Fnhp = 0.411 while the own ship was stationary with Fnho = 0. The water depth for this case is identical to that for the head-on encounter case with hlTo= 1.19, but the lateral separation distance YolBo was reduced to 1.3. Bird-eye views of two time instances of the chimera grid system used in the computations of the overtaking case are shown in Figure 4. Due to the small lateral separation, a significant portion of the ship grids was found to fall within the hulls of another ship during the overtaking encounter. During the computations, these hole-points were automatically removed from the computational domain using the PEGSUS program of Subs and Tramel (1991). It is quite obvious that the use of grid overlapping and embedding techniques 3

greatly facilitates the simulation of ship-ship interactions during close encounters without tedious regeneration of numerical grids at each time step. Figure 3. Chimera grids for head-on encounter case (case 1) Figure 4. Chimera grids for overtaking case (case 2) Computations were performed for both the head-on encounter case (case 1) and overtaking encounter case (case 2) using a constant time increment of t/T = 0.1. In both simulations, the passing ship was accelerated from zero speed to V= Fnhp between t/T = 0 and 4, and then maintained a constant speed for the remaining excursion. The same starting condition was also specified for the own ship for the head-on encounter case. In both numerical simulations, both the passing ship and the own ship were held fixed in sway, heave, roll, pitch, and yaw directions. The ships were moving in straight courses and no equations of motions were solved. Figure 5 shows the computed longitudinal velocity and pressure contours at several time instants to illustrate the general flow characteristics during the head-on encounter (case 1~. It is noted that the pressure field generated by the passing and own ships propagated in front of the ship bows and induced strong ship-ship interactions well before the ships meet at Xo/Lo = 0. The reflection of the pressure waves from the tank walls produced nearly two-dimensional wave fronts as seen in Figure S(b)-(c). The collision of these wave fronts produced a sharp rise in pressure field near the middle of the towing tank at t/T = 36. These wave fronts continue to propagate towards the other ends of the wave tank and produced a complex wave pattern before the ships meet at t/T = 72. A close examination of the pressure contours shown in Figure S(c) clearly showed that the two pressure wave fronts met near the center of the towing tank even though the passing ship travels at a significantly higher speed than the own ship. This indicates that the propagation speed of the pressure waves is nearly independent of the actual ship speed. For the present head-on encounter case, the two ships met at t/T = 71.9 (Xo/Lo = 0) and completely passed each other at t/T = 114.7 (Xo/Lo = 2~. It can be seen from the pressure and velocity fields shown in Figure S(e)-(h) that there is a very strong interaction between the passing and own ships during the encounter. A low-pressure region was developed in the narrow clearance region between the two ships due to the Bernoulli effects. After the ships completely passed each other, the pressure interactions reduced gradually but the ship wakes remain clearly visible at the end of the simulation. For completeness, we shall present also the longitudinal velocity and pressure contours for the overtaking case at several time instants as shown in Figure 6. It is clearly seen from Figure 6(b)-~0 that the pressure waves generated by the passing ship travel considerably faster than the ship itself. The wave front is again nearly two-dimension due to the tank wall effects. The bow of the passing ship crossed the stern of the stationary own ship at t/T = 40 (XJLo = 1.0) and completely overtook the own ship at t/T = 110 (XJLo = -0.8387~. As noted earlier, the lateral separation distance YolBo for the overtaking case is only 1.3. This resulted in a sharp reduction of pressure in the narrow passage between the two ships. After the passing ship completely overtook the own ship, the pressure field around the own ship returns gradually to zero. However, the reflection of pressure waves from the tank wall was still quite significant at t/T = 150. In order to facilitate a direct comparison with the experimental measurements of Dand (1981), the computed forces and moments on the own ship (model 5233) were non-dimensionalized as follows: Surge force coefficient: Cx = FX 1( 2 pBoTou2 ) Sway force coefficient: Cy = Fy 1~2 pBoToUp ) Yaw moment coefficient: Cn = Mz 1~2 pBoToUp) 4

- - ~ _11 c - - _11 - - - - - - _11 - - _11 - - _11 - ~ - ll - - - 1 l 1 b 1 i i ~ . _ _ _ l _ _—r _ _ __ at_ _ - l - _ _ _ _ - _ 3_ _ 1 _ 1 Hi l -~1 -uw -O.OB ¢04 002 0 002 004 006 008 01 _ _ ~' _ 4~02 001 0 001 0~02 003 004 Figure 5. Longitudinal velocity (left) and pressure (right) contours for head-on encounter case (case 1) 5

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -0.06 -0.05 -0.04 403 -0= -0.01 0 0.01 0.02 0.03 0.04 Figure 6. Longitudinal velocity (left) and pressure (right) contours for overtaking encounter case (case 2) 6

~ 0.4 (a) Cx ~-02 ~ ~0.4 Cakuladons : ~ Experkr~ (Filleted ~0.6 -2 -1 xiLo 2 . (b) Cy 5 o ,~\: of -2 . Cartons 4r 2 E ~ O . ~ -2 ~ ~ ._ ,. es_w, ~ . . . . . . . . . . . . . . . . . -2 -1 x IL O 1 : ~ ~ o _ T/\~/~{ I-/— Cakubdons ~ Eat rr~ (Fllb~ -2 -1 XolLo 0 1 2 Figure 7. Force and moment coefficients on the own ship for head-on encounter case (case 1) : (a)cy 50.4~.-~ ~ ~0. 4 - -- CakubUons U) ~ Expert (FIlbred) -1 .5 -1 -0.5 XJLo ~ 1.5 2 is: 1 ~ U o ,, -1 . o 2 (b) Cy N~ ,' Cakuh~ons ~ E~tperltr~ (Fllbred} -1.5 -1 -0.5 0 XJLo05 1 1.5 4, 2~(C) Cn ,~ ~ it' ~ ]-2 Is Cakubffons : ~ E~tperbr~ (Fllbre~ -41 .5 -1 -0-5 0 xJ~ 05 i 1 5 Figure 8. Force and moment coefficients on the own ship for the overtaking case (case 2) Figure 7 and Figure 8 show the comparisons of the computed and measured Cx, Cy' and Cn for the head- on encounter case (case 1) and overtaking encounter case (case 2), respectively. The blue solid lines represent the numerical results and the red dash lines represent the filtered experimental measurements by Dand (1981~. As mentioned before, the model test data went through a curve-fitting procedure to fit the data into a modified sine function format. The original measured data was not available in the original report. In general, the numerically predicted force and moment coefficients are in fairly good agreement with the filtered model test data, although the numerical results contain more oscillatory components. The observed discrepancy may be attributed to the following factors: . . · Due to the removal of the wild data points and the curve fitting of test data, the more oscillatory components of the interactions might have been lost in the figures presented by Dand (1981~. · The forward speed of the ships were assumed to be constant (except the initial ramp start) in the numerical calculations, while the actual speed of the ship models might not be constant, as reported in Dand (1981), due to increase of resistance while one ship passes another. · The ship acceleration during ramp start produced strong pressure waves that is responsible for the initial oscillations in the surge forces. The motions of the ship models in the numerical calculations were constrained, while the test model was allowed to pitch and roll. In addition, the tank walls were assumed to be perfectly reflective in the simulations. · The more oscillatory components observed in the computations could also be attributed to vortex shedding, the free wave system of each vessel, or a local wave system induced by the interacting vessels. These phenomena have been reported and discussed by Norrbin (1985), as well as by Kaplan and Sankaranarayanan (1986~. The physical test models were equipped with rudder and propeller. Dand (1977) showed that the effects of rudder and propeller are recognizable although not overwhelming. SHIP-SHIP INTERACTION IN CHANNEL With validations established for ship-ship head-on encounter and overtaking cases, the chimera RANS method was applied to study a more complicated navigation channel design related problem. In this study, the effect of one and two passing ships on a ship moored next to a pier was examined. A total of 44 simulations were performed to examine the effects of passing ship speed, wharf line distance from the channel boundary (i.e., lateral separation distance), crabbing angle of the passing ship (a manifestation of the wind effects), bottom clearance, and sheltering effects in two passing ships cases. A complete listing of the numerical simulations performed in this study is shown in Table 1. Note that h represents the reference water depth. Figure 9 shows the cross- sectional view of the navigation channel. In most of 7

the simulations performed, the wharf line (denoted shape A) consists of a 2:1 bank slope which terminates at a water depth of 0.5h. A second wharf line shape (shape B) was considered in case 17 to examine the sensitivity of the bank shape on the ship- ship interactions. In this case, the 2:1 bank slope was extended further up to the shallow water region with a minimum water depth of 0.125h. It should also be noted that the water depth in Case 36 is 0.2h deeper than the other cases everywhere. In addition, there is no 2:1 slope at the wharf line and the vertical quay wall is located at a distance of 2h away from the moored ship-C (to the starboard side). In all simulations, the North (N) is going into the paper, East wind (E) means the wind is from the East, and West wind (W) means the wind is from the West. The wind effects were represented by a 5° crabbing angle of the ships to counter the wind action. ~ ! 1 1 West Side ' N ~ CL Channel, I Boundary ' 8h _ . . East Side Channel Boundary ~1 Distance to, i , Constant Wharf Line ' ; ' Depth , ; , Figure 9. Navigation channel cross section The coordinate system convention used in the present study is shown in Figure 10, with the Taxis going into the water. The forces and moments shown in all Run# 01-03 04-06 07-09 10-12 13-16 17 18 19 20-21 22-24 25-27 28-30 31-33 34 35 36 37-38 39-41 42-43 44 Wharf Line Distance 10h 10h 10h 16.6h 6h,8h,12h,14h 10h (Shape B) Open Water 10h 10h 16.6h , 10h l 16.6h 10h 12h 10h 16.6h . 10h (Deep) 10h,12h 10h 12h 16.6h the plots are consistent with this coordinate convention and represent the hydrodynamic forces and moments acting on the moored ship. As noted in Table 1, three types of ships were considered in the present simulations. The dimensions (i.e., length, beam, and draft) for these three ships are (19.3h, 2.112h, 0.86h) for ship-A, (l9.Oh, 2.9h, h) for ship-B, and (25.92h, 3.6h, 0.94h) for ship-C, respectively. In all simulations, ship-A is the inbound vessel moving from North to South and ship-B is the outbound vessel moving from South to North. Two different moored vessels were considered with ship-A moored at the pier in the first 27 cases while the much larger ship-C was the moored vessel for the remaining 17 simulations. In order to facilitate the comparison of the forces and moments, the length of ship-A was chosen as the reference length L = 19.3h for all the figures presented in this paper. The forces and moments obtained by the chimera RANS method were non-dimensionalized as follows: For Forces: 1/2p(VV p )l'3 Up (4) For Moments: 1/2p(VVp)~'2U2 (5) where p is the water density, V is the displacement of the moored ship, Vp is the displacement of the passing ship, and Up is the speed of the passing ship. When more than one passing ship is present, the ship that tracks closer to the moored ship is regarded as the reference passing ship for non-dimensionalization purpose. Table 1. Simulated Cases Ship A A A A A A A A A A A A A A - A A A 8 . ~ Sneed Wind U. 1.33U 1.67U Calm , , U,1.33U,1.67U E U,1.33U,1.67U W . U,1,33U,1,67U E 1.33U E U W 1.67U Calm W W W W Calm . W W W W W U,1.33U,1.67U U. 1.33U 1.67U , , U,1.33U,1.67U U,1.33U,1.67U 1.67U 1.67U 1.67U U. 1.33U, 1.67U U. 1.33U _ U

An overview of the computational grid for a two-way passing calculation is shown in Figure 11. As can be seen in the figure, both inbound and outbound ships have a 5-degree crabbing angle to counter the wind from the West (west wind condition). Both the inbound and outbound ships travel approximately 5 ship lengths (depending on ship speed) and the longitudinal extent of the computational domain is about 150h. For the two-way passing cases, the numerical grid consists of 15 computational blocks (3 blocks for each ship, 4 blocks for the harbor, and two phantom grids) with 995,776 volume grid points and 10 free-surface blocks with 43,055 free surface grid points. A close up view of the grid surrounding the moored ship-C is shown in Figure 12, and a bird-eye view of the blanking scheme between grid blocks is shown in Figure 13. In the present simulations, the numerical grids for the inbound and outbound ships were allowed to move in prescribed translational motion relative to the moored vessel and the stationary harbor grids. Other than the forward direction, motions of the passing ships were fixed in all directions. The moored ship was also assumed to be fixed. No equations of motions were solved in the current study. Figure 10. Simulation coordinate system Figure 11. Numerical Grid for Two-Way Passing Configuration :'~^'~-~--~: . ~ ~~ :: ~ ~ Figure 12. Close-up view of grids for moored Ship-C next to a pier ~ i: ~ i :i:::i~: i :~i :lR~ i-l-!-i-E ~1 ~ ~ -~-~-~--~-1-l ~ I i-t ~ it ~ 1.~ 1~4 1 ~ I Lt til ! ~ li - ~:~f~ Figure 13. Bird-eye view of blanking between grid blocks Detailed Flow Field Time-domain simulations were performed for each case listed in Table 1 with various combinations of ship speed, wind direction, and wharf line distance. A total of 2000 time steps were used for each simulation with a constant non-dimensional time increment of 0.0025. The initial positions of the passing ships are about 2.0L from the center of the moored ship. In all simulations, the passing ship (or ships) was accelerated from zero velocity to the designed speed between tlT= 0 and 0.2, where T = LlUp is the characteristic time. The passing ship travels 0.1 ship length during the initial acceleration and then maintain a constant speed for the remaining excursion. The Reynolds number is 7.78 x 108 and the Froude number is 0.0575 based on the length of ship-A and the reference speed U. In all simulations, the time histories of the surge, sway, and heave forces, as well as the roll, pitch, and yaw moments 9

acting on the moored ship were recorded. In addition, a 500-frame movie was made for each simulation to provide more detailed understanding of the flow field induced by the passing ships). It should be noted that the center of the passing ship (closest to the moored ship) was aligned with the center of the moored ship at tlT= 2.1 for all the simulations considered. Due to the large number of simulations performed, it was not possible to present detailed velocity and pressure fields for every case. For the sake of brevity, we shall present the free surface velocity and pressure contours for Case 39 to illustrate the general flow features for two passing ships in the navigation channel. In this case. the inbound ship-A is moving from the left to the right, and the outbound ship-B is moving from the right to the left. Figure 14 shows the computed longitudinal velocity and pressure contours for Case 39 at several different time instants of tlT= 0.25, 0.5, 0.75, 1.0, 2.1, 2.25, 3.0, 4.0, and 5.0. It is seen that the pressure field generated by ship-B is considerably stronger than that induced by the ship-A. This is clearly due to the fact that ship-B has significantly wider beam and larger draft compared to ship-A. The rather blunt bow shape for ship-B also produces larger pressure waves in front of the bow. Furthermore, the bottom clearance for ship-B is only 10% of its draft and is significantly smaller than the 18% bottom clearance for ship-A. A close examination of the movie file for this simulation showed that the pressure waves induced by the passing ships reached the moored ship at around tlT= 0.6 with each passing ship travel about 0.5 ship length. It is also interesting to note that ship acceleration during the initial ramp start produced very strong pressure waves that are significantly higher than those observed after tlT= 1 when both the inbound and outbound ships were traveling at constant speeds. As noted earlier, the ship-ship interactions in navigation channel is very complicated due to the shallow water effects and the presence of channel bank and mooring piers. In addition to the interactions between the passing and moored ships, it is also clearly seen that there are very strong interactions between the inbound and outbound ships when both ships were in the vicinity of the moored ship. This is particularly evident at tlT= 2.35, shortly after the inbound and outbound ships passed the center of the moored ship. The influence of the passing ships diminished gradually after tlT= 3.0 with the free surface pressure returns slowly to its ambient value around the moored ship. However, the trajectories of the wakes behind the passing ships were still clearly visible at tlT= 5.0. Forces and Moments In order to analyze the mooring line forces induced by moving traffic vessels, the computed pressure and shear stresses were integrated over the hull surface of the moored vessel. Figure 15 shows the computed hydrodynamic forces and moments on the moored ship-C. For all cases, the force and moment coefficients were plotted as a function of XIL instead of tlT with the initial position of ship-A located at XIL = -2.0 and the center of the moored ship located at the origin (XIL = 0 and tlT= 2.1~. It is noted that the moored ship experienced a large downward heave force after the inbound and outbound ships passed the center of the moored vessel. This is clearly associated with the depression of water elevation in the narrow navigation channel due to ship motions. It is also noted that the pitch moment history exhibited some high frequency oscillations. This high frequency oscillation was not caused by the numerical instability since the oscillation period (about 0. 11) was found to be significantly longer than the time increment used in the simulation. Furthermore, the high frequency oscillations were present only when the ship-C is the moored vessel and was completely absent for ships A-A-B configuration (Case 22) shown in Figure 16 when ship-A was moored at the pier. It is believed that these high frequency oscillations were caused by the small underkeel clearance of the moored vessel and will be discussed in more details later when the effects of bottom clearance is considered. In the mooring line analysis, the sway forces and yaw moments are of particular concern because they produce lateral displacements and rotations perpendicular to the wharf line. It is seen from Figure 15 that the sway force reached a maximum value at XIL = 0.6 (i.e., tlT = 2.7), long after the ships passed the center of the moored ship. In the quasi- steady, semi-empirical formula for deep water or shallow open water ship maneuvering, the sway force typically reached a maximum value at XIL = 0 when the vessels meet on parallel course. The shift of the maximum sway in the present simulation is most likely due to the combined effects of wharf line (bank) shape and small underkeel clearance that were not included in the traditionally open water simulations. Finally, it is also worthwhile to note that both the sway force and yaw moment diminish gradually after XIL = 1.0. Therefore, it is reasonable to terminate the simulations after the passing ships have traveled 3 ship lengths beyond the center of the moored ship if our primary interest is to determine the maximum mooring line forces. 10

.0.4 0.2 0.0 0.2 0.6 0.8 Figure 14. Longitudinal velocity (left) and pressure (right) contours at t/T= 0.25, 0.5, 0.75, 1, 2.1, 2.25, 3, 4, 5. 11

2.5' ' ~ Surge force Sway to roe Heave force Roll moment ~ Pitch moment be, Yaw Nero T ent ! 2 1.5 ~ 1 E 0.5 - .~' I--! E o ~_~ _ _ ~ ~ ~~#~ —_ ,,h~ , . i. -1 .~ ~ . -2 -1 o 1 2 3 X/L Figure 15. Force and moment coefficients for Case 39 .:, 2 ._ 0 1 E ~ : 0 0 E 8 o -1 ~ Surge force Sway force I Heave force . Pitch rnornent ,~ %N Yaw moment ,' -" t_~_~ "A ........................ -2 -1 0 1 2 3 X/L Figure 16. Force and moment coefficients for Case 22 The present study seeks to determine the effects of passing ship speed, wharf line distance, wind direction, bottom clearance, and ship sheltering while there are more than two ships in the navigation channel. In the following sections, we shall compare the sway force andfor yaw moment time histories to quantify these effects. Passing Ship Speed Effects In the present simulations, three different ship speeds of U. 1.33U and 1.67U were considered for the inbound ship-A while the speed for the outbound ship- B is always the same. It is seen from Table 1 that the speed effects were investigated for various combinations of wind directions, wharf distance, and ship types. Cases 1-12 examined the speed effects for ships A-A in calm, east, and west wind conditions at two different wharf line distance of s = lOh and 16.6h. The speed effects were also investigated in Cases 22- 33 and 39~1 for other ship types involving A-A-B, C- A, and C-A-B configurations. Figure 17 shows the sway force and yaw moment coefficients for Cases 1-3 with ships A-A under calm wind condition at a wharf distance s = lOh. The results are normalized by the corresponding forward speed of the moving ship Up. It is seen that both the phase and magnitude of the peak sway force and yaw moment change with the ship -0.1 0.2 speed. The phase lag for higher speed cases is to be expected since the faster ship will be closer to the moored ship when the pressure waves generated by initial ship acceleration reach the moored vessel. Although the yaw moment coefficient is higher for the lower speed case, the magnitude of the yaw moment actually increases with the ship speed since the dimensional yaw moment is obtained by multiplying Up to the coefficients. Similar trends were also observed for other cases although the effect varies somewhat with different wind directions, wharf distance, and ship types. 0.2 0.1 ~ O 4''V`;! In · Up= 1.67U Up= 1.33U Up=U (''1'.\'~ ~ (a) Sway Force ~ — Up - 1.67U - - - - Up= 1.~JU Up=U Q 0.1 E O 1 'l' i -0.1 (b) Yaw Moment 2 ... ........... -2 -1 ° X'L 1 2 3 Figure 17. Speed effects for ship A-A in calm wind: (a) sway force, (b) yaw moment Further insight was gained when the numerical results were examined in dimensional quantity. Table 2 summarizes the peak sway forces when a ship A is passing a moored ship A or ship C at U. 1.33U, and 1.67U in the s = lOh wharf line configuration. The moving vessel crabbed to the west in simulating a transit to cope with a 3 U westerly wind. It was noticed that the speed effects on the ships A-A interactions are different from the speed effects on the ships C-A interactions. In the ships C-A cases, the speed effect is approximately proportional to the square of speed. This is what one would expect from a fluid flow phenomenon. However, the speed effect is almost negligible on the ships A-A interactions. It is hypothesized that when the moored vessel is a ship C, the much wider beam, longer length, deeper draft and very small underkeel clearance (1.064 water depth to draft ratio) have made the pressure pulse contribution 12

to the interaction dominant over other contributions to the interaction. With smaller dimensions and more space under the keel for a moored ship A (1.163 water depth to draft ratio), the predominant contribution to the interaction seemed to come from ship wave trains, local waves induced by interacting vessel pressure fields, vortex shedding or other physical mechanisms. This finding shows that in a complicated setting of a ship next to a pier environment we cannot rely only on one generic set of non-dimensionalized ship-ship interaction curves for sway force calculations. Significant errors could be introduced in estimating the real effects of passing ships if the sway forces are used in dimensional form based on instantaneous passing ship speed. Table 2. Speed Effects with Wind from West CFD Run 7 8 9 21 22 23 M-P Ships . A-A A-A A-A C-A C-A C-A Max Sway Coeff 0.0725 0.1100 0.1800 0.2050 0.2010 0.2150 Wind Direction Effects Passing Speed, U 1.67 1.33 1.00 1.67 1.33 _ 100 Max Sway Force, pU2 . 0.2014 0.1955 0.1800 0.5695 0.3555 0.2150 The wind effects for a moored ship-A with a passing ship-A or ship-B were investigated in Cases 1-9 and 19-20, respectively. For the East (E) wind condition, the bow of the inbound ship will turn 5° in the counterclockwise direction, and the bow of the outbound ship will turn 5° in the clockwise direction. On the other hand, the bow of the passing ships will turn 5° in the opposite direction under West (W) wind conditions. In Figure 18, the wind effects for ship speed Up = 1.67U cases are shown for the ships A-A configuration at a wharf line distance s = 10h. Figure 19 shows the wind effects for ships A-B configuration with Up = U and s = 1 Oh. For the ships A-A configuration, the wind directions and the related crabbing angles show varying yaw moments from the calm wind cases, but the effects are relatively small. The yaw moment coefficients shown in Figure 19 for ships A-B case are more sensitive to the wind directions. In addition, the yaw moments and sway forces induced by the larger ship-B are also considerably higher than those observed in Figure 18 for ships A-A configuration. In general, it was observed that crabbing angle introduced additional interaction forces and moments depending on the passing ship speed and crabbing direction. A typical simulation adjustment of crabbing angle on passing ship effects is to multiply the parallel course testing data with a cosine function of the relative heading. However, the finding from present calculation does not support that approach. 0.2 t c' 0.1 E° o 3 -0.1 CalmWind East Wind ------- West Wind -2 -1 ° XL 1 2 3 Figure 18. Wind effects for ship A-A at Up=1.67U 0.2 t West Wind <, 0.1 ~ 0 ~N f ~ take], o 3 ~.1 fY I'm ~ /\ -3 -2 -1 X'L ° 1 2 Figure 19. Wind effects for ships A-B at Up= U Wharf Line Distance Effects The wharf line distance effects for single and two ship passing configurations were examined in Cases 4-6, 10-16, 20-33 and several other runs listed in Table 1. For the sake of brevity, we will present mainly the results for ships A-A configuration under East (E) wind condition at a speed Up= 1.33U. Figure 20 shows the sway force and yaw moment histories for moored ship-A at six different wharf distances of s = 6h, 8h, 10h, 12h, 14h, and 16.6h, respectively. It is interesting to note that the peak sway force, in general, reduces with increasing wharf distances. On the contrary, the peak yaw moment tends to increase with increasing wharf distances. However, the phase and magnitude variations with wharf distance are not completely monotonic. It should be noted that the wharf line is still within one ship length of the passing ship even for the longest wharf distance considered. The hydrodynamic interactions between two ships can be attributed to at least two major physical 13

phenomena: the pressure pulse and the ship generated wave train. The pressure pulse can be explained by Bernoulli's effects in the fluid flow surrounding the vessels. The contribution of this mechanism is very strong and dominant in close separation distance. But the effect decreases rapidly when the vessels move apart from each other. The effect of a ship-generated wave train is of smaller order of magnitude, but the effect can reach greater distance. There may be contributions from other mechanisms that can also reach greater distance, such as the "local waves" mentioned by Dand (1981) and the vortex shedding pointed out by Kaplan and Sankaranarayanan (1986~. 0.2 ~0 O. ~ O 3 u' E E s= 16.6 h ------ s=14h --- s= 12 h . s= 10 h — s= 8 h ~ -— such 0.2 0.1 c ~ 1 ~ ~ h s = 14 h s = 12 h A .~ s=10h ------- s=8h s=6h -0.1 (b) Yaw Moment -2 -1 0 X/L 1 2 3 Figure 20. Wharf line distance effects: (a) sway force, (b) yaw moment Most of the towing tank tests only cover small lateral separation distances where pressure pulse dominates the ship interactions. For example, the test range of separation distance covered by Dand (1981) was 1.1 to 3.66 of own ship beam widths. It is noticed that at 3.66 beams lateral distance, the interaction curves of overtaking started to show more oscillatory behavior. When a moored vessel is moored at a pier that is built on top of a sloped bank, the physics of ship- ship interaction hydrodynamics becomes more complicated, especially with small underkeel clearance for one or all of the interacting vessels. There might be even "resonant phenomenon" when snipes) passes a moored ship in the presence of a sloped bank because the wave train could reflect and refract. This is an area needs more research to understand the underlying mechanism and harness the knowledge for simulation applications. The results of SSPA model tests reported by Li (2000) on bank effects clearly demonstrate the complex hydrodynamic phenomena when a vessel is sailing near a bank of various configurations. Bottom Clearance Effects The comparisons of pitch yaw moment coefficients in Figure 21 for Cases 30 and 36 provide evidence for the physical nature of the high-frequency force oscillations detected in the ship-C runs discussed earlier. Such oscillations are induced when a small clearance between the ship and the bottom of the channel is present. As noted earlier, Case 36 has the same conditions as those in Case 30 except that the water depth is 0.2h deeper throughout the channel and the 2 to 1 bank slope at the wharf line was replaced by a vertical wall at a distance s = 2h away from the ship-C. For the shallow water cases, the underkeel clearance is only 0.06h or 6.4% of the ship draft. Therefore, the water flow induced by the passing ships was forced to move back and forth across the bottom of the ship hull through small clearance. When the bottom clearance was increased from 0.06h to 0.26h for Case 36, the water was able to move more freely around the moored vessel and the high frequency oscillations were completely eliminated. 2 ~ 1 0 o -1 0.06h underkeel clearance -- 0.26h underkeel clearance -2 -1 2 3 Figure 21. Sway force and yaw moment for bottom . . c ~earance varlatlon Ship Sheltering Effects Traditionally, the superposition of forces and moments has been used as a simplified way of obtaining the influence of multiple ships passing the moored ship in a simulator environment. In order to determine the validity of superposition and the influence of sheltering effects, a comparison between the superposition of Case 7 and Case 20 and the combined simulation of the same ships in Case 22. Figure 22 shows a comparison of the free surface pressure contours for Cases 7, 20 and 22 at two different time instants t/T= 2.0 and 2.5, respectively. The addition of forces and moments from Cases 7 and 20 are shown in Figure 23 and the comparison of Case 22 to the 14

superimposed solutions is shown in Figure 24. The comparison of the superimposed solution to the direct multi-ship solution shows that there are significant sheltering effects from Case 22 that are neglected in the superimposed solution. The free surface pressure contours shown in Figure 22 clearly illustrates the complexity of the nonlinear interactions between the (a) Case 7, Ship A-A, t/T = 2.0 inbound ship, outbound ship, moored ship, and the bank. These highly nonlinear multi-ship interactions highlight the inaccuracy of a superposition method for the multi-ship interaction problem needed to establish the lashing loads on the moored ship. (b) Case 7, Ship A-A, tot = 2.5 (c) Case 20 Ship A-B t/T = 2.0 (d) Case 20, Ship A-B, tlT = 2.5 (e) Case 22 Ship A-A-B t'T = 2.0 (f) Case 22, Ship A-A-B, tfT = 2.5 . ~ .... . _ _ _ _ . _ _ _ . _ _ _ _ . -0.4 -0.2 0.6 0.4 `~ 0.2 ID o o -0.2 -0.4 , , - .... o 0.6 t 0.4 0.2 E o ~ 5 -02 -04 -0.6( ) Shlps 1A ~ A1 ~ - - Ships 1A ~ B] _ - - - - - Superpose ~ ~ Ad fib ~1 (a, sway Force . . . . . . j ... - Ships ~ ~ ~ | - ships CAT ED - ---- -- Supelposiffon [A ~ A] ~ p~ ~ ED l' ~ ~ ~ ~ -a' it; , . . . . 1 2 tlT 5 (b) Yaw Moment , . . . . . 3 4 ~ Figure 23. Superposition of (a) sway forces and (b) yaw moments for Case 7 and 20 0.0 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0 4 Figure 22. Sheltering effects at t/T=2.0 (left) and 2.5 (right), respectively 0.6 0.4 ID `' 0.2 ~ O LL ~ -0.2 u, -0.4 0.6 -0.4 O.B 0.8 ~ - Superposl~don [A+AJ ~ [A ~ B] lo Ships IA ~ A ~ B1 0 1 ) ', ........ 0 1 2 VT 3 '\1~ (M rob F x" , . . . . . . . . . . . . . . 2 tlT 3 4 ' i Superposition [A+Al + Urn E] alps pa ~ A ~ ~ - V~'~ (b) Yaw Moment . 4 1 Figure 24. Ship sheltering effects: (a) sway force, (b) yaw moment 15

CONCLUDING REMARKS In this paper, the unsteady chimera RAN S method has been validated against some available experimental measurements and employed to study the effects of multiple-ship passing effects in a navigation channel. In the validation study, the time-domain surge force, sway force, and yaw moment results track the experimental measurements very well even though there are minor differences in the settings of the numerical study and the original experiments. In the navigation channel design study, the numerical results were systematically organized and compared to investigate the ship speed effect, wind direction effect, wharf line distance effect, bottom clearance effect, and ship sheltering effect while there are more than two ships in the channel. For safe operation of a ship while operating in the proximity of other ships and near obstacles with variable-depth bottom topography, accurate interaction forces and moments acting on the ships are required. At the current time, this vital information is typically obtained from extrapolation of model test data and from operator's experiences over the years. There are very few systematic and reliable data exist at the full scale. Advanced physics-based computation method allows us to model the actual viscous flow phenomena of the problem at a level not previously possible. The current computation method offers the ability and flexibility of modeling complex ship geometry, multiple ships operating in the proximity of each other and near obstacles, and channel bottom topography, etc. In addition, computations can be done at full scale Reynolds number where the viscous effects on the ships are very difference from those at the model scale. Although a great deal of improvements and further studies are still required, the results presented in this paper clearly demonstrate the potential of using the chimera RANS method for ship-ship interaction and navigation channel design related problems. ACKOWLEDGEMENT The authors would like to thank Mr. Kenneth Weems, Mr. Paul Jones, Dr. Daniel Liut, and Mr. Michael Meinhold for their help on geometry generation and data processing of the results in this paper. REFERENCES Chen, H.C. and Chen, M., "Chimera RANS Simulation of a Berthing DDG-51 Ship in Translational and Rotational Motions," International Journal of Offshore and Polar Engineering 1998, Vol. 8, No. 3, pp. 182- Chen, H.C. and Korpus, R., "A Multi-block Finite- Analytic Reynolds-Averaged Navier-Stokes Method for 3D Incompressible Flows," ASME FED-Vol. 150, pp. 113-121, Proceedings of the ASME Fluids Engineering Conference, 1993, Washington, D.C., June 20-24. Chen, H.C., Liu, T., Chang, K.A., and Huang, E.T., "Time-Domain Simulation of Barge Capsizing by a Chimera Domain Decomnosition ~Ol)It)~! Domain Decomposition Approach," Proceedings of the 12th International Offshore and Polar Engineering Conference, KitaKyushu, Japan, May 26-31,2002. Chen, H. C., Liu, T., and Huang, E.T., "Time-Domain Simulation of Large Amplitude Ship Roll Motions by a Chimera RAN S Method," Proceedings of the 11th International Offshore and Polar Engineering Conference, Vol. III, 2001, pp. 299-306, Stavanger, Norway. Chen, H.C., Liu, T., Huang, E.T. and Davis, D.A., "Chimera RANS Simulation of Ship and Fender Coupling for Berthing Operations," International Journal of Offshore and Polar Engineering, Vol. 10, No. 2, 2000, pp. 112-122. Chen, H.C. and Patel, V.C., "Near-Wall Turbulence Models for Complex Flows Including Separation," AIAA Journal, Vol. 26, No. 6, 1988, pp. 641-648. _. . . .. Chen, H.C., Patel, V.C. and Ju, S., "Solutions of Reynolds-Averaged Navier-Stokes Equations for Three-Dimensional Incompressible Flows," Journal of Computational Physics, Vol. 88, No. 2, 1990, pp. 305- 336. Dand, I.W., "Ship-Ship Interaction in Shallow Water," Report R 8, 1977, National Maritime Institute, Feltham, Middlesex, United Kingdom. Dand, I.W., "Some Measurements of Interaction between Ship Models Passing on Parallel Courses," Report R 108, 1981, National Maritime Institute, Feltham, Middlesex, United Kingdom. Li, D-Q., "Experiments on Bank Effects under Extreme Conditions," SSPA Report No. 113, 2000, Goteborg, Sweden. Norrbin, N., "Model Tests for CAORF Panama Canal Study- Part 7: Ships in Straight Channels," S SPA Report 3062-7, 1985, Goteborg, Sweden. Kaplan, P. and Sankaranarayanan, K., "Analysis of Asymmetric Channel Hydrodynamic Interaction of Ships." Report No. VPI-AERO-156, 1986, Virginia Polytechnic Institute and State University, Blacksburg, VA. Subs, N.E. and Tramel R.W., "PEGSUS 4.0 Users Manual," Report AEDC-TR-91-8, 1991, Arnold Engineering Development Center, Arnold Air Force Station, TN. 16

DISCUSSION Hoyte C. Raven MARIN, The Netherlands Your Figure 14 and the animations in your presentation show large waves being generated at the initial startup of the vessels. There are solitary waves running ahead, interacting with ships and reflecting at the far boundaries. How did you separate the true interaction forces and those caused by the transient waves? Would it not be more accurate to neglect free surface effects all together for these cases? AUTHORS' REPLY We would like to thank Dr. Raven for his valuable discussion. If the ship starts from rest in the real scenario, then the true interaction should obviously include the transient wave effects. Depending on the test condition, it is actually quite common to observe solitary waves running in front of the ship in shallow water ship model tests. In the current study, the ship speed was ramp up from zero to a constant value in a fairly short distance. In an ideal setting, we should allow the ships to run for a while so that it can reach steady-state condition before them encounter each other. However, this approach will require a fairly large computation domain and the large amount of computation time. The ship speed is not high (less than 5 knots in most cases), and the wave effect should be small. I would not say the calculations without the free surface effects would be more accurate. But, based on the fact of low ship speed, we should be able to neglect the free surface effects and expect the error to be small. However, if the ships are very close to each other during passing, a rigid surface condition on the free surface (neglect the free surface effects) will create an unrealistic pressure between the ship since the fluid will be forced to escape in the horizontal direction rather than both the horizontal and the vertical directions. DISCUSSION Tao Jiang VBD European Development Centre for Inland and Coastal Navigation, Germany I would like to congratulate the authors for their comprehensive study on a practically relevant subject. The proposed method seems to work well for such complex interaction problems near a harbor basin. Looking at Figure 5, one can observe that there is a strong interaction of bow-wave systems of the two ships subjected for head-on encounter case. However, bow-wave systems do depend on the start contributions which are quite different from the real situation. Could the authors comment on their experience regarding the influence of numerical accelerations of the ship for truly unsteady problems? AUTHORS' REPLY We would like to thank Dr. Jiang for his valuable comments. As discussed in the previous reply, in an ideal setting, we should allow the ships to reach a steady-state condition before they pass each other. In that case, the bow waves would not be an issue. However, there are practical limitations of this approach based on domain size and computation effort considerations. Our observation indicated that the effect of the bow waves were relatively minor comparing to the actual interact between the ships. However, more study is required to quantify the true impact.

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