Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Unsteady RANS Simulation of a Surface Combatant with Roll Motion Robert Wilson and Fred Stern (University of Towa, USA) ABSTRACT An unsteady RANS method is developed to compute the flow and wave field for surface ships with general non-linear 6DOF motions. The method is based on an extension of CFDSHIP-IOWA (a general-purpose code for computational ship hydrodynamic) to predict ship motions with large amplitude and non-slender geometry. The flow solver uses hi~her-order upwind discretization, PISO method for pressure-velocity coupling, k-m two-equation turbulence model, free- surface tracking approach, and structured multi-block grid systems. As an initial step, unsteady simulations of a surface combatant with prescribed sinusoidal roll motion are performed over a range of three frequencies. The response of the boundary layer and free-surface is described and quantified using instantaneous results and Fourier analysis. The hydrodynamic roll moment is post-processed to give added moment of inertia and roll damping coefficient, which is compared to previous experimental measurements and computational predictions for 2D and 3D rolling bodies. Preliminary results for the surface combatant without forward speed and with free roll decay motion are given where the model is released from an initial angular displacement and the resulting roll motion predicted. The final paper will include simulations with larger roll angles where non-linear effects are important. Also, free-roll decay simulations with forward speed will be performed and the effect of ship appendages on roll damping will be investigated by performing simulations of the 5512 geometry with bilge keels. 1 INTRODUCTION Recent progress in RANS CFD code development and application is making simulation based design an imminent reality. Development of such a tool will allow the merging of traditionally separate naval architecture sub-disciplines for resistance and propulsion, maneuvering, and seakeeping, and when combined with CFD-based optimization, will likely revolutionize the ship design process. Of the three sub-discipline areas, application of RANS methods to resistance and propulsion is the most advanced with nearly two decades of experience. Existing approaches are able to predict ship resistance with reasonable accuracy as shown from results for three steady flow test cases at the recent Gothenburg 2000 Workshop on CFD in Ship Hydrodynamics, Larsson et al., (2000~. Methods have recently been applied to optimize hull forms for a variety of objective functions for ships with steady forward speed in calm seas (Tahara et al., 2000~. In comparison, application of RANS methods to maneuvering and seakeeping is less mature due to obstacles from unsteady flows, ship motions, complex environment (e.g., incident waves, wave breaking, bubbly flow) and increased required computer resources. Also, methods developed for resistance and propulsion may not be easily extended into these areas. Application of RANS methods for steady maneuvers (e.g., off-design yaw, steady turn) can be found in Tahara et al. (1998~; Alessandrini and Delhommeau (19981; and Di Mascio and Campana (19991. However, application for more complex unsteady maneuvers is rare and most investigations rely on motion simulation programs with empirically derived coefficients. Typical seakeeping solution techniques are based on assumptions of small amplitude motions and potential flow so that the general 6DOF non-linear equations of motions are reduced to two separate sets of linear equations (i.e., vertical plane motions become uncoupled from horizontal plane motions) and are solved in the frequency domain. Within those assumptions, predictions show good agreement for vertical plane motions. For horizontal plane motions, seakeeping codes based on potential flow methods model viscous effects by incorporating empirically derived roll damping data. For example, Taz Ul Mulk and Falzarano (1994) used a linear frequency- dependent hydrodynamics model augmented by empirically derived linear and non-linear roll

damping data from Himeno (1982) and non-linear roll and heave/pitch restoring forces. Predictions with these methods are limited to the range of geometry, frequency, and operating parameters from the empirical data and suffer from scale effects. Simulations in the time-domain have been performed to predict larger non- linear roll motions, but most suffer from use of potential flow methods with empirical damping data (Tanizawa and Naito, 1998~. Thus, there is a critical need for development of numerical methods for viscous flows and prediction of large amplitude motions for surface ships with appendages. In an effort to develop a physics-based approach, RANS methods have recently been applied to prediction of roll motion of oscillating bodies. These methods have the potential to produce superior results since effects due to viscosity, creation of vorticity in the boundary layer, vortex shedding, and turbulence are naturally included. RANS methods were used to study the flow around 2D oscillating cylinders by Korpus and Falzarano (1997), Sarkar and Vassalos (2000), and Yeung et al. (1998~. Accurate prediction of forces and moments on a 3D submerged cylinder fitted with bilge keels and with prescribed roll motion was demonstrated in Kim (2001~. Prediction of pitch and heave motions for the Wigley hull and Series 60 cargo ship advancing in regular head waves with a RANS method was demonstrated by Sato et al. (1999~. Free roll decay motion of a barge in calm sea and roll motion in incident waves were performed using a Chimera RANS method by Chen et al. (2001~. Development of unsteady RANS methods within the ship hydrodynamics group at IIHR Hydroscience and Engineering Lab (IIHR) enables extensions from previous applications for resistance and propulsion to applications for seakeeping and maneuvering. A step- by-step approach was followed towards this goal by initially performing unsteady RANS simulations for the forward speed diffraction (ship advancing in waves but constrained from motions) and radiation (prescribed ship motions in calm water) problems. In the former case, simulations were performed for the Wigley hull for a wide range of conditions (Rhee and Stern, 2001) and for the surface combatant, DTMB model 5512 (5512), for medium speed/long wave and high speed/short wave conditions (Wilson and Stern, 2002) including comparisons with IIHR towing tank data (Gui et al., 2002; Longo et al., 2002~. In the latter case, simulations were performed for a double body 5512 model with prescribed vertical (pitch and heave) and horizontal (roll) plane motions as part of a DoD challenge project on 6DOF motions and maneuvering for surface ships (Kim, 20011. The objective of the present work is to extend previous work to unsteady RANS simulations of general 6DOF ship motions and maneuvering, but with focus on applications for prescribed and predicted roll motions for which viscous effects are predominant. As an initial step towards this goal, simulations are performed for 5512 with and without bilge keels for prescribed and predicted roll motions. Simulations with prescribed motions are used to investigate the unsteady response of the turbulent boundary layer, wave, and wake fields to roll motion. Simulations with free roll decay are used to predict roll damping and resonant frequencies for the surface combatant. Simulations are performed using the RANS CFD code, CFDSHIP-IOWA, which was shown to be one of the better codes at the recent Gothenburg 2000 Workshop on CFD in Ship Hydrodynamics (2000) for the surface combatant test case (Wilson et al, 2000~. The CFD results will be used to guide IIHR towing tank measurements, which are then used to validate simulations in a complementary fashion. Verification and validation of simulations follow Stern et al. (2001) and Wilson et al. (2001) and are based on the detailed study for steady flow simulations of 5415 (Wilson and Stern, 2002~. This paper is organized as follows. In Section 2, the RANS solution methodology is presented, while capabilities for prescribed and predicted ship motions are discussed in Section 3. In Section 4 and 5, a description of geometry, data, conditions, and grids is given. In Section 6, verification and validation of the simulations is addressed and results are presented in Section 7. Finally, concluding remarks are presented in Section 8. 2 RANS SOLUTION METHODOLOGY Unsteady RANS simulations are performed with CFDSHIP-IOWA, which is a general-purpose, multi-block, high-performance parallel computing code developed for computational ship hydrodynamics and applied to surface ships and complex propulsors without ship motions. The basic methodology is given in detail in Paterson et al. (2002), and the reader is referred to this paper when solution details are omitted. CFDSHIP-IOWA was recently extended to predict general 6DOF ship motions with large amplitude, non-linear, non-slender geometry for viscous flows. In a mathematical framework, the ship motions problem is cast as an initial boundary value problem (IBVP), which is solved using a RANS solution methodology. Details of the methodology are

presented in this section, including governing equations, initial and boundary conditions, and numerical method. 2.1 Governing Equations In non-dimensional and Cartesian tensor notation form, the unsteady incompressible RANS and continuity equations are written for an inertial reference frame as computational domain in non-orthogonal curvilinear coordinates if, it, (,4. A partial transformation is used in which only the independent variables are transformed, leaving the velocity components Ui in Cartesian coordinates. The continuity (1) and momentum (2) equations in the transformed space are given by aUi =0 t~y J by, (6iiUj)=o (5) axi -O (1) at + Uj Oxj = - Ox; + Re Oxjaxj - Oxj a juj + fib; (2) where Ui = (U. V, W) are the Reynolds-averaged velocity components, xi = (XY,ZJ are the Cartesian coordinates, p is the piezometric pressure ~ p+Z/Fr2 ), ujuj are the Reynolds stresses, fb; are the body-force terms, which represent the effects of the propeller, Fr = U0/~/~; is the Froude number, and Re = UoL/v is the Reynolds number. Equations are normalized by reference velocity UO, ship length L, and density, p. The Reynolds stresses are related to the mean rate of strain through an isotropic eddy viscosity vt ; (aUi aUj ) 2 ,~ k (3) where did is the Kronecker delta, v' is the isotropic eddy viscosity, and k is the turbulent kinetic energy. Substituting (3) for the Reynolds-stress term in (2), the momentum equations become auj +U auj _ aP + ~ abut at ~axj axi Ree~axjaxj +axj (axj + axj )+fbi where P=p+3k 1 1 = _+V. Reek Re (4) Eddy viscosity is calculated using Menter's blended k- ^-e model with the standard model. The equations are transformed from the physical domain in Cartesian coordinates (x,y,z,t) into the aui+ak aUi_ ~ gii 32Ui = an u ark Red Ii 1 k aP --hi k +SU where (6) k I bk (U l by a v, ax ) f <7' sum = R2 (gl2 a{ta;2 + gl3 a3'a¢3 + 23 a a + - b k ~ Go) ( A) (8) . . . and bt', gel, and J are the geometric coefficients, conjugate metric tensor, and Jacobian, respectively. 2.2 Initial and Boundary Conditions Solution of the IBVP requires specification of initial and boundary conditions. For the unsteady simulations with prescribed roll motion, initial conditions are specified using a fully converged steady state solution for the 5512 geometry with uniform forward speed and zero roll angle. Details of the steady state simulation are given in Wilson and Stern (2002) where detailed verification and validation is performed to estimate simulation errors. The steady state solution provides a consistent initial condition at t=0, after which the prescribed roll motion is gradually increased to full magnitude over the first period, thus reducing the time required to damp initial transients and reach a periodic response. For simulations with free roll decay, the ship is given an angular displacement as initial condition and then released and allowed to freely rotate about the roll axis. Specification of boundary conditions are also required for t > 0. Twenty-seven different boundary condition types are available in CFDSHIP-IOWA and can be organized into physical (e.g., no-slip, free- surface), computational (e.g., inflow, far-field, exit),

and multi-block (e.g., patched and overset). Only the free-surface boundary condition and modifications to boundary conditions required for the ship motions problem will be discussed in detail here. Figure 1 shows the starboard side of the full computational domain including coordinate system, grid, and boundary conditions (body SB; free-surface SFS; far-field SFF; exit SE; and multi-block SMB) for the simulations of the 5512 geometry. The free surface boundary conditions are based upon the exact nonlinear kinematic and approximate dynamic free-surface boundary conditions, both of which are applied on the actual free surface. The kinematic equation is a 2D hyperbolic wave equation for the wave elevation ~ (~+U(x+V<y=W (9) Details of the numerical solution of Eqn. (9) are given in Sect. 2.3.3. The dynamic condition requires that stress across the free surface is constant. The three components of stress provide boundary conditions for U. V and P and are derived by neglecting external stress and surface tension and by assuming small curvature and small gradients of normal velocity. Under these assumptions, the following approximate dynamic boundary conditions for pressure and tangential velocity components are given by Fr' (10) 3(U V) ' =0 (11) Liz Lastly, a zero-gradient condition is used for W. which is consistent with the approximations employed for the dynamic condition =.. where n is the ship normal vector and the subscript 'S' is used to denote boundary conditions on the ship hull. Fig. 1 Starboard side of the full computational domain, coordinate system, grid, and boundary conditions for the 5512 surface combatant. 2.3 Numerical Method The continuous momentum equation Eqn. (6) is reduced to algebraic form through the use of finite- differences. The continuity equation is enforced through a pressure-based method where a pressure- Poisson type equation is derived. Details of the numerical method are given in this section. 2.3.1 Discretization For temporal discretization, a second-order accurate, Euler backward difference is used at time level 'n' to achieve time-accurate simulations a I = ~ ( 2 ~ 2¢ + 2 ~ ) For spatial discretization, the convective terms in Eqn. (6) are discretized with a second-order upwind-biased formula LOW A (12) U; ark 2( U' I uil)~ i ( au —| au | ) s; Ui Liz The usual velocity and pressure boundary conditions for a stationary no-slip surface (i.e., Us = 0 and OPs /On = 0 ~ are modified to account for the motion of the ship hull US (t) Us = gas (13) apS = aUS.n (14) On Ot 2 s~j¢=2¢i-2-2¢i-,+2¢i (16) All other first-derivative and viscous terms are discretized using standard second-order central differencing.

Applying the temporal and spatial discretizations given by Eqns. (15) and (16) to the continuous momentum equations (6) gives treatment to avoid odd-even decoupling. Fourth- order artificial dissipation is implicitly added by using a half-cell operator YOU, + ~ AnbU, nb = SU ——b, ~e P (17) EP + NP = age Bali (23) where Aijk and Anb denotes the central and neighboring coefficients of the discretized momentum equations, respectively. The source term, Sue, contains the velocity at the previous time step and the mixed derivative terms originating from the viscous terms which are lagged to the previous iteration. 2.3.2 Pressure-Velocity Coupling The PISO method is used for the pressure- velocity coupling, which uses a predictor-corrector approach to advance the momentum equation in time while enforcing the continuity equation. In the predictor step, the momentum equation (17) is advanced implicitly using the pressure field from the previous time step A' ~ ~ Dn-1 Ai U * ~ A U * S ~ bk or ( 1 8) nb J ~ where superscript '*' is used to denote advancement to an intermediate time level. In the corrector step, the velocity is updated explicitly Ui = Hi JO bi auk (19) using a pressure obtained from a derived Poisson equation and where the psuedo-velocity is defined as d = _(S —~ ~ U ) (20) A Ink nb A pressure-Poisson equation is derived by taking the divergence of equation (19) ~ and bi'U,** =-aS~j bind .~ a; j ( JO i ask ) (21) and by realizing that the LHS of equation (21) goes to zero upon convergence a2;j(~4,,~k ask) ask, ~ i (22) Because a regular, or collocated, grid approach is used, solution of equation (22) requires special where L is the half-cell operator and N is the operator containing mixed-derivative terms ~ {aid (a §~t ) + 3~2 (a §~2 ) + 3~3 (a333~3 )} (24) N = J God, (a]23~2 + ai33~3 ) + 342 (2, + a233~3 ) +~3 (3 + a323~2 )} with 3;i (I) = ((t)i+y2 - (t)i-~/2 ) ' §(i (A = ((hi+ - ¢)i-~ ~ / 2, and aij = JgiiI47k . 2.3.3 Kinematic Free-Surface Boundary Condition Solver (KFSBC) The KFSBC Eqn. (9) is discretized using Eqns. (15) and (16) and solved on the faces of the free-surface blocks. Given the solution for wave elevation A, the volume grid is conformed to the new wave elevation for each free-surface block through the use of linear interpolation. For stability reasons, however, several numerical concerns must be addressed. First, a combination of highly clustered near-wall spacing (i.e., 10-6 or less) and lack of either physical or numerical dissipation (i.e., for the higher-order schemes) results in an unstable numerical system. Secondly, the KFSBC becomes singular at the no- slip/free-surface intersection due to the "contact line" problem. At the intersection point, the fluid velocity is zero due to the no-slip condition and at the same time the wave elevation on the hull is in motion: an obvious contradiction. To address both of these problems, near-wall blanking and solution filtering are used. An approximate contact-line model is implemented wherein the highly-clustered near-wall region is blanked out for solution of A. Values of ~ in this region are obtained using linear extrapolation. To maintain stability and eliminate spurious oscillations, {is filtered with a sixth-order filter after each iteration. Solution procedures for the KFSBC equation have been shown to be stable and accurate for simulation of the surface combatant without motions

In calm sea and in regular head waves. However, for simulations with ship motions, additional damping of the free-surface is required to stabilize the solution in the region behind the transom face. In this localized region, a weighted average of the wave elevation from the steady solution is used to stabilize the simulation. 2.3.4 Summary With a description of the numerical method given, the overall solution procedure is summarized below for a given time step: 1. Define the six ship motion components as described in Section 3. 2. Translate and rotate all computational blocks using components from step 1 and conform the free surface blocks to the ship hull and current wave elevation. 3. Solve the KFSBC Eqn. (9) and conform grid to the wave field. 4. Calculate transformation metrics including grid velocity terms mica . 5. Solve two-equation turbulence model equations and calculate eddy viscosity. 6. Execute the predictor step by solving the momentum Eqn. (181. 7. Execute the corrector step by solving the pressure Eqn. (23) and correcting the velocity field using Eqn. (19~. The corrector step is repeated until velocity corrections are negligible. For the unsteady roll simulations, five corrections were used to reach convergence for the corrector step. 8. Post-process results and output to file for diagnostic and visualization purposes. 9. Advance to the next time step. The method is fully implicit and therefore requires iterative solvers. Currently, a line-ADI scheme with pen/a-diagonal solvers and under relaxation is used to solve the algebraic equations that arise from discretization of the momentum, pressure Poisson, and KFSBC equations. To achieve time-accurate simulations, the iterative solvers are run until a pre- selected convergence criterion is met. The tolerances for solution of the KFSBC in step 3, the momentum equation in step 6, and the pressure equation in step 7 are 1x10-6, 1x10-5, and 1x10-5, respectively. On average 5, 10, and as many as 500 iterations are required to reach convergence for solution of the KFSBC, momentum, and pressure equations, respectively. The simulations were performed using 24 processors of the SGI Origin 3000 parallel machine at ARL and typically required 800 total CPU hours to simulate three periods of rolling motion. 3 SHIP MOTIONS Ship motions are either prescribed from an input file or predicted by integrating the equations of rigid body motions as described in this section. 3.1 Prescribed Motions For simulations with prescribed 6DOF ship motions, the time history of the translation of the center of rotation (OR' YR, ZR) and the ship orientation is specified from an input file. For simulations presented here, only pure roll motion is considered so that the grid coordinates at time level 'n+1' (xn+~,yn+7,zn+~) are computed from the base coordinates (XB,YB,ZB) using a solid body rotation inn+! = XB Y YR + ~ YB YR ~ COS (,1?')—(\ ZB—ZR ~ Sin (~) zn+l = z + (I YB—YR ~ sin (mat) + (\ZB ZR N) ~ ~ A sinusoidal function is used to specify the roll angle O(t) = A sin (2~ f t + ~) (26) where A, f = 1/:, a, and ~ are the amplitude, non- dimensional frequency, non-dimensional period, and phase angle of the roll motion, respectively. 3.2 Predicted Motions Prediction of ship motions is accomplished through development of software modules to solve the full, non-linear rigid body equations of motion for the ship trajectory and orientation. Prediction of fully coupled motions will require flow solution in a ship- fixed, non-inertial reference frame where ship geometric properties (e.g., moments and products of inertia) are independent of time. For this case, the dynamic rigid body equations in the ship-fixed reference frame must be supplemented with kinematic equations which relate velocities and angular rates in the body-fixed frame with ship trajectory and orientation is the earth fixed frame. However, for pure roll simulations considered here, these issues do not arise so that an inertial coordinate system is used (i.e., the grid coordinates are a function of time). Roll motion is predicted by numerically integrating the rigid body equation of motion where the time rate of change of angular momentum about the roll axis is balanced by external roll moments L acting on the ship

.. L=Ix~ where Ix is the moment of inertial about the roll axis and the roll moment L is composed of contributions from hydrodynamic pressure and friction buoYancY. and shin weight. art---,, A-- - -- r 4 GEOMETRY, DATA, AND CONDITIONS 4.1 Geometry Unsteady RANS simulations are performed for the Model 5512 geometry which is representative of a modern full-scale naval combatant with sonar dome and transom stern (Fig. 21. Geometric properties for this model are given in Table 1. Currently, the moment of inertia about the roll axis Ix has not been measured experimentally. To perform free roll decay simulations a value of Ix = 5.75x10-5 is used until an experimentally measured value is available. Table 1. Geometric properties of Model 5512. Parameter Model 5512 Beam/length, B/L 0.1266 Draft/length, T/L 0.0702 Wetted surface area, S/L2 0.1475 Displacement, S/L3 4.502x10-3 Block coefficient CB = V I (LBT) 0.5060 Fig. 2 Model 5512 surface combatant geometry. 4.2 Data For the steady flow over the 5512 without ship motions, a large experimental database exists for the surface combatant, which was procured as part of an international collaboration between IIHR, Istituto Nazionale per Studi ed Esperienze di Architettura Navale (INSEAN), and DTMB. Facility and scale effect biases were evaluated for Fr=0.28 (design condition) through overlapping tests at all three institutes as documented in Stern et al. (20001. Recently, this collaboration has been extended to perform experiments for the surface combatant with roll motion where forces and moments, wave elevation, mean velocity, turbulence quantities, and ship motions will be measured. Most of the roll experiments are in the planning phase and currently there is little available data to compare with the present CFD simulations. For the final paper, simulation results will be compared with all available experimental data. 4.3 Conditions Unsteady simulations are performed in calm water with roll motion. Flow conditions for the unsteady simulations are based on EFD steady flow experiments for the 5512 geometry with Fr=0.28 and Re=4.65xl06. Simulations are performed with the ship in the static orientation. Simulations are performed over a range of three rolling frequencies where the roll center of the model was fixed at the design waterline and ship symmetry plane, i.e., (OR' OR, ZR) = (0,0,01. Initially, two frequencies were selected with the lower non- dimensional frequency JO having a roll period equal to the ship time scale T (i.e., T=L/UO and based on ship length L and forward speed UOJ and the higher frequency f=2 selected to be twice that of the lower frequency. Subsequently, the roll natural frequency of the 5512 geometry was estimated to be roughly f~1.18 by scaling the full-scale appended ship value reported in Bishop (1983), which is intermediate between the originally selected lower and higher frequencies. As a result, simulations performed at these three frequencies will be referred to as the sub- resonant, resonant, and super-resonant cases as shown in Table 2. All simulations are performed with roll amplitude and phase angle, A=5 degrees and B=0, respectively. Selection of final frequencies for simulations with prescribed roll motion will be based on CFD predictions and EFD measurements of the resonant frequency for 5512 geometry with free roll decay motion. Table 2. Conditions for unsteady roll simulations for Model 5512. Sub- resonant Resonant Super- resonant 1.0 1.0 1.18 0.85 2.0 0.5

5 COMPUTATIONAL GRIDS AND TIME STEP The computational grid for the unsteady roll simulations is based on the coarsest of three grids used in a study to assess simulation errors and uncertainties for steady free-surface flow around the surface combatant. Details of the verification and validation study and generation of the three grids are given in Wilson and Stern (2002~. A brief description of the three grids and relationship to the grid used for the unsteady roll simulations is provided here. 5.1 Verification and Validation of Steady Flow Steady flow solutions for the surface combatant were performed in Wilson and Stem (2002) so that the flow was symmetric about the ship centerline (i.e., only the starboard side of the ship was simulated). The finest grid was generated using the commercial code GRIDGEN and consists of a hyperbolically generated near-hull grid and algebraic far-field grid (Fig. 11. Grid topology was selected so that the block fixed to the transom face could be conformed around the bottom edge of the transom face. A coarse-grain parallel approach was used where the base grid is decomposed into 24 blocks of varying sizes and each block was mapped to a separate processor on an Origin 2000/3000 machine. The decomposed grid contained 12 free- surface blocks that are dynamically conformed to the wave elevation and ship hull and 12 keel blocks that do not require conforming. For systematic grid refinement with rG = ~/i, the coarsest grid is obtained simply by removing every other point from the fine grid. 5.2 Unsteady Roll Simulations The grids used for the current unsteady roll simulations and the coarsest of the three-grid study contain the same grid number and distribution in the axial and transverse computational directions. For the keel to free-surface direction, the keel and free-surface blocks of the steady coarse grid are joined to improve the grid quality during the conforming process while the ship is rolling. The grid is then mirrored about the free- surface plane to provide an adequate amount of grid for the conforming process as described in Section 2.3.3. The grid is also mirrored about the ship centerline since both port and starboard sides are required to simulate the unsteady asymmetric flow due to the roll motion. Modifications to the grid number in the keel to free- surface direction and mirroring processes results in a 24-block grid system with 0.8M total grid points. 5.3 Time Step The time step is selected so that the temporal evolution of each period of unsteady motion is resolved with 100 time steps, i.e., At = a/ 100, where ~ is the period of the roll motion. 6 VERIFICATION AND VALIDATION Verification and validation of simulations follow Stern et al. (2001) and Wilson et al. (2001) and will be based on the detailed study for steady flow simulations of 5415 (Wilson and Stern, 2002~. 7 RESULTS Discussion and analysis of unsteady results for the surface combatant with prescribed sinusoidal roll motion are provided in this section for boundary layer (Sect. 7.1), free-surface (Sect. 7.2), and forces and moments (Sect. 7.3~. Preliminary results for 5512 with free roll decay are given in Sect. 7.4. Figure 3 shows the time history of the prescribed roll angle and angular velocity normalized by maximum values. During the first period and a half (0 < t < 1.5~), the amplitude of the roll angle is gradually increased to its maximum value of A=5 degrees. The resulting solution undergoes a transient response for the first two periods of prescribed motion, after which the transients decay and a periodic response is achieved for t > 2T. Results are analyzed and presented for one typically cycle of the periodic response (i.e., 2T < t < 3r ). Time sequences at each quarter-phase are used to describe the unsteady results with quarter-phases indicated in Fig. 3. Time sequences of instantaneous results and harmonic analysis are used to quantify the unsteady response of the boundary layer and wave-field to the roll motion. The effect of roll frequency is investigated by performing simulations at three frequencies, although most of the focus in this section is on the sub- and super-resonant cases. In ._ ct a) ._ by o ._ O ._ ct ~ -1 A i Roll Angle, 4)/(P'MAX -1~ Angular Velocity, dO/dt Time, t/: Fig. 3 Motion kinematics for unsteady 5512 simulation with quarter-phase indicated by symbols.

7.1 Boundary Layer Since the volume grid is undergoing both solid- body rotation about the roll axis and deformation (due to grid conforming to free-surface and ship), the solution is transformed from the inertial reference frame to the ship-fixed reference frame and interpolated onto a grid that is fixed in time for analysis purposes. This procedure provides a time history for the flow field at fixed spatial locations so that the harmonic content can be analyzed using Fast Fourier Transforms. Figures 4 and S show a time sequence for axial velocity and vorticity with roll motion for the super- resonant case. In the figure, the ship is advancing from bottom to top with under free-surface perspective. The rolling motion of the ship induces cross flow velocity resulting in unsteady asymmetric axial velocity contours (Fig. 4) and a serpentine motion of vorticies emanating from the sonar dome at ~c/L=O.1 (Fig. S). These unsteady features are in contrast to the steady state solution without ship motions, which shows symmetric and axial velocity and sonar dome vortices with straight vortex cores. The wavelength Jv and period tv for the motion of the sonar dome vortices is estimated from the product of the forward speed of the ship UO and period of the roll motion ~ (i.e., by = UOT ). The motion is analogous to a vibrating string with fixed ends (i.e., an oscillating standing wave with zero displacement at nodal points and maximum displacement between nodal points). The wavelength Jv corresponds to the distance between nodal points, as shown below. The effect of roll frequency is examined by comparing instantaneous results for the super-resonant case (Fig 4a and Sa) with that from resonant and sub- resonant cases at t=~/4 as shown in Fig. 6. The results show the presence of nodal points at x/L=0.5, 0.85, and 1.0 for super-resonant, resonant, and sub- resonant cases, respectively which correspond to wavelength estimates, As = UOT . Contours of first and second harmonic amplitudes from a Fourier analysis of the axial velocity are shown in Fig. 7 for super- and sub-resonant cases. For both cases, the results show minimum amplitudes near nodal points and maximum amplitudes midway between nodal points. The analysis shows that higher harmonics (i.e., third and higher) are at least one order of magnitude smaller than first harmonic amplitudes and thus negligible. The unsteady evolution of the sonar dome vorticies is analyzed by tracing the port and starboard vortex core locations during one period of motion as shown in Fig. 8. A harmonic analysis of the transverse vortex core locations shows axial locations for amplitude maximums and minimums (Fig. 9), which are consistent with those for axial velocity. Results for first harmonic phase angle shown in Fig. 9 are used to quantify phase lags in the boundary layer vortex and is consistent with wavelength estimates (i.e., the phase angle completes one cycle from O to 2~ within one spatial wavelength). 7.2 Free-surface Instantaneous total g and fluctuating ~ - {M~ wave elevation at quarter phases are shown in Figs. 10 and 11, respectively. Recall that at t = if/ 4 ~ t = 3~/ 4 ), the ship has rolled to maximum positive (negative) roll angle as shown in the Fig. 11 insert. Generation and propagation of fluctuating wave contours are described in terms of viscous and pressure effects at the hull surface due to the roll motion. Due to the viscous no-slip condition, the rolling motion of the ship induces a cross-plane velocity VNO_S,IP at the ship hull, which is in phase with angular velocity d ~?/dt (Fig. 3) - VNO_SLIP= d~ xr (28) where r is the position vector from roll axis to the point on the hull. The magnitude |VNO-5LU| is given |VNO-SLIP | = | al |~|r| = A0)|r|cos (cot) (29) with roll angle ~ from Eqn. (26~. Therefore, viscous effects generate a vertical velocity WNO_SLIP at the intersection of the ship hull and free-surface that is a wave elevation source on one side of the ship and a sink on the other, depending on the phase of the motion. For example, a source exists on the port side for theinterval, r14<t<3~14. Pressure forces at the surface of a rolling body generate waves due to transverse displacement of fluid during the roll motion. This concept is used to generate waves with plunging- and paddle-type wavemalcers. In contrast, a 2D circular rolling cylinder cannot generate waves through pressure effects.

~ `: - A ~ (d) t=t | ~ ' -,,)_ ., a_ '~_ ~ ,! _ _ I'',,. _ ;~ '' I,.. 1'—~ Fig. 4 Time sequence of axial velocity contours U for 5512 simulations with prescribed roll motion (f=2J: (a) t=~4; (b) t=~2; (c) t=3~4; and (d) t=r _ Few Fin ~ : - - he [ (d) to= | it. ... he, .. .. - .... ..... as: . 5.00 9.00 3.00 3.00 -9.00 5.00 Fig. 5 Time sequence of axial vorticity contours (x for 5512 simulations with prescribed roll motion (f=2): (a) t=~4; (b) t=~2; (c) t=3~4; and (d) t=z:

| (a) U(,~l ~ \W'~ .1& ~ ': ;~ ~ - 1 ~ ~ ~ l_ ;~ ~ tic' u t=T/4 ~ i, i' v_...: en' be.; _ ~,.J in' Fig. 6 Instantaneous axial velocity U and vorticity {x at t= ~4 for (a), (b) resonant case (f=1.18) and (c), (d) sub-resonant case ~1~. - , ~ I(c)W - - a ~ 0.1` 0.1( O.O' ~ 0.0, En o.o~ 0.0' ~ 0.0 — 0.0 . ~ ., _ . ~ (d)(X,t=~/41 ...,..,,,_ ''' - 'I I''' me,. - 1' z _ r - ~ . _ _~U(2) A_ Z _] _ l_ E_ _. _ . _ _ At.. .......... 0.12 0.10 ~ o.os EN 0.07 ~ 0.06 1~1 0.04 0.03 0.01 Fig. 7 Harmonic analysis of axial velocity for (a), (b) super-resonant case ~2) and (c), (d) sub-resonant case (f=11. (a), (c) first harmonic and (b), (d) second harmonic amplitudes.

0.04 _ ~0 t=0.2T .— — — · t=0.4T t=0.6T t=0.8T .—- _ t=T ,~ .: _ S'~A - (a? ~ MA -u us O2 ORAL 0.6 0 8 0.04 _ : ~,0 On4 , , , -.- 0.2 04xIL 06 o 8 Fig. 8 Time sequence for transverse location of sonar dome vortex cores for (a) super-resonant case ~-2) and (b) sub-resonant case ~1~. 0.02 0.01~ O v 0 0.2 0.4 0.6 Axial Distance, xlL · Amplitude (f=2) - - - - - - Phase Angle (f=2) · Amplitude, (f=1) - - - - - - Phase Angle (f=1 ) In< - _ ~ ~ / 540 450 360 270 180 90 Fig. 9 Harmonic analysis of transverse location of sonar dome vortex cores. Examination of a time sequence at one typical ship cross-section shows generation of waves due to frictional and pressure effects. Focusing on the starboard side, a wave crest is generated during the first quarter period due to viscous effects since the starboard side is rolling up and generating positive vertical velocity at the free-surface. During the second and third quarter period, the roll motion reverses and pressure effects cause the crest to propagate away from the hull in the transverse direction, during which time viscous effects act to create a wave trough at the free-surface, which propagates in the transverse direction during the fourth quarter period. Waves are generated in a similar manner on the port side, but are 180 degrees out of phase with the starboard side. The importance of viscous effects is mainly determined by Re and roll speed |VNO-SIU | which depends on roll amplitude A, frequency lo, and cross-sectional size through r as shown by Eqn. (29~. Importance of pressure effects is determined by the transverse displacement of fluid at the free-surface, which depends on roll frequency and shape of the cross-section at the free-surface. The above discussion applies to 2D rolling bodies, while the problem considered here is more complicated due to the effects of varying cross- sectional shape with axial distance, presence of transom stern, and forward speed, which generates a steady wave system. Excluding the transom region, the roll motion and variation in cross-sectional shape results in a transverse propagating wave system traveling upstream and originating on the forebody (0.3 < C/L < 0.5) as shown in Fig. 11. To facilitate a harmonic analysis of the free- surface, the wave elevation is interpolated at every time step onto a 2D grid which is fixed in time. A Fourier analysis is performed on time histories at each point to quantify the response of the wave-field to the roll motion. Results from the harmonic analysis of the wave elevation are shown in Fig. 12 where first harmonic amplitude and phase angle are given for the super- and sub-resonant cases. First harmonic amplitude in Fig. 12a shows maximums on the forebody due to the upstream traveling wave system as discussed above. The gradient of phase angle contours V(3 in Fig. 12b is used to indicate wave propagation direction and speed, in the same manner that the temperature gradient is used to visualize heat flux vectors. Since the gradient is mainly increasing in the transverse direction, the figure confirms that waves propagate from the ship

Fig. 10 Wave elevation contours ~ for 5512 simulations with prescribed Fig. 11 Fluctuating wave contours, ~ - (M~ J=21: (a) t=~4; (b) t= ~2; roll motion (f=2): (a) t=~4; (b) t=~2; (c) t=3~4; and (d) to=: Cross-section (c) t=3~4; and (d) t= ~ through sonar dome showing ship orientation (insert). 1 ~ 1 -360-300-240-180-120 60 0 60 -0-~ - 1 (b) 7(1) 1 l l l Fig. 12 Harmonic analysis of wave elevation for (a),~b) super-resonant case ~2) and (c),gd) sub-resonant case (f=1~. First harmonic amplitude A(1J (a),~c) and phase angle 71J (b),td).

hull towards the far-field, except at the forebody (~/L=0.25), where phase angle contours are curved and more closely spaced indicating upstream propagation at larger speed. Examination of higher harmonics shows that the free-surface response is largely first harmonic at the roll frequency. Comparison of first harmonic amplitude contours for super- (Fig 12a) and sub-resonant (Fig 12c) cases show similar peak values at the forebody (x/L=0.251. In comparison with the super-resonant case, the second region of peak amplitude (0.3 < C/L < 0.5) is absent for the sub- resonant case and a large region of near-zero amplitude exists on the afterbody (0.5 < C/L < 1.0) near the hull. Comparison of phase angle contours show straight contours for the sub-resonant case (Fig. 12d), indicating uni-directional wave propagation at an angle of 23 degrees to the ship centerline, in contrast to curved contours for the super-harmonic case (Fig. 12b). Also, phase angle contours indicate wavelengths of 0.2L and 0.4L for super- and sub-resonant cases, respectively. 7.3 Forces and Moments Elemental hydrodynamic pressure and frictional forces and moments about the roll axis are integrated over the hull surface at each time step to give a time history of resultant forces and moments. Total roll moment MT is shown in Fig. 13 as well as frictional and pressure components (e.g., MT=MF+MP). Figure 13 shows that the total roll moment is dominated by the pressure component for both cases (i.e., MF < 5%MT) and a Fourier analysis of MT shows first harmonic response. Larger roll angles will be required to excite higher harmonics with non-linear response. For the super-resonant case, the pressure and frictional components lag the roll motion by 22 and 76 degrees, respectively, since the maximum roll angle occurs at 90 degrees of phase. Trends for the sub-resonant case are similar with a roll moment amplitude of 45% of the super-resonant value. - c Velocity and pressure are interpolated onto equally spaced y-z planes so that the cross-sectional roll `~ moment MT can be computed to show the contribution O of each station to the total roll moment MT. This procedure gives a time history for the roll moment at each axial station which is analyzed to yield Fourier amplitude and phase angle as shown in Figs. 14a and 14b, respectively. The first harmonic amplitude shows large peaks at the front (~=0.01) and rear ('c/L=0.08) of the sonar dome, with contributions from the sonar dome (O<x/L<O.11) and afterbody (0.4<'c/L<1) being 14%MT and 60%MT, respectively. The pressure component for the afterbody region is roughly constant with mean value equal to the total roll moment MT. Figure 14b shows that the pressure component on the front of the sonar dome lags the roll motion by 90 degrees and that on the rear by 180 degrees. Aft of the sonar dome at x/L=0.25, the pressure component leads the roll motion by 45 degrees and decreases with axial distance, indicating that upstream stations lag downstream values. This is consistent with conclusions from analysis of the free-surface which showed upstream propagating wave elevation contours. 0.0002 F a' to ol c o in: . , . ...................................... ~ . ~ ——~ ~ - MF tf=1 ) - MP (f=1 ) = .- ,. -n none .. ...... i.- . _ I I . i I I 0 0.25 ——E—-— MF (f=2) _~ ~ MP (f=2) \ // O . MT (f=2) \~ '/ . ~ me, , . . . . , 0.5 0.75 1 time, to Fig. 13 Time history of roll moment MT for 5512 simulation with roll motion for super- and sub- resonant cases. 0.0008 O) 0.0006 ._ Q 0.0004 .O o ct I, 0.0002 - 'em O( ) 0.2 ~ 0-4 AL O.6 ~ ~ 0.8 '' 'a 1 90 , 45 o -__,,—(b) 'A—e' ''53,~. ~ f ~~ I -~VI ) 0.2 0 4 XIL 0.6 0.8 1 Fig. 14 Harmonic analysis of cross-sectional roll moment MT for super and sub-resonant cases first harmonic (a) amplitude and (b) phase angle.

Although the approach presented here is applicable to fully non-linear ship motions, traditional solution procedures for seakeeping problems use linear assumptions so that the problem can be split into the two separate problems of forward speed diffraction and radiation. For the radiation problem with linear assumptions, the response of forces and moments are assumed to be purely first harmonic and proportional to the motion and its first and second time derivative, with the expression for pure roll given by .. . MT = -A44~? - B44~ (30) The constants of proportionality for the first and second derivative terms in Eqn. (30) are referred to as the roll damping B44 and added roll moment of inertia A44, respectively. To facilitate comparison with EFD and CFD results for 2D and 3D rolling bodies, the total hydrodynamic roll moment is transformed to the frequency domain where the imaginary MT, and real MTR parts are used to compute the A44 and B44 values, respectively A44 = ~ MT B. - ~ M Normalized coefficients are given by (32) a = A44 `33' b = B44 ~ `34> where V and B are the ship displacement and beam, respectively. Normalized values are shown in Fig. 15 for the three frequencies along with comparison of the current results with those from EFD and CFD for 2D and 3D rolling bodies. In Fig. 15a, added moment values from the present CFD results and experimental measurements from Tanaka, et al. (1983) for a 3D fishing vessel show large differences when compared to results for 2D rolling bodies. Present values for the surface combatant and measurements for the fishing vessel show a similar trend of decreasing am with frequency. Roll damping coefficients for the fishing vessel, cargo ship, and surface combatant all show increased damping with frequency. However, large differences in value and slope exist between the three geometries. Experimental measurements for the 5512 geometry will be required to quantitatively validate the present results. The cross-sectional roll moment MT was also post-processed to give cross-sectional added moment A A A44 and roll damping B44 coefficient with normalized values given by a44 and b44 . Normalized values are shown in Fig. 16 for the super- and sub- resonant cases. The results show positive roll damping over most of the ship with cross-sectional values consistent with those for total roll damping except at the rear of the sonar dome. Negative damping occurs at this location since the pressure component leads the roll angle by 90 degrees as discussed above. ~ G ~ t::1 O Young (1975), 2D Square Body (Expt) ~ Young (1975), 2D Square Body (Inviscid) _ Sarkar (2000), 2D Square Body (RANS) , ~: Tanaka (1983), 3D Fish. Vessel (Expt) _~ Present, 3D Surface Combalam (PANS) <A ~~ Id (a) o 0 0.5 1 1.5 2 (31) ~* O Young, ED Square Body (Expt) l —- - Young, 2D Square Body (Inviscid) 0 05 sew ED Square Body (IRONS) n Tanake(1983), Fish. V—so (Expt) n ~ Ikoda (1978), 3D Cargo, w/o BK (Expt) . p Ike (1978), 3D CA`90, w/BK (Expt) ~ + Present, 3D Surface Combs tent ( rlANS) ~ i~ - /o0° ~ ~ lo ~ 570 /' / ~ - / O <>a ~ ~ - 0.04 0.0% D on' 0.01 ~ ~ a - (b) u 0 0.5 1 1.5 2 Fig. 15 (a) Added moment of inertia am and (b) damping coefficient b44 for 2D and 3D bodies. no OF -O.z -o.' -0.60— 0 ~ ¢~ ,~ 4^' ~ 6` ':~~~-~' 8.' O O O G , . ~ a44 (f=1) _ O b44 (f=1) a44 (f=2) _ ~ b,, (f=2) _ 0.2 0 ^/L 0 6 0.8 1 Fig. 16 Cross-sectional added moment a44 and A damping b44 for super- and sub-resonant cases.

7.4 Free Roll Decay Preliminary results for simulation of the surface combatant without forward speed and with free roll decay are given in this section. To simulate and predict roll decay motion, the ship is given an angular displacement as initial condition and then released and allowed to freely rotate about the roll axis. The ship restoring moment damps the resulting sinusoidal motion, which eventually decays to a steady state condition. Figure 17 shows the time history for the total roll moment and predicted roll angle after the model is released from an initial angular displacement of 5 degrees at t=0. ~ 2 - - <5: o ~ -2 _A 6 1 _ Roll Angle O.- ~ Roll Moment _ 0.001 O E _ 0 _ o it, ,vv~~ ~ . T! _ ~ \'v'' ':.,'jl 3;" I' '' I '-- _ ·,', \ ~ ail+.+ , ~ 'hi V, rj ~ '~ ·'~'~,, of%<,< .~ %~' _ :~ !< ~ ~~ -0.001 -6, ~ O Time, VT 1 5 Fig. 17 Time history of predicted roll angle and moment from free roll simulation. 8 CONCLUDING REMARKS A RANS based method for prediction of flow around ships with rolling motion was presented in this paper. Unsteady simulations are used to investigate the response of the boundary layer and wave-field to rolling motion. The effect of roll frequency was demonstrated by performing simulations at three frequencies corresponding to estimates of sub-resonant, resonant, and super-resonant natural roll frequencies for the 5512 geometry. It was shown that the rolling motion of the ship resulted in an unsteady oscillatory motion of the boundary layer dependent upon position along the axial direction. The unsteady motion was described using a standing wave analogy with reduced transverse displacement at nodal points and maximum displacement between nodal points. The distance between nodal points (i.e., the wavelength) was found to be dependent on the product of the forward speed and roll period. Analysis of the free-surface showed that the fluctuating wave pattern was dominated by upstream traveling waves with amplitude, direction, and velocity dependent on rolling frequency. Time histories of axial force and roll motion show linear response to roll motion at this modest roll amplitude (A=5 degrees). Analysis of the hydrodynamic rolling moment was used to compute added moment of inertia and damping from a linear theory allowing the present results to be compared with computational and experimental values for other 2D and 3D rolling bodies. Preliminary results were presented which demonstrated prediction of free roll decay motion of the surface combatant without forward speed. The final paper will include simulations with larger roll angles where non-linear effects should be important. Free-roll decay simulations with forward speed will be performed and the effect of ship appendages on roll damping will be investigated by performing simulations of the 5512 geometry with bilge keels. 9 ACKNOWLEDGEMENTS This research was sponsored by Office of Naval Research grant N00014-01-1-0073 under the administration of Dr. Patrick Purtell. The authors would like to acknowledge the DoD High Performance Computing Modernization Office and the DoD Challenge Program. Simulations were performed at the Army Research Lab Major Shared Resource Center using both the SGI Origin 2000 and 3000 machines. REFERENCES Alessandrini B. Delhommeau G. "Viscous Free Surface Flow past a Ship in Drift and in Rotating Motion," Proceedings of the 22nd ONR Symposium on Naval Hydrodynamics, Washington, DC, Aug. 1998. Bishop, R.C., "On the Roll Decay and Seakeeping Performance of the DDG 51 Hull as Represented by Model 5415," Report DTNSRDC/SPD-200-07, 1983, David Taylor Research Center, Washington D.C. Chen, H.C., Liu, T., and Huang, E.T., "Time-Domain Simulation of Large Amplitude Ship Roll Motions by a Chimera RANS Method," Proceedings of the 11th International Offshore and Polar Engineering Conference, Stavanger, Norway, June 17-22, Vol. III, 2001, pp. 299-306. Di Mascio A., Campana E.F., "The Numerical Simulation of the Yaw Flow of a Free-Surface Ship," Proceedings of the 7th International Conference on Numerical Ship Hydrodynamics, Nantes, France 1999. Gui, L., Longo, L., Metcalf, B., Shao, J., and Stern, F., "Forces, Moment, and Wave Pattern for Naval Combatant in Regular Head Waves: Part II:

Measurement Results and Discussions," Experiments in Fluids, 2002, Vol. 32, pp. 27-36. Himeno, Y., "Prediction of Ship Roll Damping State-of- the-Art," Report No. 239, 1982, University of Michigan, Dept. of Naval Architecture, Ann Arbor, MI. Lkeda, Y., Himeno, Y., and Tanaka, N., "Components of Roll Damping of Ship at Forward Speed," Journal of Society for Naval Architects. Japan Vol. 143 1978 p. , , , 113. Kim, K.-H., "Unsteady RANS Simulation for Surface Ship Dynamics," Proceedings of the DoD HPCMP Users Group Conference, Biloxi, MS, June 2001. Korpus, R. and Falzarano, J., "Prediction of Viscous Roll Damping by Unsteady Navier-Stokes Techniques," Journal of Offshore Mechanics and Arctic Engineering, Vol. 119, 1997, pp. 108-113. Larsson, L., Stern, F., Bertram, V., editors, Proceedings of the Gothenburg 2000: A Workshop on Numerical Ship Hydrodynamics, Chalmers University of Technology, Gothenburg Sweden, Sept. 2000. Longo, J., Shao, J., Irvine, M., Stern, F., "Phase- Averaged PIV for Surface Combatant in Regular Head Waves," Proceedings of the 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, July 2002. Paterson, E., R. Wilson, and F. Stern, "General Purpose Parallel Unsteady RANS for Ship Hydrodynamics," to be submitted to Computers and Fluids, June 2002. Rhee, S. and Stern, F., "Unsteady RANS Method For Surface Ship Boundary Layer And Wake And Wave Field" Int. J. Num. Meth. Fluids Vol. 37 2001, pp. , , , 445-478. Sarkar, T. and Vassalos, D., "A RANS-Based Technique for Simulation of the Flow Near a Rolling Cylinder at the Free-Surface," Marine Science and Technology, Vol. 5, 2000, pp.66-77. Sato, Y., Miyata, H., and Sato, T., "CFD Simulation of 3-Dimensional Motion of a Ship in Waves: Application to an Advancing Ship in Regular Head Waves," Marine Science and Technology, Vol. 4, 1999, pp.108-116. Stern, F., Longo, J., Penna, R., Olivieri, A., Ratcliffe, T., and Coleman, H., "International Collaboration On Benchmark CFD Validation Data For Surface Combatant DTMB Model 5415," Proceedings of the 23rd ONR Symposium on Naval Hydrodynamics, Val de Reuil, France, Sept. 2000. Stern, F., Wilson, R.V., Coleman, H., and Paterson, E., "Comprehensive Approach to Verification and Validation of CFD Simulations-Part 1: Methodology and Procedures," ASME J. Fluids Eng., Vol. 123, 2001, pp. 793-802. Tahara, Y., Longo, J., Stern, F., Himeno, Y., "Comparison of CFD and EFD for the Series 60 CB=0.6 in Steady Yaw Motion," Proceedings of the 22nd ONR Symposium on Naval Hydrodynamics, Washington, DC, Aug. 1998. Tahara, Y., Paterson, E., Stern, F., and Himeno, Y., "CFD-Based Optimization of Naval Surface Combatant," Proceedings of the 23rd ONR Symposium on Naval Hydrodynamics, Val de Reuil, France, Sept. 2000. Tanaka, N., Himeno, Y., and Sakaguchi, S., "Study on Roll Motion Characteristics of Small Fishing Vessel, Part 3 Effects of Over-Hung Deck," Journal of Kansai Society of Naval Architects, Japan, No. 196, 1985. Tanizawa, K. and Naito, S., "An Application of Fully Nonlinear Numerical Wave to the Study of Chaotic Roll Motions," Proceedings of the 8th International and Offshore Polar Engineering Conference, Honolulu, Hawaii, Vol. 3, 1998, pp. 280-287. Taz U1 Mulk, M. and Falzarano, J., "Complete Six- Degree-of-Freedom Nonlinear Ship Rolling Motion," Journal of Offshore Mechanics and Arctic Engineering, Vol. 116, 1994, pp. 191-201. Wilson, R., Paterson, E., and Stern, F., 2000, "Verification and Validation for RANS Simulation of a Naval Combatant" Proceedings of Gothenburg 2000: A Workshop on Numerical Ship Hydrodynamics Chalmers University of Technology, Gothenburg Sweden, Sept. 2000. Wilson R., Stern, F., Coleman, H.' and Paterson, E., "Comprehensive Approach to Verification and Validation of CFD Simulations-Part2: Application for RANS Simulation of A Cargo/Container Ship,'' ASME J. Fluids Eng., Vol. 123, 2001, pp. 803-810. Wilson, R. and Stern, F,4'Verification and Validation for RANS Simulation of a Naval Surface Combatant'' 40'h Aerospace Sciences Meeting and Exhibit, Reno, Nevada, AIAA 2002-0904' 2002a. Wilson, R. and Stern, F., "Unsteady RANS Simulation of a Surface Combatant in Regular Head Waves," in preparation, 2002b. Yeung, R. W., Liao, S.-W., Roddier, D., "On Roll Hydrodynamics of Rectangular Cylinders," Proceedings of the 8th International Offshore and Polar En~aineerinta Conference, Montreal, Vol. III, 1998, pp.445-453.

DISCUSSION Ernie O. Tuck University of Adelaide, Australia In your forced-roll cartoon of the fluctuating part of the disturbance, the waves just seem to disappear as they approach the edge of the computational domain. Should they not continue until they reach (and pass across or be reflected by) the edge of the computational domain? This would surely be so unless the viscosity is being overestimated relative to the real-world value, so that too large a damping of waves occurs. In real life such waves would not be so strongly damped. AUTHORS' REPLY The focus of the current paper is on accurate prediction of near field viscous and pressure forces, which is required for accurate prediction of general 6DOF ship motions. Resolution of the turbulent boundary layer at the ship hull requires fine grid spacing in the wall normal direction with expansion of the grid towards the far-field (see Fig. 1), leading to some damping of the waves as they approach the far-field. In Wilson and Stern (2002b), verification and validation is performed for the current application to access numerical error and uncertainty by performing simulations on three systematically refined grids. The results show that the current grid accurately resolves the free-surface in the near-field, with uncertainties for the wave profile on the hull of less than 1% of the bow wave height and 5.5% at a transverse wave cut at 0.1 7L from the ship hull. Thus, grid resolution in the far-field must be sacrificed to accurately resolve the viscous forces at the ship hull and to predict general 6DOF ship motions.