Journal of Ship Research

Transcription

1 Journal of Ship Research, Vol. 53, No. 4, December 009, pp Journal of Ship Research Model- and Full-Scale URANS Simulations of Athena Resistance, Powering, Seakeeping, and 5415 Maneuvering Shanti Bhushan, Tao Xing, Pablo Carrica, and Frederick Stern IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa, USA This study demonstrates the versatility of a two-point, multilayer wall function in computing model- and full-scale ship flows with wall roughness and pressure gradient effects. The wall-function model is validated for smooth flat-plate flows at Reynolds numbers up to 10 9, and it is applied to the Athena R/V for resistance, propulsion, and seakeeping calculations and to fully appended DTMB 5415 for a maneuvering simulation. Resistance predictions for Athena bare hull with skeg at the model scale compare well with the near-wall turbulence model results and experimental fluid dynamics (EFD) data. For full-scale simulations, frictional resistance coefficient predictions using smooth wall are in good agreement with the International Towing Tank Conference (ITTC) line. Rough-wall simulations show higher frictional and total resistance coefficients, where the former is found to be in good agreement with the ITTC correlation allowance. Self-propelled simulations for the fully appended Athena performed at full scale using rough-wall conditions compare well with full-scale data extrapolated from model-scale measurements using the ITTC ship-model correlation line including a correlation allowance. Full-scale computations are performed for the towed fully appended Athena free to sink and trim and the boundary layer and wake profiles are compared with full-scale EFD data. Rough-wall results are found to be in better agreement with the EFD data than the smooth-wall results. Seakeeping calculations are performed for the demonstration purpose at both model- and full-scale. Maneuvering calculation shows slightly more efficient rudder action, lower heading angle overshoots, and lower roll damping for full-scale than shown by the model scale. Keywords: resistance (general); powering estimation; sea keeping; maneuvering 1. Introduction IN THE past few years, tremendous advances have been made in the development and validation of computational fluid dynamics (CFD) for ship hydrodynamics, which has the potential of assisting in ship design (Larsson 1997, Gorski 00, Stern et al. 006a). As documented in the previous symposiums on naval hydrodynamics, most of the CFD and EFD studies have focused on model-scale ship flows; that is, Reynolds number Re 10 7, including resistance, propulsion, seakeeping, and maneuvering (e.g., Crook 1981, Miller et al. 006, Carrica et al. 006a, Longo et al. 007). In contrast, full-scale ship flow computations (Re 10 9 ) Manuscript received at SNAME headquarters January 8, 008; revised manuscript received October 6, 008. are limited (Gorski et al. 004, Hanninen & Mikkola 006). This is the result of the challenging issues with regard to both numerical methods and turbulence modeling. The overall objective of this study is to extend the unsteady Reynolds averaged Navier-Stokes (URANS) solver, CFDShip-Iowa (Carrica et al. 007a, 007b), to simulate full-scale ship flows using wall functions for ship hydrodynamics applications including resistance, propulsion, seakeeping, and maneuvering. Also of interest is the application of wall-functions at model scale to reduce the number of grids points near the wall. Numerical issues for full-scale simulations include several aspects. Near-wall turbulence models require high grid density near the wall, usually unaffordable for ship geometries. The extremely small grid spacing in the wall normal direction near the wall causes huge aspect ratios of grid cells. This significantly increases the DECEMBER /09/ $00.00/0 JOURNAL OF SHIP RESEARCH 179

2 errors in mass and momentum flux calculations and may lead to solution divergence. Implementation of lower-order numerical schemes such as first order will likely overcome this problem but with the penalty of significant loss of accuracy. Development and validation of near-wall turbulence models for the Re of full-scale ship flows are lacking because EFD data are sparse even for canonical flows and difficulties of maintaining fixed environmental conditions to conduct practical full-scale ship measurements (Patel 1998, Aupoix 007). The use of wall functions avoids the numerical limitations of the near-wall turbulence models and significantly reduces the computational cost. Wall functions are based on the near-wall behavior of nonseparating two-dimensional turbulent boundary layers, also valid along the streamwise direction of three-dimensional (3D) turbulent boundary layers with small cross flow (Bradshaw & Huang 1995). Thus the obvious limitations of the wall functions are in accurately predicting separated flows and 3D boundary layers with significant cross flow. For example, in a backward facing step flow the reattachment length is often underpredicted in wall-function simulations. Nevertheless, near-wall turbulence models also suffer from the same deficiency as the model constants are derived under similar turbulent boundary layer assumptions (So & Lai 1988). Some of the issues addressed in the literature although not fully resolved are: Sensitivity to the spacing of the first grid point away from the wall (matching point herein) Inclusion of pressure gradient effects for better prediction of separated flows Modeling of wall roughness Different implementation approaches for determination of the velocity direction at the matching point. The standard wall-function approach (Launder & Spalding 1974) is based on the stringent criteria that the matching point lies in the log layer (one layer only). However, different flow regimes make it difficult to place the matching point in the log layer, for example, laminar-turbulent transition zone, variation of the boundary layer thickness along the ship hull, or when the ship is slowly accelerated from a static condition. This limitation is addressed by a two-layer model, in which boundary conditions for the velocity and turbulent quantities are switched between the sublayer and log-layer analytic profiles in the near-wall region, depending on the local y + value of the matching point (Grotjans & Menter 1998, Esch & Menter 003). A three-layer model includes an additional formulation for the buffer layer (Temmerman et al. 003), but still suffers from the deficiency that the velocity profile is not smooth across the near-wall region. This deficiency was resolved by Shih et al. (003) who proposed a generalized multilayer model using curve fitting to provide a continuous function to bridge the sublayers and log layers. Kalitzin et al. (005) developed a multilayer model along with look-up tables for evaluating the friction velocity. The look-up tables are obtained from a separate zero-pressure gradient smooth flat-plate simulation using a near-wall turbulence model. This approach leads to an accurate calculation of the friction velocity than the analytic equations used in above models. However, the applicability of this approach for high Reynolds number flows would require examination of the look-up table. The models discussed previously are based on Dirichlet-type boundary conditions. Utyuzhnikov (005) proposed a differential form for the boundary condition valid for the entire boundary layer. Overall, the multilayer wall-function model provides the most flexibility in the placement of the matching point. Evaluation of the pressure gradient effect on wall functions for separated flows with mild pressure gradients was conducted by some previous studies. For a backward-facing step flow, the twolayer models with pressure gradient effect predict the separation, recirculation, and reattachment regions better than that without pressure gradient effect (Wilcox 1993, Kim & Choudhury 1995). Knopp et al. (006) extended the Kalitzin et al. (005) approach for nonequilibrium flows by including pressure gradient effects. Their main conclusion is that in the regions of stagnation and strong pressure gradients, a near-wall solution is the best strategy for which they applied flow-based grid adaptation. In the strong pressure gradient regions, usually encountered in ship flows, clipping of the pressure gradient magnitude to # 75% of the friction velocity is suggested to avoid numerical instability or divergence (Wilcox 1993). Thus the benefit of including pressure gradient effect for wall functions in separated flows is questionable. The effect of surface roughness is more important for full-scale computations than for those at the model scale as it leads to significant increase in frictional and total resistances. The most commonly used model for surface roughness is based on the downshift of the log-layer profile (White 008). Aupoix (007) recently provided a formulation of the downshift of log-law that is in better agreement with EFD data, which could be of future interest. Several studies have investigated the effect of roughness type on the boundary layer profile (Jimenez 004). Schultz (00) performed model-scale experiments on several sanded and unsanded painted surfaces encountered in ship flows. The studies have shown that the roughness type does not show significant effect in the transitional roughness regime (5 # k +, 70) or the downshift of log-layer profile. Thus surface roughness modeling based on the downward shift of the log-layer profile can be used with relative confidence for full-scale ship calculations where roughness length mostly lies in the transitional regime (Patel 1998, Tahara et al. 00). For one- and two-layer wall function models, modeling of roughness effect is straightforward. However, for the multilayer model appropriate correlation is not available for the buffer layer from either experimental or numerical studies. Implementation of wall-function models requires evaluation of the friction velocity, either analytically or using look-up tables (Kalitzin et al. 005), to provide boundary conditions for velocity and turbulence variables. The one-point approach proposed by Kim & Choudhary (1995) uses the flow variables at the wall neighboring cells only. This allows solutions of the momentum equations up to the matching point. The one-point approach can be implemented easily for finite-volume schemes, but introduces additional complexities and challenges for finite-difference schemes. An alternative two-point approach was introduced by Chen & Patel (1988) and extended by Tahara et al. (00) for ship flows. It uses the tangential velocity magnitude and direction at the second grid point away from the wall to obtain velocity at the matching point. Implementation of the two-point approach for finite-difference schemes is straightforward. The one-point approach has advantages over the two-point approach as it does not restrict the flow direction at the matching point to follow the flow direction at the second point away from the wall. 180 DECEMBER 009 JOURNAL OF SHIP RESEARCH

3 Applicability of wall-functions for ship flows has been demonstrated at model-scale by comparing predictions with near-wall results and EFD data. Park et al. (004) performed simulations around the Korean Research Institute for Ships and Ocean Engineering container ship (KCS) using the standard wall-function, and the results were compared with EFD data. They reported good agreement of the wave elevation pattern and resistance coefficient predictions. Tzabiras (1991) compared the standard wall-function and near-wall results for two ship models, SSPA and HSVA. The wall-function results compared well with the near-wall results for the velocity profile; however, different results were obtained for skin friction coefficients. Several studies have used the wall-function approach in ship flows to investigate the Reynolds number effects. A study conducted for a HSVA ship model (Oh & Kang 199: Re=10 9 and ) using the standard wall function showed that the fullscale Re results in much reduced skin friction but has no effect on pressure coefficients in the thin boundary layer region. In the thick boundary layer region around the stern and near the wake, the pressure coefficients for full scale are noticeably changed by the reduction of viscous-inviscid interaction and have the trend of approaching the values of the inviscid region. Similar conclusions were also drawn by Tahara et al. (00) using a two-layer wall-function model for the full-scale simulations of the Series 60 ship model. They also demonstrated the capability of the wall-function model in depicting surface roughness effect. The URANS study for a low-speed TDW VLCC (Choi et al. 003) using the standard wall-function showed that full-scale has weaker strengths of bilge vortices, which causes a smaller vortex and turbulence region and a smaller value of nominal wake fraction on the propeller plane. However, there is little scale effect on the limiting streamline and hull pressure except on the stern region. Bull et al. (00) studied full-scale effects using a two-layer model on two hull forms: the Dutch frigate the De- Ruyter and the NATO research vessel Alliance. The results compare well with EFD data for the bare hull geometry and capture the main flow features for the appended ships. None of the aforementioned studies on ship flows has emphasized the need for pressure gradient effect on wall-functions, probably because of the issues already discussed. To achieve the overall objective, two-layer (TL) (Esch & Menter 003) and multilayer (ML) (following Shih et al. 003) wall-function models with the ability to account for the wall roughness and pressure gradient effects are developed and implemented in CFDShip-Iowa. The downshift of buffer layer is tentatively assumed to be the same as White s (008) downshift of the log layer. As CFDShip-Iowa is based on finite-difference schemes, wall-function models are implemented using the twopoint approach (Tahara et al. 00). In the following section, computational methods and the wall-function implementation approach are discussed. In section 3 the wall-function models are first validated for smooth flat-plate flows at high Re equivalent to full-scale ship flows. In section 4 the experimental and simulation conditions used for Athena R/V and 5415 applications are summarized. Resistance computations for Athena bare hull with skeg at model- and full-scale, with and without roughness and pressure gradient effects using both TL and ML models are presented in section 5. Self-propelled simulations, and boundary layer and wake profile comparison with EFD data for fully appended Athena at full scale using both TL and ML models with smooth- and rough-wall conditions are presented in sections 5 and 6, respectively. Seakeeping calculations at model- and fullscale for Athena and maneuvering simulations for full-scale DTMB 5415 (5415) are performed using smooth-wall TL and ML models for demonstration purposes in sections 8 and 9, respectively. Finally in section 10 conclusions and future works are discussed.. Computational method The solver CFDShip-Iowa solves the URANS equations in the liquid phase of a free-surface flow. The free surface is captured using a single-phase level set method..1. Equations of motion The governing equations of motion are solved in either absolute inertial earth-fixed or relative inertial coordinates for an arbitrary moving but nondeforming control volume. Xing et al. (008) showed that the solution of the equations in the absolute inertial earth-fixed (or relative inertial) coordinates has several advantages over the noninertial ship fixed coordinate system such as simplicity in the specification of boundary conditions, savings of computational cost by reducing the solution domain size, and allowing straightforward implementation of ship motions. The governing equations for the water phase in dimensionless form are: U i ¼ 0 ð1þ x i U i þðu j U Gj Þ U i ¼ ^p þ 1 U i u i u j ðþ t x j x j Re x j x j x j where U i =(U,V,W) are the Reynolds-averaged velocity components, U Gj is the local grid velocity in either the absolute inertial earth-fixed or relative inertial Cartesian coordinates x i =(x,y,z). ^p = p abs /ru0 + z/fr +k/3 is the dimensionless piezometric pressure where p abs pffiffiffiffiffi is the absolute pressure, u i u j are the Reynolds stresses, Fr = U 0 / gl is the Froude number, and k is the turbulent kinetic energy (TKE). U 0 is the free stream velocity, L is the ship length, and Re is the Reynolds number based on L... Turbulence modeling..1. Blended k-v/k-«and DES model. Two-equation closure is used for the Reynolds stresses, modeled as a linear function of the mean rate-of-strain tensor through an isotropic turbulent eddy viscosity (n t ), U i u i u j ¼ n t þ U j x j x i 3 d ijk ð3þ where d ij is the Kronecker delta. The unknown turbulent eddy viscosity is evaluated from the TKE and the specific dissipation rate (v). Additional transport equations, presented below, are solved following Menter s (1994) blended k-v/k-e (BKW) approach. k t þ ð v s k rn t Þrk 1 r k þ s k ¼ 0 ð4aþ P k v t þ ð v s v rn t Þrv 1 r v þ s v ¼ 0 P v ð4bþ DECEMBER 009 JOURNAL OF SHIP RESEARCH 181

4 The turbulent eddy viscosity and the effective Peclet numbers are 1 n t ¼ k=v; P k=v ¼ ð5þ 1=Re þ s k=v n t and the source terms in k and v equations constitute the production and dissipation terms (refer to Carrica et al. 006b for details). The model constants, say a, are calculated from the standard k-v (a 1 ), and k-e (a ) values using a blending function F 1 (see Menter 1994 for the model constants values): a ¼ F 1 a 1 þ ð1 F 1 Þa ð6þ F 1 is designed to be unity in the near-wall regions of boundary layers and gradually switches to zero in the wake region to take advantage of the strengths of the k-v and k-e models, respectively.... Wall-function (WF) models. In the TL model, the velocities in the sublayer and log-layer regions are: U ;3 u t ( ¼ y;3 þ ; y þ ;3 # 11:67 : sublayer k 1 lnðy þ ;3 ÞþB DB 1:13yþ ;3 Pþ ; y þ ;3. 11:67 : log-layer ð7þ where subscripts and 3 represent the first and second grid point away from the wall, respectively. The superscript p + quantities ffiffiffiffiffiffiffiffiffiffi are nondimensionalized by the friction velocity (u t = t w =r ), where t w is the wall shear stress and Re. The von Karman constant (k) and B are chosen to be 0.4 and 5.1, respectively (Knobloch & Fernholz 00). The factor DB in equation (7) accounts for the effect of wall roughness, resulting in the downshift of the loglayer region (White 008): DB ¼ k 1 lnð1 þ k þ Þ 3:5 ð8aþ where k + is the roughness parameter based on nondimensional roughness length k s. For naval applications, the wall roughness lies in the transitional roughness regime (Patel 1998), that is, 5 # k +, 70. P þ ¼ n ^p ut 3 ð8bþ x is a dimensionless parameter that is used to include the effect of pressure gradient (PG) tangential to the wall in the WFs (Wilcox 1989). The effect of PG is clipped such that y þ Pþ, 3/4 as proposed by Wilcox (1989). The TKE and v in the sublayer and log-layer regimes are defined based on the analytic solution (Wilcox 1993): 8 < 3:3 k ¼ k j¼3 Dy Dy 3 : y þ # 11:67 u t 0:3 1 þ ð9þ : 1:16yþ Pþ : y þ. 11:67 ( v ¼ 6n 0:075Dy : y þ # 11:67 1 0:3y þ Pþ : y þ. 11:67 u t 0:3kDy ð10þ where Dy is the distance normal to the wall. A ML model is developed by using curve fitting (fourth-order polynomial as adopted by Shih et al. 003) that blends the sublayer and log-layer velocity profiles in the TL model and provides the buffer-layer profile. 8 þ y ;3 : y þ ;3 # 5 U >< ;3 a 0 þ a 1 y;3 þ ¼ þ a ðy;3 þþ þ a 3 ðy þ ;3 Þ 3 þ a 4 ðy þ ;3 Þ 4 DB u t : 5, y þ ;3 # 30 >: 1 k lnðy ;3 þ ÞþB DB : y þ ;3. 30 ð11þ The five unknown model coefficients in the buffer-layer region are determined by satisfying continuity of the velocity and its first derivative across the sublayer and buffer-layer and buffer-layer and log-layer intersections. An additional equation is obtained by allowing the curve to pass through the analytic buffer-layer curve u + = 5 ln (y + ) 3.05 at y + = 0. This yields the model constants: a 0 ¼ 1:875736; a 1 ¼ 1: ; a ¼ 0: ; a 3 ¼ 0: ; a 4 ¼ 3: e 005 ð1þ In the absence of any numerical model or experimental data showing the effect of wall roughness on the buffer layer, downshift of buffer layer is assumed to be same as that of the log layer in this paper as shown in equation (11). As there is no analytic solution available for turbulence quantities in the buffer-layer regions, approximated functions are used. The TKE in the buffer-layer region is obtained by using Kalitzin s et al. (005) approximation and for v the blending function proposed by Esch & Menter (003) is used: 8 Dy 3:3 k j¼3 : y þ Dy # 5 3 >< k ¼ dy þ du þ 1 v þ : 5, y þ # 30 ð13þ >: u t 0:3 : y þ n : y þ 0:075Dy # 5 >< sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ 6n u t 0:075Dy þ : 5, y þ 0:3kDy # 30 >: u t 0:3kDy : y þ. 30 ð14þ..3. Wall-function implementation. TheWFsareimplemented using the two-point approach by Tahara et al. (00). The TL model is implemented using the following steps (also refer to Fig. 1): 1. The friction velocity is computed from the relative tangential velocity of the second point ( j = 3) away from the wall ( j = 1), either from thepsublayer ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi or log-layer equation (7). For the sublayer, u t = U 3 =ðredy 3 Þ. For the log layer, the friction velocity is obtained by iteratively solving the log-layer equation using the Newton-Raphson method.. The computed friction velocity is then used in equation (7) to obtain the magnitude of the tangential velocity at j =, with the same direction of the tangential velocity at j = 3. The normal velocity at j = is approximated using the linear interpolation, based on the normal velocity at j = 3 and wall 18 DECEMBER 009 JOURNAL OF SHIP RESEARCH

5 Fig. 1 Flow chart of the wall-function implementation considering j = 1 as the wall point. Text in the shaded area is for ML implementation. distances at j =(Dy ) and j =3(Dy 3 ). Coordinate transformation is needed to give velocity boundary conditions in the physical coordinates. 3. The boundary conditions for the turbulence quantities k, v are specified at j = using equations (9) and (10). For the ML model a similar approach is used, where the friction velocity computation is obtained from sublayer, buffer-layer, or loglayer regions as shown in equation (11). In the buffer-layer region the frictional velocity is obtained using the Newton-Raphson iteration. For the turbulent quantities boundary conditions equations (13) and (14) are used. As the buffer layer is neglected in the TL model, the implementation of the two-point approach would result in the following possibilities: (y þ : sublayer, yþ 3 : sublayer; yþ : sublayer, yþ 3 : loglayer; or y þ : log-layer, yþ 3 : log-layer). If either yþ or yþ 3 is placed in the buffer layer (5, y + # 30), modeling errors will be large. This limitation is overcome by the multilayer wall function as illustrated using smooth flat-plate simulations (discussed later). In the current study the Re 10 7 to 10 9, for which the sublayer is thin and log-layer extent is large. Thus, both j = and j = 3 can be conveniently placed in the log-layer region, for example, Tahara et al. (00) used y þ to be 103 in Series 60 ship model calculations. To resolve the boundary layer as best as we can and still save sufficient grid points near the wall, y þ 30 is applied for most simulations in the current study. However, in the lowpressure regions of the ship, such as the stern, the local y þ is lower than rest of the hull and lies in the buffer-layer region. The ML model is thus expected to perform better than the TL model for the overall calculations..3. Free-surface modeling The location of the free surface is given by the zero value of the level set function (f) positive in water and negative in air. Since the free surface is a material surface, the level set function follows a simple advection equation. For stability purposes, a small artificial diffusion term is added to the equation. Negligible shear stress in the air phase is assumed that provides the jump condition at the free surface. As a good approximation for air/ water interfaces, the pressure in the air is assumed equal to the atmospheric pressure. The velocity and turbulent quantities k and v are extended from the air/water interface to air by solving an equation similar to f over the whole air domain. Readers are referred to Carrica et al. (007a) for details..4. Propeller model A simplified body force model for the propeller is used to prescribe axisymmetric body force with axial and tangential components (Stern et al. 1988). The propeller model requires thrust, torque, and advance coefficients as input and provides the torque and thrust forces. These forces appear as a body force term in the momentum equation for the fluid inside the propeller disk. The location of the propeller is defined in the static condition of the ship and moves according to the ship motion..5. Six degrees of freedom (6DOF) module The total force and moment vectors are computed in the absolute inertial earth-fixed coordinates from the nonstatic pressure, hydrostatic pressure (buoyancy), and frictional forces acting on the ship surfaces, and propeller thrust and torque. In this study, it is assumed that the center of rotation is coincident with the center of gravity. The forces and moments are then projected into the noninertial ship-fixed coordinates. These forces and moments are used to evaluate the surge, sway, and heave velocities, and roll, pitch, and yaw angular velocities assuming rigid body motion. The equations are solved using a predictor/corrector implicit solver. The velocities are then transformed back to the absolute DECEMBER 009 JOURNAL OF SHIP RESEARCH 183

6 inertial earth-fixed coordinates to predict the evolution of the location and attitude of the ship. The time marching is done using the Euler second-order backward differences. The readers are referred to Carrica et al. (006a) and Xing et al. (008) for more details of the implementation..6. Numerical methods and high-performance computing The governing equations are discretized using finite difference schemes with body-fitted curvilinear grids. The convection terms are discretized using second-order upwind scheme, and the diffusion terms are discretized using second-order central difference scheme. A pressure-implicit and splitting of operators (PISO) (Issa 1985) algorithm is used to obtain a pressure equation and satisfy continuity. The pressure Poisson equation is solved using the PETSc toolkit (Balay et al. 00). The software SUGGAR (Noack 005) runs as a separate process from the flow solver to get the interpolation coefficients required for the dynamic overset grids. Message Passing Interface (MPI) based domain decomposition is used, where each decomposed block is mapped to a single processor. For more technical details readers are referred to Carrica et al. (007a). 3. Smooth flat plate simulations The TL and ML models are first validated for smooth flat-plate simulations for a range of Re (10 6 to 10 9 ). Results are compared with BKW solutions, analytic log-layer profiles, and EFD data at large Re (Watson et al. 000). The grids used for BKW and WF simulations consist of and gridpointsinthestream- wise, wall-normal, and spanwise directions, respectively. At first, simulations are performed to study the effect of y þ and y þ 3 on the prediction of the boundary layer profiles at Re = As shown in Fig., both TL and ML results are in close agreement with BKW results when both y þ and yþ 3 lie in the loglayer or the sublayer region (not shown in figure). However, the ML results are better than those of TL, especially for the turbulent eddy viscosity predictions when y þ lies in the buffer-layer region. Flat-plate boundary layer profiles obtained from the WF simulations for a range of Re (10 6 to 10 9 ) using y þ 30 were compared with the analytic log-layer profiles (Bhushan et al. 007). The comparison shows good agreement for u +, k +, v +, and n + profiles. The variation of the skin friction coefficient along the plate for Re =10 9 is shown in Fig. 3. The results agree well with the Karman-Schoenherr equation,* which is suitable for high Re flows (Watson et al. 000), and the EFD data for y þ = 30 to 600. Results are found to be slightly dependent on the y þ for lower Re x, that is, toward the leading edge of the plate. The above results demonstrate that the WF is capable of simulating high Re flows equivalent to full-scale ship Re. Fig. Smooth flat plate boundary layer profiles are compared with the BKW results at Re x = a Streamwise velocity profile. b Turbulent eddy viscosity (n + =Re+n T) appended surface combatant 5415 was used in the SIMMAN (008) workshop for verification and validation of ship maneuvering simulations methods. This study extends previous modelscale BKW simulations using CFDShip-Iowa for Athena (Wilson et al. 006, Miller et al. 006, Xing et al. 008) and 5415 (Carrica et al. 008) to full scale. 4. Application to Athena and 5415 The Athena geometry used herein for resistance, propulsion, and seakeeping calculations was previously used at the 005 ONR wave-breaking workshop to assess the capability of CFD methods in predicting bow and transom wave breaking. The fully p ffiffiffiffi *Karman-Schoenherr equation: c f = (0.558c d )/( c d) where 0.4/ p ffiffiffiffi = log10 (c d Re x ) = log 10 (Re u ). c d Fig. 3 Variation of the skin friction coefficient along the flat plate compared with experimental data of Watson et al. (000) and Karman- Schoenherr equation 184 DECEMBER 009 JOURNAL OF SHIP RESEARCH

7 4.1. Experimental data and simulation conditions The EFD data for Athena bare hull with skeg (BH) are obtained from the towing tank experiments conducted on a 1/8.5-scale ship model (Jenkins 1984). Measurements of the total resistance, wave resistance, bow and stern sinkage, and wave elevation along the hull at several stations were made for Frs ranging from 0.8 to Miller et al. (006) performed resistance calculations for fixed sinkage and trim BH at model scale using BKW for six Frs. Resistance, sinkage and trim calculations performed by Xing et al. (008) at model scale using BKW included a single run full Fr curve, and two steady-state computations for Fr = 0.48 and 0.8. Simulations performed here for BH using WF are: (1) resistance calculations for fixed sinkage and trim at model scale (Fr = 0.8, 0.48, and 0.8) and full scale (Re =+10 9, Fr = 0.48) using the smooth-wall TL model, () verification study at full scale (Re =+10 9, Fr =0.48)forthe resistance coefficients, sinkage, and trim using the smooth-wall TL model, (3) resistance, sinkage and trim predictions at model- and full-scale (Fr = 0.48 and 0.8) using both ML and TL models assuming smooth- and rough-wall conditions. Effect of PG is evaluated for full-scale simulation using the smooth-wall TL model, and (4) sensitivity study of the ML and TL model resistance, sinkage, and trim predictions on y þ (yþ = 30, 100, and 300) at full-scale for smooth- and rough-wall (k + = 8) conditions. The EFD data for the self-propelled fully appended Athena (AH) (Crook 1981) include resistance and powering characteristics, sinkage and trim at 17 different Frs ranging from to 0.839, open water curves for thrust and torque coefficients for the propeller model 4710, with the corresponding propeller revolution per seconds (RPS), thrust deduction, and wake factors. The EFD data were presented at full-scale extrapolated using the ITTC ship-model correlation line including a correlation allowance** C A = Xing et al. (008) performed self-propelled simulations for AH free to sink and trim at model scale using BKW. Their simulations included full Fr powering curves and steady-state computations at Fr = 0.43, 0.575, and Herein, ship speeds and consequently Frs are predicted at full scale for AH free to sink and trim using operational propeller conditions at Fr = 0.43, 0.575, and in the EFD. The effect of wall roughness is demonstrated for the Fr = 0.43 and cases using roughness length of 100 micron with both TL and ML models. The full-scale boundary layer and wake profiles were measured on AH by Day et al. (1980). The ship was self-propelled by the operating propeller on the port side. The course of the ship was maintained with a small port side rudder angle between and 5. On the starboard side, the rudder was set at 0 and the propeller blades were removed. Measurements on the starboard side included (1) boundary layer profiles at four rake locations, and () wake profiles including axial, tangential, and radial velocities on the forward propeller plane at three radial locations: 0.417, and 0.75R, where R is the radius of the propeller, and on the propeller plane at four radial locations: 0.456, 0.633, 0.781, and 0.963R. The measurement locations are shown in Fig. 4. The trim angle was reported to be less than 3, whereas the sinkage was not documented. The wall roughness length was reported to be between 50 and 130 mm. To approximate the EFD conditions on the starboard side, the ship was towed at the speed corresponding to Re = and Fr = 0.36, for which the most comprehensive **C A ¼ð105ks 1=3 0:64Þ 10 3 Fig. 4 Towed fully appended Athena EFD measurement locations (Day et al. 1980). a Wake profile measurement planes (A B: propeller plane; C D: forward propeller plane). b Boundary layer measurement locations EFD data are available. The simulations are performed using both smooth- and rough-wall conditions. Three roughness lengths are considered, 50 (k + = 15), 100 (k + = 3), and 130 (k + = 43) mm covering the reported roughness range. Seakeeping EFD data are not available for Athena; thus simulations are performed for demonstration purposes using the smooth-wall TL model for Fr = 0.3. An incoming regular head wave is specified with nondimensional wave length 1. and amplitude The encounter period of the wave is 0.488L/U 0. BKW and WF calculations are performed for towed BH at model scale (Re = ). Full-scale (Re =+10 9 ) calculations are performed using TL model for towed BH and AH. For the SIMMAN (008) workshop EFD data were acquired for captive and free-sailing (6DOF) model tests for tanker, container ship, and surface combatants hull forms. The free-sailing model EFD data were not provided before the workshop; thus the comparisons were blind. Carrica et al. (008) performed steady turn and zigzag maneuvers for self-propelled fully appended Steady turn computations were performed at model scale using BKW for 0 and 35 rudder deflections for Fr = 0.5 and 0.41, in calm and regular waves, and for constant torque and constant RPS. 0/0 zigzag maneuvers were performed using BKW for constant RPS for Fr = 0.5 and 0.41 at model scale (Re = ). An additional simulation was performed for the Fr =0.41caseatfullscale (Re = ) using the smooth-wall ML model. The zigzag DECEMBER 009 JOURNAL OF SHIP RESEARCH 185

8 maneuver is controlled by the rudder angle, which changes from 0 starboard to 0 port. The rudder angle is changed at the rate of 9 deg/s (in full scale) to the other side when the heading angle is 0, which is referred to as rudder check point. Here, results of the Fr = 0.41 model- and full-scale 0/0 zigzag maneuvers are presented for demonstration purposes. Table 1 summarizes the geometry, y þ, simulation conditions, and comparison/validation data used for various WF simulations performed in this paper. 4.. Domains, grids, and boundary conditions Figures 5 and 6 show the grid topology used for the Athena and 5415 geometries. The simulations performed using Athena geometries are for half domain only, taking advantage of the symmetry of the problem about the center plane y = 0. The AH consists of a skeg, rudder, stabilizer, propeller shaft, and struts. The fully appended 5415 consists of skeg, split bilge keels, stabilizers, shafts and struts, rudder seats, and rudders. Body-fitted double-o type grids are generated for the BH. Overset body-fitted O type grids are generated for the ship appendages except the rudder, stabilizer, and shaft caps, which are open topology. A Cartesian background grid, with clustered grid near the free surface to resolve the wave flow pattern, is used for specifying boundary conditions away from the ship hull. Model-scale BH simulations are performed using a grid consisting of 360K points. A grid consisting of 1.7 million (M) points is found to be appropriate for the full-scale WF simulations from the verification study (performed later in the paper). In the model-scale simulations y þ 30 is used, whereas for full-scale calculations y þ = 30 to 300. The AH grid consists of.m points split into 4 blocks (Xing et al. 008). For the fullscale WF simulations the grid design is such that y þ = 30 to 60. The grid used for the 5415 simulation consists of 7M grid points decomposed into 7 blocks (Carrica et al. 008). The y þ values for the WF simulations are shown in Table 1. Towed and seakeeping simulations are performed using relative inertial coordinates, whereas self-propelled simulations are in absolute inertial earth-fixed coordinates. The ship-hull, skeg, and appendages have no-slip boundary conditions, and the WF boundary conditions are applied at the first grid point away from the wall. The boundary conditions for the Athena geometries are shown in Fig. 5. In the 5415 simulation, boundaries are located 10 ship lengths from the ship hull where inlet boundary conditions are specified. The details of the boundary conditions are presented in Table. In the self-propulsion simulations, propeller RPS are specified to provide the required thrust force allowing the ship to accelerate from static condition to the target surge velocity. A proportional, Table 1 Model- and full-scale simulation conditions for Athena resistance, powering, and seakeeping, and 5415 maneuvering using WFs Geometry Ship motions Re Fr y þ Wall-Function Model, Wall Condition Results and Compare Resistance coefficients, BKW results TL, smooth-wall(xing et al. 008) and EFD Fixed sinkage (Jenkins 1984) data and trim TL, smooth-wall Resistance, ITTC line Towed Athena 0.48 ML and TL, Resistance coefficients, sinkage and trim, bare hull with smooth-wall BKW results (Xing et al. 008) and EFD skeg Predict sinkage 0.8 TL, smooth-wall (Jenkins 1984) data at model-scale. and trim , 100, 300 ML and TL, smooth Compare ML and TL results for and rough-wall smooth-wall. 6- grids TL, smooth-wall Study sensitivity of results on y + and TL, smooth-wall wall-roughness for both ML and TL models. Verification study using TL model Self-propelled fully appended Athena Towed fully appended Athena Pitch and heave in waves Predict sinkage and trim Predict sinkage and trim Pitch and heave in waves TL, smooth-wall TL, smooth and rough-wall ML and TL, smooth and rough-wall BKW solution at model-scale Resistance, power, and motion BKW results (Xing et al. 008) and EFD (Crook 1981) data at model scale extrapolated to full-scale TL, smooth-wall Compare TL and ML results and wall roughness effect TL, smooth and rough-wall Boundary layer and wake survey, full-scale EFD (Day et al. 1980) TL, smooth-wall Bare hull with skeg in waves Self-propelled fully appended DOF ML, smooth-wall BKW solution at model scale 186 DECEMBER 009 JOURNAL OF SHIP RESEARCH

9 integral, and differential (PID) speed controller is used that controls the ship velocity via the propeller RPS (Huang et al. 007). In the maneuvering calculation, only surge, heave, and pitch are allowed during the acceleration stage. Once the target speed is achieved the propeller RPS is kept constant throughout the maneuver. The constant RPS obtained by the controller in the model- and full-scale calculations is and in full-scale dimensions, respectively. The thrust, torque, and advance coefficients for the given RPS are evaluated using the propeller open-water curves obtained from Crook (1981) for Athena and SIMMAN (008) for Bare hull resistance simulations for towed Athena with skeg 5.1. Fixed sinkage and trim Fig. 5 Boundary conditions for the simulation domain, represented for Athena bare hull with skeg grid topology The model- and full-scale results obtained using smooth-wall TL models are presented in Fig. 7. The total resistance coefficients (C tot,x ) at model scale are within 5% of the EFD data for Frs = 0.8 and 0.48, but is overestimated by as much as 6.75% for Fr = 0.8. Model- and full-scale (Re =10 7 to 10 9 ) frictional resistance coefficients (C f,x ) follows the ITTC { line. However, at model-scale WF predictions are larger than BKW results by 5% to 7%. Figure 8 a shows that the full-scale boundary layer is thinner than that in model scale. The free-surface elevation pattern is not significantly affected by Re as seen in Fig. 8 b. A closer inspection of the transom wave elevation shows that the wave slope at full scale is slightly lower than at model scale (Fig. 8 c), which is consistent with the findings of Starke et al. (007). 5.. Verification study for resistance, sinkage, and trim Solution verification study is performed following the quantitative methodology and procedures proposed by Stern et al. (006b) (ST) and recent modifications proposed by Xing & Stern (008) (XST) for situations when estimated order of accuracy (P G )is larger than the theoretical order of accuracy (P G,th ) and correction factor 1, C G,. C G solutions are too far from the asymptotic range and also regarded as divergent in the XST method. For the verification study, six grids are designed with a systematic refinement ratio of r G = 1/4, summarized in Table 3. Six sets of verification studies are possible, four with r G = 1/4 (1,,3;,3,4; 3,4,5; and 4,5,6) and two with r G = 1/ (1,3,5 and,4,6). Results have been plotted in Figs. 9 a and c, and the converged results are shown in Table 4. Convergence condition is defined by R G which is the ratio of solution changes for medium-fine and coarse-medium solutions. Correction factor 1 C G indicates how far the solution is from asymptotic range where C G =1. Figures 9 b and d show the relative change of the solution (e N ) between two successive grids along with iterative errors (U I ) obtained using five nonlinear iterations on each grid for resistance coefficients and motions, respectively. U I are of the same order of magnitude for all the grids, which suggests that they are determined primarily by the iterative method and independent of grid resolution. The results are discussed in the light of recent study by Xing et al. (008) (XET) for BH at model scale (Re = , Fr = 0.48) using BKW where an additional grid consisting of 8.1M grid points was considered. C tot,x monotonically converges on grids (,4,6) and (,3,4), oscillatorially converges on grids (1,3,5), oscillatorially diverges on grids (1,,3) and (4,5,6) and monotonically diverges on grids (3,4,5). The pressure resistance coefficient (C p,x ) monotonically converges on grids (,4,6) and (3,4,5), oscillatorially converges Fig. 6 Grid topology for a fully appended Athena and b fully appended 5415 { ITTC: C f = 0.075/(log Re ) DECEMBER 009 JOURNAL OF SHIP RESEARCH 187

10 Table Boundary conditions for all the variables Boundary f p k v u n w p diverged solutions involve coarsest grid 6. XET also obtained diverged solutions involving this grid, which is likely caused by insufficient grid resolution. C tot,x shows large values of 1 C G,.45 for grids (,4,6), and 74 for grids (,3,4). The grid uncertainty (U G ) is 1.55% on grids (,4,6) and 0.13% on grids (,3,4) using the ST method. U G is 5.53% on grids (1,3,5). Grids (,4,6) are closest to the asymptotic range for C p,x for which 1 C G is C G for grids (3,4,5) is.79. U G for C p,x is below 4% on grids (,4,6) for both the ST and XST methods, 13.59% on grids (3,4,5) based on the ST method, and 11% and 3.4% on grids (1,3,5) and (,3,4), respectively. 1 C G for C f,x is 0.81, and U G is.91% and 7.8% on grids (1,,3) based on the ST and XST methods, respec- Towed f = z n ¼ 0 k fs ¼ 10 7 v fs ¼ 9 U inf =1 V inf =0 W inf =0 p Inlet Selfpropelled f = z n ¼ 0 k fs ¼ 10 7 v fs ¼ 9 U inf =0 V inf =0 W inf =0 Seakeeping Eq. (43) Eq. (4) k fs ¼ 10 7 v fs ¼ 9 U inf = Eq. (40) V inf =0 W inf = Eq. (41) (Carrica et al. (Carrica et al. (Carrica et al. (Carrica et al. 007a) 007a) 007a) 007a) f Exit n ¼ 0 p n ¼ 0 k n ¼ 0 v n ¼ 0 U n ¼ 0 V n ¼ 0 W n ¼ 0 f Far field No. 1 n ¼ 0 0 k n ¼ 0 v n ¼ 0 U n ¼ 0 V n ¼ 0 W n ¼ 0 f Far field No. n ¼ 0 p n ¼ 0 k n ¼ 0 v n ¼ 0 U inf V inf W inf f Symmetry n ¼ 0 p n ¼ 0 k n ¼ 0 v n ¼ 0 U n ¼ 0 0 W n ¼ 0 6n No-slip v Carrica et al. 0 0:075Dy (007a), u ship n ship w ship Eq. (37) Wall-function ( j =) f n ¼ 0 p n ¼ 0 TL: Eq. (9) ML: Eq. (13) TL: Eq. (10) ML: Eq. (14) Transformation of tangential velocity and normal velocity to physical coordinate system. Eq. (7) for TL model or Eq. (11) for ML model. on grids (1,3,5) and (,3,4), oscillatorially diverges for (4,5,6), and monotonically diverges on grids (1,,3). C f,x monotonically converges on grids (1,,3), oscillatory converges on grids (1,3,5), (,4,6), and (,3,4), oscillatorially diverges on grids (4,5,6), and monotonically diverges on (3,4,5). Sinkage monotonically converges on grids (,3,4), oscillatory converges on grids (1,,3), (3,4,5) and (4,5,6), oscillatorially diverges on grids (1,3,5), and monotonically diverges on grids (,4,6). Trim monotonically converges on grids (1,,3), oscillatorially converges on grids (3,4,5) and (4,5,6), oscillatorially diverges on grids (,4,6) and (1,3,5), and monotonically diverges on grids (,3,4). It must be noted that the monotonically converged solutions involve the three finest grids 1,, or 3, and most of the Fig. 7 Athena bare hull with skeg and fully Appended Athena frictional resistance and total resistance coefficients are compared with ITTC line and EFD (Jenkins 1984), respectively 188 DECEMBER 009 JOURNAL OF SHIP RESEARCH

11 Fig. 8 Local flow field for model- and full-scale Athena bare hull with skeg. a Boundary layer profiles colored by streamwise velocity. b Transom free surface wave elevation contour. c Free surface elevation profile at y = 0.01 L for Fr =0.48 tively. For other grids on which oscillatory convergence is observed, U G is below 3.5%. A large range of.79, 1 C G, 0., neglecting the largest value, for resistance coefficients suggests that all solutions are still far from the asymptotic range. XET also observed significant variation in 1.4 # 1 C G # 0.93; however, the range of variation was smaller than the present case. The variations of C f,x in the WF simulations are within 4% for all the grids, whereas in the BKW simulations by XET significant grid dependence up to 8% is observed. e N of C p,x and C tot,x linearly increase when grid is refined from 6 to 5, which suggests that the coarsest grid is too coarse, then linearly decrease when grid is refined from 5 to. Further refinement to grid 1 leads to a rebound of e N, which is caused by the problem of separating U I from U G as they are of the same order of magnitude. e N for C p,x in the WF simulations are about 7%, which is significantly larger than % in XET. Overall, U I around % in the present simulations are higher than that reported by XET around 0.6%. Thus, to achieve the asymptotic range more nonlinear iterations and/or implementation of more accurate and efficient iterative methods are required. Grid study (,3,4) is closest to asymptotic range for sinkage as 1 C G is U G is 15.48% on grids (,3,4), about 4% on grids (1,,3), and large values of 7.44% and 11.64% are observed on grids (3,4,5) and (4,5,6), respectively. The solutions of trim are far from the asymptotic range as evident from the large value of 1 C G around 39 on grids (1,,3). U G is.56% on grids (1,,3) using the ST method, and less than 6% on grids (3,4,5) and (4,5,6). Compared with resistance coefficients, motions are difficult to converge. e N for sinkage decreases almost linearly when grid is refined from 6 to 1. e N for trim also shows a linear decrease with grid refinement from 6 to 3. A sudden rise in e N is observed for grids and 3 when U I and U G are of same order of magnitude. Maximum U I for both sinkage and trim are obtained between grids and 3, which are.7% and 3.4%, respectively, which are larger compared to 0.% in XET. Of all the converged grid studies, the lowest grid uncertainty for the resistance coefficients and motions are obtained on grids (,3,4) and (1,,3), respectively. The XST method predicts more reasonable estimate for U G compared with ST method for C p,x on grids (1,,3) where P G. P G,th. Grid studies (,3,4) and (,4,6) are closest to asymptotic range for sinkage and C p,x, respectively. These calculations show that to reach the asymptotic range further grid refinement is required. To maintain an affordable computational cost with a reasonable accuracy, grid No. 3 is used for fullscale calculations. The full-scale results are consistent with the XET results as in both the cases: (a) the converged and diverged solutions involve finest and coarsest grids, respectively, (b) large variations in Table 3 Grids used for verification study for Athena bare hull with skeg (Re = +10 9, Fr = 0.48) Grid number Ratio 1 1/4 1/ 3/4 5/4 Ship Background Total 360,58 59,416 1,017,046 1,7,644,884,4 4,845,456 y þ DECEMBER 009 JOURNAL OF SHIP RESEARCH 189

12 Fig. 9 Verification for resistance coefficients and motions for Athena bare hull with skeg (Re = +10 9, Fr = 0.48). a Resistance coefficients. b Relative change e N = (S N S N+1 )/S and iterative errors for resistance coefficients. c Sinkage and trim. d Relative change en and iterative errors for sinkage and trim 1 C G are observed suggesting that further grid refinement is required, (c) a significant increase in e N is observed when U I and U G are of the same order of magnitude. There are specific differences such as: (a) The variations of C f,x in the WF simulations are 50% of that in near-wall simulations in XET, which is expected as boundary layer in the WF is treated using the log-layer formulation. (b) The variations of C p,x in the WF simulations are significantly larger compared with BKW solutions in XET. This could be due to the zero pressure gradient boundary condition at the wall used in the two-point approach. (c) Overall higher levels of grid uncertainties are observed in WF simulations compared to BKW solution in XET. The verification study presented here could be significantly affected by: (a) contamination of U G due to higher levels of U I, (b) sensitivity of the solution on y + as discussed in section 5.4, and (c) the pressure boundary condition at the wall used in the WF Predicted sinkage and trim In Table 5 and Fig. 7, smooth- and rough-wall TL and ML results at model and full scale are presented. In model-scale simulations, TL and ML models overpredict C f,x by 8% and 5% Table 4(a) Verification study for resistance coefficients and motions of Athena bare hull with skeg at full-scale (Re = Monotonically converged solutions, U G is %S 1,%S,or%S , Fr = 0.48). U G (%) Parameter Grids,4,6 C tot,x,3,4,4,6 C p,x 3,4,5 C f,x 1,,3 Sinkage,3,4 Trim 1,,3 Refinement Ratio (r G ) R G P G 1 C G Stern et al. (006a) Xing and Stern (008) p ffiffiffi ffiffiffi p p ffiffiffi ffiffiffi p p ffiffiffi ffiffiffi p ffiffiffi p DECEMBER 009 JOURNAL OF SHIP RESEARCH

13 Table 4(b) Verification study for resistance coefficients and motions of Athena bare hull with skeg at full-scale (Re = +10 9, Fr = 0.48). Oscillatory converged solutions, U G is %S 1,%S,%S 3,or%S 4 Parameter Grids C tot,x 1,3,5 1,3,5 C p,x,3,4 Refinement Ratio (r G ) R G U G (%) p ffiffiffi p ffiffiffi ffiffiffi p ,3,5 p ffiffiffi C f,x,4,6 ffiffiffi ,3,4 4p Sinkage Trim 1,, p ffiffiffi 3,4, ,5, ,4,5 4p ffiffiffi ,5, compared with BKW, respectively. C p,x is within 3% of the BKW results for both the WF models. C tot,x is within 6% of the BKW results and.5% of the EFD data for both the WF models. The WFs overestimate sinkage for the higher speed when compared with the EFD data. Similar trend is observed for the BKW results. Notice that the absolute error is small, but the low value of sinkage increases the relative error. The trim values obtained using WFs are within 5% of the BKW results, and 3.% of the EFD data for Fr = 0.8. Larger differences around 10% are observed between the WF results and EFD data for Fr = Overall, the ML results are within % of the TL results. C f,x values obtained from full-scale (Re=+10 9 ) simulations are found to be in good agreement with the ITTC line, where relative errors are less than 5%. The differences in the modeland full-scale C p,x are less than 6.5%, consistent with the previous conclusions by Oh & Kang (199). Sinkage and trim compare well with the model-scale WF results, where differences are within 4%. For this case again the TL and ML results are within %. Results obtained using TL model with PG effect are within 1% of the TL results without PG effect. This demonstrates that the effect of PG in WF models is not important for full-scale ship flows and is not included for the rest of simulations performed in this paper. An additional simulation is performed for Fr = 0.48 to demonstrate the effect of wall roughness (k + = 8). For the TL model C f,x increases by 30% over the smooth-wall results, which is consistent with the ITTC correlation allowance for this roughness length. The corresponding increase for the ML model is higher about 41%, which is discussed in the next section. C p,x is unaffected by the wall-roughness effect as the differences are within 0.1%. C tot,x increases by 10% and 15% for the TL and ML models, respectively. Sinkage and trim for both TL and ML models are within 1.5% of the smooth-wall results Sensitivity of resistance, sinkage, and trim to y + As shown in Figs. 10 a and c, for the smooth-wall TL model the C tot,x, C p,x and trim vary by as much as 15% when y þ is increased from 30 to 300. Respective variations in the C f,x and sinkage are within 5%. The variations in resistance coefficients and ship motions for the smooth-wall ML model are within 5%. Figures 10 b and c show that the resistance coefficients predicted by the rough-wall TL model are less sensitive to y þ compared with rough-wall ML model. The relative changes in TL model are only 5%, whereas ML model shows variations of as much as 10%. The wall-roughness increases C f,x by 8%, which is in good agreement with the ITTC correlation allowance of 30%. ML model overpredicts C f,x for y þ = 30 by as much as 10%, which could be the result of the tentative modeling of the Table 5 Athena bare hull with skeg simulations with predicted sinkage and trim at model- and full-scale. Error (E) is based on EFD data (D) Fr Re Sinkage Trim C tot,x * C f,x + C p,x + EFD BKW E (6.16%) ( 4.79%) (0.5%) WF TL (8.8%) ( 9.71%) (.139%) (7.78%) (.60%) E ML (10.56%) ( 10.14%) (0.870%) (4.3%) (.4%) TL E (9.97%) ( 6.06%) TL, with PG effect E (10.6%) ( 5.39%) ( 0.78%) ( 0.8%) (+0.46%) ML E (9.99%) ( 5.9%) (+1.30%) (+.08%) (+0.09%) TL k + = E (9.68%) ( 5.1%) (+11.95%) (+8.47%) (+0.05%) ML k + = E (10.6%) ( 7.46%) (+16.41%) (+41.50%) (+0.00%) EFD BKW E (+5%) ( 6.3%) ( 4.65%) WF TL E (+38%) 1.09 (+3.%) (+1.63%) (+7.7%) (1.14%) TL E (+34%) (+6.4%) (+6.5%) *Results are based on static areas. + E is based on BKW results. DECEMBER 009 JOURNAL OF SHIP RESEARCH 191

14 model-scale results. It must be noted that the absolute error is low, but relative error is high due to lower absolute value of the sinkage. The behavior is reversed for the trim, which has larger error for Fr = 0.43 than those for Fr = and For higher Frs results are within 3.5% of the EFD. The ML and TL results are in close agreement with each other with differences within %. Rough-wall simulations are performed for Fr = 0.43 and using a roughness length k s = 100 mm corresponding to k + =5 to 35 for lower and higher speeds. The rough-wall simulations lead to better prediction of the Frs that are within 1% of the EFD data. The good agreement is attributed to the increase in C f,x by for the TL and for ML model, which are in close agreement with the correlation allowance C A = used in EFD data extrapolation (Crook 1981). Similar to the BH case, no appreciable effect of wall roughness is observed for sinkage and trim values for both ML and TL models. 7. Towed fully appended Athena boundary layer and wake profiles Fig. 10 Sensitivity of results on y þ for Athena bare hull with skeg (Re = and Fr = 0.48) using TL and ML models a resistance coefficients for smooth-wall simulations, b resistance coefficients for roughwall simulations, and c sinkage and trim for both smooth- and roughwall conditions roughness effect in the buffer layer. Sinkage and trim vary only by 3% for the range of y þ considered. 6. Self-propelled fully appended Athena As shown in Table 6 and Fig. 11, the smooth-wall WF overpredicts Frs by % to 3% caused by the underprediction of C tot,x. C f,x for the AH is 10% to 15% higher than the ITTC line as shown in Fig. 7 because of the added frictional resistance from the appendages. Sinkage is within 3% of the EFD value for lower speed Fr = The differences are larger for Fr = and 0.839, similar trend is also observed in the BKW The wall roughness effect increases C f,x by 1, 31.6, and 39.5% over the smooth-wall for k + = 15, 3, and 43, respectively. This is in good agreement with the ITTC correlation allowance. C p,x variations are within 3% for all the surface conditions. The corresponding increases in C tot,x are 6.3, 10.5, and 1.3%. Sinkage and trim values are and 1.844, respectively, and show variation within 1% on the wall roughness. Trim is consistent with the EFD value, which was reported to be less than 3. The smooth-wall simulation predicts thinner boundary layer compared with the EFD data. The boundary layer thickness increases with increasing roughness, resulting in much better comparison than the smooth-wall results. The best results are obtained for k + =3. In Fig. 1, the boundary layer velocity profiles obtained from the smooth- and rough-wall (k + = 3) simulations are compared with the EFD data. The rough-wall results compare better with the EFD data than the smooth-wall results for rakes 1 and, whereas the nature is reversed for rake 4. At rake, the discrepancy between the EFD and numerical results is large. This could be the result of either the uncertainty in the EFD data as the velocity outside the boundary layer is reported to be 0.9 opposed to the expected value of 1.0 or due to grid resolution issues as probe lies in the overset region of the hull and propeller hub grids. Further investigation is required using finer grid resolutions and/or comparison with additional EFD data. In Fig. 13, results at two of the radial locations (0.417 and 0.75 R) on the forward propeller plane (C D in Fig. 4 a) are compared with EFD data. No significant differences are observed between the smooth- and rough-wall simulations for tangential and radial velocities. The axial velocity predictions using roughwall condition are better than that using the smooth-wall. Results shown in Fig. 14 are at the propeller plane (A B in Fig. 4 a) for radial locations 0.781, and R. The tangential and radial velocity components are predicted well in the rough-wall simulation when compared with the EFD data, whereas the smooth-wall results underpredict them significantly. The rough-wall results also capture the sharp decline in the axial velocity close to 0 and 360. Similar results were obtained at other radial locations for which figures are not shown. 19 DECEMBER 009 JOURNAL OF SHIP RESEARCH

15 Table 6 Self-propelled simulation at model- and full-scale Reynolds number for fully appended Athena. Error (E) is based on EFD data (D) * RPS Re Fr Sinkage Trim C f,x EFD BKW (E) (.08%) ( 5.8%) (+1.9%) WF TL (E) (+.08%) (.%) (1%) TL, k + =6(E) (+0.3%) (.78%) 0.81 (+11.8%) (+31%) EFD BKW (E) ( 1.4%) (+9%)1.54 (+5.5%) WF TL (E) (+.44%) (+17.9%) 1.43 (.10%) TL, k + =35 (E) (+0.87%) (+17.9%) 1.4 (.73%) (+3%) ML (E) (+.61%) (+18.5%) 1.46 ( 0.0%) ML, k + =35(E) (+1.04%) (+18.%) 1.45 ( 0.07%) (+30%) EFD BKW (E) ( 1.1%) (+63%) 1.75 (+9.7%) WF, TL (E) (+.0%) (+75%) 1.65 (+3.5%) * E shows the percentage change from the smooth-wall values. 8. Seakeeping for bare hull with skeg and fully appended Athena Figure 15 shows the motions for the seakeeping case. The heave has a phase lag of 0 compared with the pitch. At model scale, WF predicts the peak of pitch 6% higher than the BKW, whereas no significant differences are observed for the heave. The full-scale ship motions vary within 5% of the model-scale WF results. The transfer functions for the pitch at model scale are 0.8 and 0.85 using the BKW and WF, respectively. The corresponding value is 0.81 for full scale. The transfer function Fig. 11 Steady-state computation for total resistance coefficients, sinkage and trim for self-propelled fully appended Athena. Near-wall turbulence model solution at model-scale; + Smooth-wall calculations at full-scale using wall-function; Rough-wall simulation (k + =5to 35) at full-scale using wall-function; lines (and filled circles): EFD (Crook 1981) Fig. 1 Boundary layer velocity profiles obtained using smooth- and roughwall conditions at four rake locations, shown in Fig. 4 b, for fully appended Athena (Re = , Fr = 0.361) are compared with EFD data (Day et al. 1980). The coordinate z*/l isthedistance normal tothewall. DECEMBER 009 JOURNAL OF SHIP RESEARCH 193

16 Fig. 13 Wake profile (axial, tangential, and radial velocity components) at forward propeller plane (C D in Fig. 4 a) for fully appended Athena (Re = , Fr = 0.361), left: R and right: 0.75 R. Black line: smooth-wall; gray line: Rough wall, k+ = 3; EFD (Day et al. 1980) Fig. 14 Wake profile (axial, tangential, and radial velocity components) at the propeller plane (A B in Fig. 4 a) for fully appended Athena (Re = , Fr = 0.361), left: R and right: R. Black line: smooth-wall; gray line: rough wall, k+ = 3; EFD (Day et al. 1980) 194 DECEMBER 009 JOURNAL OF SHIP RESEARCH

17 Fig. 15 Evolution of the pitch and heave for Athena bare hull with skeg (Fr = 0.3) in presence of incoming regular head waves, wave length = 1. L, amplitude=0.01l and wave encounter frequency = L/U 0 of the heave is 0.65 for both model and full scale. Figure 16 shows that the peak of C f,x correlates with the trough of the heave, whereas the C tot,x peaks in phase with the pitch. A bimodal pattern is observed in the WF simulations at C f,x minima that occurs after the peaks of the pitch and heave. Similar bimodal pattern in the C tot,x minima is caused by the 180 phase difference in C p,x and C f,x troughs. The WF predictions of C f,x at model scale are 5% to 10% lower than the BKW. C f,x in the full-scale calculation is about 1% lower than the model-scale WF, which is in good agreement with the ITTC line. There is no significant scale effect on C p,x and C tot,x, for which the relative changes are within %. Figure 17 compares the full-scale AH results with those of the full-scale BH results. In the AH results pitch lags by 1. compared with the BH case, whereas heave is in the same phase. The amplitude of pitch is about 4% higher than the BH case, whereas the differences observed for heave amplitude are within %. The transfer functions for the pitch and heave are 0.84 and 0.66, respectively. Figure 17 b shows high-frequency oscillations in the trough of the C tot,x for both the BH and AH geometries, which coincides with the trough of the pitch. Source of these fluctuations are the pressure fluctuations and could be avoided by better convergence of the implicit motions solver (Carrica et al. 007b). The emphasis of this example is to demonstrate the capability of WFs to simulate ship motions Fig. 16 Time histories of a frictional and b total resistance coefficients for Athena bare hull with skeg (Fr = 0.3) in presence of regular head waves Fig. 17 Time histories of a pitch and heave and b total resistance coefficients for bare hull with skeg and fully appended Athena (Fr = 0.3) in presence of regular head waves DECEMBER 009 JOURNAL OF SHIP RESEARCH 195

18 Fig. 18 Time histories of a heading and rudder angle, b yaw rate, c roll angle, and d pitch and heave for model- and full-scale fully appended 5415 Fig. 19 Contours of the streamwise velocity in noninertial ship-fixed coordinates for model- (left) and full-scale (right) fully appended 5415 (Fr = 0.41) at a cross section upstream of the shaft struts in the presence of waves; thus this aspect is not further investigated. C tot,x of the AH is higher by about 10% to 15% compared with the BH case. 9. 0/0 zigzag maneuver for self-propelled 5415 Figure 18 a shows the histories of heading and rudder angles for the model- and full-scale calculations. In the full scale the rudder check point is reached faster than in model scale. The periods between rudder checks are 38 and 36 s for model and full scale, respectively. This indicates slightly more efficient rudder action for full scale. The overshoot on the heading is 5.8 for the model-scale and 5.1 for the full-scale. There are no significant differences in the yaw rate for the model and full scale as seen in Fig. 18 b. The yaw rate right before the rudder check point is 1.8 deg/s, close to the asymptotic value of 1.83 deg/s reported by Carrica et al. (008) for the turning circle case under similar conditions. Figure 18 c shows that the full scale has larger roll angle peaks and smaller damping of roll oscillations than the model scale. This is caused by the smaller effective viscosity in the full-scale computation. The peak value of the roll angle is 15.3 for the model scale and 17.5 for the full scale. Figure 18 d shows the time histories of pitch and heave. A dynamic response to the rudder direction change can be observed for both pitch and heave. The ship goes nose down and moves up just after the rudder check point and tends to recover to an asymptotic value. Pitch and heave show oscillations similar to the roll, where the damping is lower for full scale. The asymptotic value of pitch is estimated to be 0.94 for model scale and 0.98 for full scale. The minimum for the pitch is observed to be 0.4 for the model scale and 0.9 for the full scale. Heave values do not show much variation in model- and full-scale simulations. The steady value is estimated to be and maximum is for both the model- and full-scale simulations. There is no significant Re effect on free-surface elevation. However, the transom rooster tail shows slightly higher elevation in the full-scale computations compared with the model scale. Velocity contours on the cross sections immediately upstream 196 DECEMBER 009 JOURNAL OF SHIP RESEARCH