Journal of Ship Research - of Tao Xing

Journal of Ship Research, Vol. 53, No. 4, December 2009, pp. 179–198

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 109, 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 2002, Stern et al. 2006a). 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
107, including resistance, propulsion, seakeeping, and maneuvering (e.g., Crook 1981, Miller et al. 2006, Carrica et al. 2006a, Longo et al. 2007). In contrast, full-scale ship flow computations (Re
109) Manuscript received at SNAME headquarters January 28, 2008; revised manuscript received October 6, 2008. DECEMBER 2009

are limited (Gorski et al. 2004, Hanninen & Mikkola 2006). 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. 2007a, 2007b), 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

0022-4502/09/5304-0179$00.00/0

JOURNAL OF SHIP RESEARCH

179

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 2007). 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 2003). A three-layer model includes an additional formulation for the buffer layer (Temmerman et al. 2003), 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. (2003) who proposed a generalized multilayer model using curve fitting to provide a continuous function to bridge the sublayers and log layers. Kalitzin et al. (2005) 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 (2005) proposed a differential form for the boundary 180

DECEMBER 2009

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. (2006) extended the Kalitzin et al. (2005) 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 2008). Aupoix (2007) 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 2004). Schultz (2002) 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. 2002). 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. 2005), 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. (2002) 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. JOURNAL OF SHIP RESEARCH

+

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. (2004) 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 1992: Re=109 and 5 106) 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. (2002) 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. 2003) 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. (2002) studied full-scale effects using a two-layer model on two hull forms: the Dutch frigate the “DeRuyter” 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 2003) and multilayer (ML) (following Shih et al. 2003) 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 (2008) 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. 2002). 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 DECEMBER 2009

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.

2. 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. 2.1. 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. (2008) 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: ›Ui ¼0 ›xi

ð1Þ

›Ui ›Ui ›^ p 1 › 2 Ui › ¼ þ  ui uj þ ðUj  UGj Þ ›xj Re ›xj ›xj ›xj ›t ›xj

ð2Þ

where Ui = (U,V,W) are the Reynolds-averaged velocity components, UGj is the local grid velocity in either the absolute inertial earth-fixed or relative inertial Cartesian coordinates xi = (x,y,z). p^ = pabs/rU02 + z/Fr2 + 2k/3 is the dimensionless piezometric isffi the absolute pressure, ui uj are the Reynolds pressure where pabs pffiffiffiffiffi stresses, Fr = U0/ gL is the Froude number, and k is the turbulent kinetic energy (TKE). U0 is the free stream velocity, L is the ship length, and Re is the Reynolds number based on L. 2.2. Turbulence modeling 2.2.1. 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 (nt),   ›Ui ›Uj 2  dij k  ui uj ¼ nt þ ð3Þ 3 ›xj ›xi where dij 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 1 þ ðv  sk rnt Þ  rk  r2 k þ sk ¼ 0 ›t Pk

ð4aÞ

›v 1 2 r v þ sv ¼ 0 þ ðv  sv rnt Þ  rv  ›t Pv

ð4bÞ

JOURNAL OF SHIP RESEARCH

181

The turbulent eddy viscosity and the effective Peclet numbers are nt ¼ k=v;

Pk=v ¼

1 1=Re þ sk=v nt

ð5Þ

and the source terms in k and v equations constitute the production and dissipation terms (refer to Carrica et al. 2006b for details). The model constants, say a, are calculated from the standard k-v (a1), and k-e (a2) values using a blending function F1 (see Menter 1994 for the model constants values): a ¼ F1 a1 þ ð1  F1 Þa2

ð6Þ

layer and log-layer velocity profiles in the TL model and provides the buffer-layer profile. 8 þ y2;3 : y2;3 þ # 5 > > > > þ þ 2 þ 3 þ 4 U2;3 : 5 , y2;3 þ # 30 ut > > > :1 lnðy þ Þ þ B  DB : y þ . 30 k

2;3

2;3

ð11Þ

F1 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.

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+ = 20. This yields the model constants:

2.2.2. Wall-function (WF) models. In the TL model, the velocities in the sublayer and log-layer regions are:

a0 ¼ 1:875736; a1 ¼ 1:8158144; a2 ¼ 0:102066044; a3 ¼ 0:00295224178; a4 ¼ 3:3144178e  005

U2;3 ¼ u (t þ y2;3

ð12Þ ; yþ 2;3 # 11:67 : sublayer

þ þ þ k1 lnðyþ 2;3 Þ þ B  DB  1:13y2;3 P ; y2;3 . 11:67 : log-layer

ð7Þ where subscripts 2 and 3 represent the first and second grid point away from the wall, respectively. The superscript p + quantities are ffiffiffiffiffiffiffiffiffiffi nondimensionalized by the friction velocity (ut = tw =r), where tw 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 2002). The factor DB in equation (7) accounts for the effect of wall roughness, resulting in the downshift of the loglayer region (White 2008): DB ¼ k1 lnð1 þ kþ Þ  3:5

ð8aÞ

+

where k is the roughness parameter based on nondimensional roughness length ks. For naval applications, the wall roughness lies in the transitional roughness regime (Patel 1998), that is, 5 # k+ , 70.   n ›^ p ð8bÞ Pþ ¼ 3 ut ›x is a dimensionless parameter that is used to include the effect of pressure gradient (PG) tangential to the wall in the WFs  (Wilcox þ P , 3/4 as 1989). The effect of PG is clipped such that yþ 2 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  3:23 > k > : yþ j¼3 2 #5 > > Dy 3 > > 2 # 30 þ > du > > > > > : u2t : yþ 2 . 30 0:3 8 6n þ : y2 # 5 > 2 > > 0:075Dy2 > > s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2  2 <  6n ut ð14Þ v2 ¼ þ : 5 , yþ 2 # 30 > 0:3kDy2 > 0:075Dy22 > > > > : ut : yþ 2 . 30 0:3kDy2 2.2.3. Wall-function implementation. The WFs are implemented using the two-point approach by Tahara et al. (2002). 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 or ffilog-layer equation (7). For the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sublayer, ut = U3 =ðReDy3 Þ. For the log layer, the friction velocity is obtained by iteratively solving the log-layer equation using the Newton-Raphson method. 2. The computed friction velocity is then used in equation (7) to obtain the magnitude of the tangential velocity at j = 2, with the same direction of the tangential velocity at j = 3. The normal velocity at j = 2 is approximated using the linear interpolation, based on the normal velocity at j = 3 and wall JOURNAL OF SHIP RESEARCH

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 = 2 (Dy2) and j = 3 (Dy3). 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 = 2 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 þ þ þ possibilities: (yþ 2 : sublayer, y3 : sublayer; y2 : sublayer, y3 : logþ þ þ þ layer; or y2 : log-layer, y3 : log-layer). If either y2 or y3 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
107 to 109, for which the sublayer is thin and log-layer extent is large. Thus, both j = 2 and j = 3 can be conveniently placed in the log-layer region, for example, 3 Tahara et al. (2002) used yþ 2 to be 10 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, yþ 2
30 is applied for most simulations in the current study. However, in the lowpressure regions of the ship, such as the stern, the local yþ 2 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. 2.3. 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. DECEMBER 2009

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. (2007a) for details. 2.4. 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. 2.5. 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 JOURNAL OF SHIP RESEARCH

183

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. (2006a) and Xing et al. (2008) for more details of the implementation. 2.6. 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. 2002). The software SUGGAR (Noack 2005) 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. (2007a).

3. Smooth flat plate simulations

Fig. 2 Smooth flat plate boundary layer profiles are compared with the BKW results at Rex = 8 105. a Streamwise velocity profile. b Turbulent eddy viscosity (n+=Re nT)

+

+

+

+

+

+

+

The TL and ML models are first validated for smooth flat-plate simulations for a range of Re (106 to 109). Results are compared with BKW solutions, analytic log-layer profiles, and EFD data at large Re (Watson et al. 2000). The grids used for BKW and WF simulations consist of 201 91 5 and 201 71 5 grid points in the streamwise, wall-normal, and spanwise directions, respectively. At first, simulations are performed to study the effect of yþ 2 and yþ 3 on the prediction of the boundary layer profiles at Re = 8 105. As shown in Fig. 2, both TL and ML results are in close þ agreement with BKW results when both yþ 2 and y3 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 yþ 2 lies in the buffer-layer region. Flat-plate boundary layer profiles obtained from the WF simulations for a range of Re (106 to 109) using yþ 2
30 were compared with the analytic log-layer profiles (Bhushan et al. 2007). The comparison shows good agreement for u+, k+, v+, and n+ profiles. The variation of the skin friction coefficient along the plate for Re = 109 is shown in Fig. 3. The results agree well with the Karman-Schoenherr equation,* which is suitable for high Re flows (Watson et al. 2000), and the EFD data for yþ 2 = 30 to 600. Results are found to be slightly dependent on the yþ 2 for lower Rex, 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.

appended surface combatant 5415 was used in the SIMMAN (2008) 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. 2006, Miller et al. 2006, Xing et al. 2008) and 5415 (Carrica et al. 2008) 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 2005 ONR wave-breaking workshop to assess the capability of CFD methods in predicting bow and transom wave breaking. The fully *Karman-Schoenherr equation: cf = (0.558cd)/(0.558 + pffiffiffiffiffi cd = log10(cdRex) = log10(2Reu). 184

DECEMBER 2009

pffiffiffiffiffi cd ) where 0.242/

Fig. 3 Variation of the skin friction coefficient along the flat plate compared with experimental data of Watson et al. (2000) and KarmanSchoenherr equation JOURNAL OF SHIP RESEARCH

4.1. Experimental data and simulation conditions

1=3

**CA ¼ ð105ks

DECEMBER 2009

 0:64Þ · 103

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+ = 32), 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 1.2 and amplitude 0.01. The encounter period of the wave is 0.488L/U0. BKW and WF calculations are performed for towed BH at model scale (Re = 3.72 107). Full-scale (Re = 2 109) calculations are performed using TL model for towed BH and AH. For the SIMMAN (2008) 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. (2008) performed steady turn and zigzag maneuvers for self-propelled fully appended 5415. Steady turn computations were performed at model scale using BKW for 20 and 35 rudder deflections for Fr = 0.25 and 0.41, in calm and regular waves, and for constant torque and constant RPS. 20/20 zigzag maneuvers were performed using BKW for constant RPS for Fr = 0.25 and 0.41 at model scale (Re = 1.04 107). An additional simulation was performed for the Fr = 0.41 case at full scale (Re = 2.2 109) using the smooth-wall ML model. The zigzag

+

+

+

+

+

The EFD data for Athena bare hull with skeg (BH) are obtained from the towing tank experiments conducted on a 1/8.25-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 0.28 to 1.00. Miller et al. (2006) 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. (2008) 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 = 0.28, 0.48, and 0.8) and full scale (Re = 2 109, Fr = 0.48) using the smooth-wall TL model, (2) verification study at full scale (Re = 2 109, Fr = 0.48) for the 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 þ trim predictions on yþ 2 (y2 = 30, 100, and 300) at full-scale for smooth- and rough-wall (k+ = 28) 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 0.336 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** CA = 6.5 104. Xing et al. (2008) 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 = 0.432, 0.575, and 0.839. Herein, ship speeds and consequently Frs are predicted at full scale for AH free to sink and trim using operational propeller conditions at Fr = 0.432, 0.575, and 0.839 in the EFD. The effect of wall roughness is demonstrated for the Fr = 0.432 and 0.575 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 2 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 (2) wake profiles including axial, tangential, and radial velocities on the forward propeller plane at three radial locations: 0.417, 0.583 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 = 4.2 108 and Fr = 0.362, for which the most comprehensive

JOURNAL OF SHIP RESEARCH

185

maneuver is controlled by the rudder angle, which changes from 20 starboard to 20 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 20 , which is referred to as rudder check point. Here, results of the Fr = 0.41 model- and full-scale 20/20 zigzag maneuvers are presented for demonstration purposes. Table 1 summarizes the geometry, yþ 2 , simulation conditions, and comparison/validation data used for various WF simulations performed in this paper. 4.2. 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 con-

Model- and full-scale simulation conditions for Athena resistance, powering, and seakeeping, and 5415 maneuvering using WFs

Geometry

Ship motions

0.28 0.48 0.8 0.48 0.48 0.48 0.48

Resistance, ITTC line

Resistance coefficients, sinkage and trim, BKW results (Xing et al. 2008) and EFD (Jenkins 1984) data at model-scale. Compare ML and TL results for smooth-wall. Study sensitivity of results on y+ and wall-roughness for both ML and TL models. Verification study using TL model

0.8

6- grids 30

3.72 107 2 109

0.3

30–40 30–40

TL, smooth-wall

BKW solution at model-scale

3.66 108

0.432

65

4.88 108

0.575

85

7.15 108

0.839

120

TL, smooth and rough-wall ML and TL, smooth and rough-wall TL, smooth-wall

Resistance, power, and motion BKW results (Xing et al. 2008) and EFD (Crook 1981) data at model scale extrapolated to full-scale. Compare TL and ML results and wall roughness effect.

Predict sinkage and trim Pitch and heave in waves

4.2 108

0.362

50

2 109

0.3

30–40

TL, smooth and rough-wall TL, smooth-wall

Boundary layer and wake survey, full-scale EFD (Day et al. 1980) Bare hull with skeg in waves

6DOF

2.2 109

0.41

100

ML, smooth-wall

BKW solution at model scale

+ + +

+

Predict sinkage and trim

DECEMBER 2009

+

+

+

186

TL, smooth-wall

30, 100, 300

Predict sinkage and trim

Pitch and heave in waves

Self-propelled fully appended 5415

Resistance coefficients, BKW results (Xing et al. 2008) and EFD (Jenkins 1984) data

0.8 0.48

30–40

2 10

Towed fully appended Athena

TL, smooth-wall

ML and TL, smooth-wall TL, smooth-wall ML and TL, smooth and rough-wall TL, smooth-wall TL, smooth-wall

2.23 107

9

Self-propelled fully appended Athena

30–40 30 30–40 32.67 118.35 277.15

Results and Compare

+

107 107 107 108 108 109

Wall-Function Model, Wall Condition

+

Towed Athena bare hull with skeg

1.3 2.23 3.72 2 8 2

yþ 2

Fr

+

Fixed sinkage and trim

Re

+ + + + + +

Table 1

ditions 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 yþ 2
30 is used, whereas for = 30 to 300. The AH grid consists of full-scale calculations yþ 2 2.2M points split into 24 blocks (Xing et al. 2008). For the fullscale WF simulations the grid design is such that yþ 2 = 30 to 60. The grid used for the 5415 simulation consists of 7M grid points decomposed into 72 blocks (Carrica et al. 2008). The yþ 2 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 2. 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,

JOURNAL OF SHIP RESEARCH

tively. 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 (2008) for 5415.

5. 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

5.2. Verification study for resistance, sinkage, and trim Solution verification study is performed following the quantitative methodology and procedures proposed by Stern et al. (2006b) (ST) and recent modifications proposed by Xing & Stern (2008) (XST) for situations when estimated order of accuracy (PG) is larger than the theoretical order of accuracy (PG,th) and correction factor 1 , CG , 2. CG  2 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 rG = 21/4, summarized in Table 3. Six sets of verification studies are possible, four with rG = 21/4 (1,2,3; 2,3,4; 3,4,5; and 4,5,6) and two with rG = 21/2(1,3,5 and 2,4,6). Results have been plotted in Figs. 9 a and c, and the converged results are shown in Table 4. Convergence condition is defined by RG which is the ratio of solution changes for medium-fine and coarse-medium solutions. Correction factor 1  CG indicates how far the solution is from asymptotic range where CG =1. Figures 9 b and d show the relative change of the solution (eN) between two successive grids along with iterative errors (UI) obtained using five nonlinear iterations on each grid for resistance coefficients and motions, respectively. UI 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. (2008) (XET) for BH at model scale (Re = 2.23 107, Fr = 0.48) using BKW where an additional grid consisting of 8.1M grid points was considered. Ctot,x monotonically converges on grids (2,4,6) and (2,3,4), oscillatorially converges on grids (1,3,5), oscillatorially diverges on grids (1,2,3) and (4,5,6) and monotonically diverges on grids (3,4,5). The pressure resistance coefficient (Cp,x) monotonically converges on grids (2,4,6) and (3,4,5), oscillatorially converges

+

integral, and differential (PID) speed controller is used that controls the ship velocity via the propeller RPS (Huang et al. 2007). 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 195.0 and 187.8 in full-scale dimensions, respec-

The model- and full-scale results obtained using smooth-wall TL models are presented in Fig. 7. The total resistance coefficients (Ctot,x) at model scale are within 5% of the EFD data for Frs = 0.28 and 0.48, but is overestimated by as much as 6.75% for Fr = 0.8. Model- and full-scale (Re = 107 to 109) frictional resistance coefficients (Cf,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. (2007).

Fig. 6 Grid topology for a fully appended Athena and b fully appended 5415 DECEMBER 2009

{

ITTC: Cf = 0.075/(log Re 2)2 JOURNAL OF SHIP RESEARCH

187

Table 2 Boundary conditions for all the variables f

Boundary

Inlet

p

kfs ¼ 10

vfs ¼ 9

Uinf = 1

Vinf = 0

Winf = 0

kfs ¼ 107

vfs ¼ 9

Uinf = 0

Vinf = 0

Winf = 0

kfs ¼ 107

vfs ¼ 9

Uinf = Eq. (40) (Carrica et al. 2007a)

Vinf = 0

Winf = Eq. (41) (Carrica et al. 2007a)

¼0

›2 U ›n2 ¼ 0 ›U ›n ¼ 0

›2 V ›n2 ¼ 0 ›V ›n ¼ 0

›2 W ›n2 ¼ 0 ›W ›n ¼ 0

¼0

Uinf

Vinf

Winf

¼0

›U ›n

0 nship

›W ›n

Eq. (43) (Carrica et al. 2007a)

Eq. (42) (Carrica et al. 2007a)

Far field No. 1 Far field No. 2 Symmetry No-slip

Wall-function ( j = 2)

¼0 ¼0

0

¼0

›p ›n ›p ›n

¼0

v

›f ›n

›p ›n

¼0

w

¼0

f = z f = z

›f ›n ›f ›n ›f ›n ›f ›n

n

u

¼0

Towed Selfpropelled Seakeeping

Exit

v

k 7

›p ›n ›p ›n

¼0 ¼0

¼0 Carrica et al. (2007a), Eq. (37) ›p ›n ¼ 0

›k ›n ›k ›n ›k ›n ›k ›n

¼0

›v ›n ›v ›n ›v ›n ›v ›n

¼0 ¼0 ¼0

¼0

0

6n 0:075Dy22

TL: Eq. (9) ML: Eq. (13)

TL: Eq. (10) ML: Eq. (14)

on grids (1,3,5) and (2,3,4), oscillatorially diverges for (4,5,6), and monotonically diverges on grids (1,2,3). Cf,x monotonically converges on grids (1,2,3), oscillatory converges on grids (1,3,5), (2,4,6), and (2,3,4), oscillatorially diverges on grids (4,5,6), and monotonically diverges on (3,4,5). Sinkage monotonically converges on grids (2,3,4), oscillatory converges on grids (1,2,3), (3,4,5) and (4,5,6), oscillatorially diverges on grids (1,3,5), and monotonically diverges on grids (2,4,6). Trim monotonically converges on grids (1,2,3), oscillatorially converges on grids (3,4,5) and (4,5,6), oscillatorially diverges on grids (2,4,6) and (1,3,5), and monotonically diverges on grids (2,3,4). It must be noted that the monotonically converged solutions involve the three finest grids 1, 2, or 3, and most of the

¼0

uship

¼0 wship

Transformation of tangential velocity and normal velocity to physical coordinate system. Eq. (7) for TL model or Eq. (11) for ML model.

diverged solutions involve coarsest grid 6. XET also obtained diverged solutions involving this grid, which is likely caused by insufficient grid resolution. Ctot,x shows large values of 1CG, 2.45 for grids (2,4,6), and 74 for grids (2,3,4). The grid uncertainty (UG) is 1.55% on grids (2,4,6) and 0.132% on grids (2,3,4) using the ST method. UG is 5.53% on grids (1,3,5). Grids (2,4,6) are closest to the asymptotic range for Cp,x for which 1CG is 0.2. 1 CG for grids (3,4,5) is 2.79. UG for Cp,x is below 4% on grids (2,4,6) for both the ST and XST methods, 13.59% on grids (3,4,5) based on the ST method, and 11% and 3.24% on grids (1,3,5) and (2,3,4), respectively. 1 CG for Cf,x is 0.81, and UG is 2.91% and 27.8% on grids (1,2,3) based on the ST and XST methods, respec-

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 2009

JOURNAL OF SHIP RESEARCH

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, UG is below 3.5%. A large range of 2.79 , 1 CG , 0.2, neglecting the largest value, for resistance coefficients suggests that all solutions are still far from the asymptotic range. XET also observed significant variation in 1.42 # 1CG # 0.93; however, the range of variation was smaller than the present case. The variations of Cf,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. eN of Cp,x and Ctot,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 2. Further refinement to grid 1 leads to a rebound of eN, which is caused by the problem of separating UI from UG as they are of the same order of magnitude. eN for Cp,x in the WF simulations are about 7%, which is significantly larger than 2% in XET. Overall, UI around 2% 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 (2,3,4) is closest to asymptotic range for sinkage as 1  CG is 0.16. UG is 15.48% on grids (2,3,4), about 4% on grids (1,2,3), 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  CG around 39 on grids (1,2,3). UG is 2.56% on grids (1,2,3) 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. eN for sinkage decreases almost linearly when grid is refined from 6 to 1. eN for trim also shows a linear decrease with grid refinement from 6 to 3. A sudden rise in eN is observed for grids 2 and 3 when UI and UG are of same order of magnitude. Maximum UI for both sinkage and trim are obtained between grids 2 and 3, which are 2.7% and 3.4%, respectively, which are larger compared to 0.2% in XET. Of all the converged grid studies, the lowest grid uncertainty for the resistance coefficients and motions are obtained on grids (2,3,4) and (1,2,3), respectively. The XST method predicts more reasonable estimate for UG compared with ST method for Cp,x on grids (1,2,3) where PG . PG,th. Grid studies (2,3,4) and (2,4,6) are closest to asymptotic range for sinkage and Cp,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

+

DECEMBER 2009

3/4

+ +

2 187 49 94 187 49 94 1,722,644 50

+ +

+ +

2 157 41 79 157 41 79 1,017,046 62

+ +

2 132 34 66 132 34 66 592,416 78

2

1

2 222 58 112 222 58 112 2,884,224 40

25/4 264 69 133 264 69 133 4,845,456 30

JOURNAL OF SHIP RESEARCH

+ +

3

1/2

+ +

4

1/4

+ +

5

+ +

+ +

1 111 29 56 111 29 56 360,528 94

+ +

Ratio Ship Background Total yþ 2

6

+ +

Grid number

109, Fr = 0.48)

+ +

Table 3 Grids used for verification study for Athena bare hull with skeg (Re = 2

189

+

+

Fig. 9 Verification for resistance coefficients and motions for Athena bare hull with skeg (Re = 2 109, Fr = 0.48). a Resistance coefficients. b Relative change eN = |(SNSN+1)/S1| 100 and iterative errors for resistance coefficients. c Sinkage and trim. d Relative change eN and iterative errors for sinkage and trim

1  CG are observed suggesting that further grid refinement is required, (c) a significant increase in eN is observed when UI and UG are of the same order of magnitude. There are specific differences such as: (a) The variations of Cf,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 Cp,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 UG due to higher levels of UI, (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. 5.3. 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 Cf,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 = 2 Monotonically converged solutions, UG is %S1, %S2, or %S3

109, Fr = 0.48).

UG (%) Parameter

Grids

Ctot,x

2,4,6 2,3,4

Cp,x

2,4,6 3,4,5

Cf,x Sinkage Trim

1,2,3 2,3,4 1,2,3

190

DECEMBER 2009

Refinement Ratio (rG) p ffiffiffi 2 p ffiffi2ffi 4 2 p ffiffiffi 2 ffiffi2ffi p 4 2 p ffiffiffi 4 p ffiffi2ffi 4 2ffiffiffi p 4 2

RG

PG

1  CG

Stern et al. (2006a)

Xing and Stern (2008)

0.224 0.0313

4.31 20.1

2.45 73.84

1.55 0.132

0.555 0.389

1.7 5.45

0.2 2.79

3.98 13.59

3.98 –

0.571 0.743 0.057

3.23 1.72 16.5

0.81 0.164 38.71

2.91 15.48 2.56

27.80 15.48 –

– –

JOURNAL OF SHIP RESEARCH

+

Table 4(b) Verification study for resistance coefficients and motions of Athena bare hull with skeg at full-scale (Re = 2 109, Fr = 0.48). Oscillatory converged solutions, UG is %S1, %S2, %S3, or %S4

RG

UG (%)

0.77

5.53

1,3,5 2,3,4

Refinement Ratio (rG) p ffiffiffi 2 2 p ffiffiffi 2 ffiffi2ffi p 4 2

0.80 0.28

10.59 3.24

Cf,x

1,3,5 2,4,6 2,3,4

ffiffiffi p 2 2 p ffiffiffi 4 2

0.97 0.30 0.55

1.39 3.42 2.74

Sinkage

1,2,3 3,4,5 4,5,6

ffiffiffi p 4 2

0.35 0.75 0.68

4.09 7.44 11.64

Trim

3,4,5 4,5,6

ffiffiffi p 4 2

0.59 0.67

3.57 5.47

Parameter

Grids

Ctot,x

1,3,5

Cp,x

+

compared with BKW, respectively. Cp,x is within 3% of the BKW results for both the WF models. Ctot,x is within 6% of the BKW results and 2.5% 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 3.2% of the EFD data for Fr = 0.8. Larger differences around 10% are observed between the WF results and EFD data for Fr = 0.48. Overall, the ML results are within 2% of the TL results. Cf,x values obtained from full-scale (Re=2 109) simulations are found to be in good agreement with the ITTC line, where relative errors are less than 5%. The differences in the model-

and full-scale Cp,x are less than 6.5%, consistent with the previous conclusions by Oh & Kang (1992). 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 2%. 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+ = 28). For the TL model Cf,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. Cp,x is unaffected by the wall-roughness effect as the differences are within 0.1%. Ctot,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. 5.4. Sensitivity of resistance, sinkage, and trim to y2+ As shown in Figs. 10 a and c, for the smooth-wall TL model the Ctot,x, Cp,x and trim vary by as much as 15% when yþ 2 is increased from 30 to 300. Respective variations in the Cf,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 yþ 2 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 Cf,x by 28%, which is in good agreement with the ITTC correlation allowance of 30%. ML model overpredicts Cf,x for yþ 2 = 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) Sinkage

Trim

Ctot,x*

0.48

2.23 107 2.23 107 E

0.00341 0.00320 (6.16%)

0.710 0.676 (4.79%)

0.00575 0.00578 (0.522%)

0.002480

0.00223

0.00310 (8.82%) 0.00305 (10.56%) 0.00307 (9.97%) 0.00305 (10.62%) 0.00307 (9.99%) 0.00308 (9.68%) 0.00306 (10.26%)

0.641 (9.71%) 0.638 (10.14%) 0.667 (6.06%) 0.664 (5.39%) 0.668 (5.92%) 0.673 (5.21%) 0.657 (7.46%)

0.005873 (2.139%) 0.005800 (0.870%) 0.00385 0.00382 (0.78%) 0.00390 (+1.30%) 0.00431 (+11.95%) 0.00454 (+16.41%)

0.002673 (7.78%) 0.002585 (4.23%) 0.00144 0.001436 (0.28%) 0.00147 (+2.08%) 0.00185 (+28.47%) 0.00208 (+41.50%)

0.002288 (2.60%) 0.002284 (2.42%) 0.00216 0.00217 (+0.46%) 0.00218 (+0.09%) 0.00217 (+0.05%) 0.00216 (+0.00%)

3.72 107 3.72 107 E

0.00194 0.00145 (+25%)

0.997 0.934 (6.3%)

0.00430 0.00411 (4.65%)

0.00220

0.00176

3.72 107 E 2.0 109 E

0.00120 (+38%) 0.00128 (+34%)

1.029 (+3.2%) 1.061 (+6.4%)

0.00437 (+1.63%) 0.00341

0.00236 (+7.27%) 0.00141

0.00178 (1.14%) 0.00187 (+6.25%)

WF TL ML TL TL, with PG effect ML TL k+ = 28 ML k+ = 28

WF TL TL

+

2.23 107 E

+ + + + +

2.0 2.0 2.0 2.0 2.0 0.8

109 E 109 E 109 E 109 E 109 E

+ +

EFD BKW

+ +

Re

+ +

EFD BKW

Fr

Cf,x+

Cp,x+

–

–

–

–

*Results are based on static areas. + E is based on BKW results. DECEMBER 2009

JOURNAL OF SHIP RESEARCH

191

+

+

+

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 = 0.432 than those for Fr = 0.575 and 0.839. 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 2%. Rough-wall simulations are performed for Fr = 0.432 and 0.575 using a roughness length ks = 100 mm corresponding to k+ = 25 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 Cf,x by 6.0 104 for the TL and 5.5 104 for ML model, which are in close agreement with the correlation allowance CA = 6.5 104 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 y2þ for Athena bare hull with skeg (Re = 109 and Fr = 0.48) using TL and ML models a resistance coeffi2 cients 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 yþ 2 considered.

6. Self-propelled fully appended Athena As shown in Table 6 and Fig. 11, the smooth-wall WF overpredicts Frs by 2% to 3% caused by the underprediction of Ctot,x. Cf,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 = 0.432. The differences are larger for Fr = 0.575 and 0.839, similar trend is also observed in the BKW 192

DECEMBER 2009

The wall roughness effect increases Cf,x by 21, 31.6, and 39.5% over the smooth-wall for k+ = 15, 32, and 43, respectively. This is in good agreement with the ITTC correlation allowance. Cp,x variations are within 3% for all the surface conditions. The corresponding increases in Ctot,x are 6.3, 10.5, and 12.3%. Sinkage and trim values are 0.00233 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+ =32. In Fig. 12, the boundary layer velocity profiles obtained from the smooth- and rough-wall (k+ = 32) 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 2, whereas the nature is reversed for rake 4. At rake 2, 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 0.92 opposed to the expected value of 1.0 or due to grid resolution issues as probe 2 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 0.963 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. JOURNAL OF SHIP RESEARCH

Sinkage

Trim

Cf,x*

– 15.37

1.61 107 1.61 107

+ +

0.432 0.423 (2.08%)

0.00360 0.00381 (5.8%)

0.726 0.740 (+1.9%)

– –

15.18

3.67 108

0.441 (+2.08%) 0.433 (+0.23%)

0.00368 (2.2%) 0.00370 (2.78%)

0.813 (12%) 0.812 (+11.8%)

1.95 103 2.56 103 (+31%)

– 20.08

2.15 107 2.15 107

0.575 0.567 (1.4%)

0.00335 0.00305 (+9%)

1.46 1.54 (+5.5%)

WF TL (E) TL, k+=35 (E) ML (E) ML, k+ = 35 (E)

19.85

8

4.88 10

0.589 (+2.44%) 0.580 (+0.87%) 0.590 (+2.61%) 0.581 (+1.04%)

0.00279 (+17.9%) 0.00279 (+17.9%) 0.00273 (+18.5%) 0.00274 (+18.2%)

1.43 (2.10%) 1.42 (2.73%) 1.46 (0.02%) 1.45 (0.07%)

EFD BKW (E) WF, TL (E)

– 27.72 27.43

3.13 107 3.13 107 7.15 108

0.839 0.830 (1.1%) 0.856 (+2.02%)

8.0 104 3.0 104 (+63%) 2.0 104 (+75%)

1.595 1.75 (+9.7%) 1.65 (+3.5%)

– –

+ + + +

1.9 2.5 1.85 2.41

103 103 (+32%) 103 103 (+30%) – –

1.76 103

+

EFD BKW (E)

+

WF TL (E) TL, k+ = 26 (E)

+ + +

EFD BKW (E)

+ +

Fr

+ + +

Re

+ +

RPS

+

Table 6 Self-propelled simulation at model- and full-scale Reynolds number for fully appended Athena. Error (E) is based on EFD data (D)

*

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 20 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

DECEMBER 2009

Fig. 12 Boundary layer velocity profiles obtained using smooth- and roughwall conditions at four rake locations, shown in Fig. 4 b, for fully appended Athena (Re = 4.2 108, Fr = 0.3621) are compared with EFD data (Day et al. 1980). The coordinate z*/L is the distance normal to the wall.

+

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+ = 25 to 35) at full-scale using wall-function; lines (and filled circles): EFD (Crook 1981)

JOURNAL OF SHIP RESEARCH

193

+

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 = 4.2 108, Fr = 0.3621), left: 0.417 R and right: 0.75 R. Black line: smooth-wall; gray line: Rough wall, k+ = 32; ○ 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 = 4.2 108, Fr = 0.3621), left: 0.456 R and right: 0.963 R. Black line: smooth-wall; gray line: rough wall, k+ = 32; ○ EFD (Day et al. 1980) 194

DECEMBER 2009

JOURNAL OF SHIP RESEARCH

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 = 1.2 L, amplitude = 0.01 L and wave encounter frequency = 0.488 L/U0

Fig. 16

Fig. 17

of the heave is 0.65 for both model and full scale. Figure 16 shows that the peak of Cf,x correlates with the trough of the heave, whereas the Ctot,x peaks in phase with the pitch. A bimodal pattern is observed in the WF simulations at Cf,x minima that occurs after the peaks of the pitch and heave. Similar bimodal pattern in the Ctot,x minima is caused by the 180 phase difference in Cp,x and Cf,x troughs. The WF predictions of Cf,x at model scale are 5% to 10% lower than the BKW. Cf,x in the full-scale calculation is about 12% lower than the model-scale WF, which is in good agreement with the ITTC line. There is no significant scale effect on Cp,x and Ctot,x, for which the relative changes are within 2%. Figure 17 compares the full-scale AH results with those of the full-scale BH results. In the AH results pitch lags by 1.2 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 2%. 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 Ctot,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. 2007b). The emphasis of this example is to demonstrate the capability of WFs to simulate ship motions

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

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 2009

JOURNAL OF SHIP RESEARCH

195

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. Ctot,x of the AH is higher by about 10% to 15% compared with the BH case.

9. 20/20 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. (2008) 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 196

DECEMBER 2009

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 0.24 for the model scale and 0.29 for the full scale. Heave values do not show much variation in model- and full-scale simulations. The steady value is estimated to be 0.0059 and maximum is 0.0035 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 JOURNAL OF SHIP RESEARCH

and downstream of the shaft strut are shown in Fig. 19. Since the propeller is modeled as a body force, this leads to higher velocities at the propeller plane and higher velocity reaching the rudders. This would explain the slightly faster reaction of the full-scale ship in the zigzag maneuvers and smaller overshoot.

propulsion, seakeeping, and maneuvering. However, there are issues and questions that need to be resolved in the future: • To relax the sensitivity of rough-wall simulations using ML

•

10. Conclusions and future work This study develops and implements TL and ML models with wall roughness and pressure gradient effects using a two-point approach in the URANS solver CFDShip-Iowa. The TL and ML models are first validated for smooth flat-plate simulations at large Re equivalent to full-scale ship Re. Results show that the þ TL model is accurate when both yþ 2 and y3 are placed in either sublayer or log layer, whereas ML has no such limitations. The WF models are further validated for BH simulations at both model and full scale with fixed or predicted sinkage and trim. Resistance, sinkage, and trim predictions at model scale agree well with the BKW results and EFD data. Quantitative verification of resistance coefficients and motions at full scale shows that the grids considered are still far from the asymptotic range. A sufficiently fine grid is identified and used for fullscale simulations. In the WF simulations the iterative errors and relative changes between two successive grids of Cp,x are higher compared with the BKW results of Xing et al. (2008) at model scale. For full-scale simulations Cf,x predictions are in good agreement with the ITTC line. Rough-wall simulations show higher Cf,x and Ctot,x, where the former agrees well with the ITTC correlation allowance. As expected, sinkage and trim are not significantly affected by the Re or the wall roughness. In full-scale simulations, boundary layer is thinner and the slope of the transom wave is slightly lower compared with the model scale. It is shown that the TL model is more sensitive to yþ 2 # 300 than the ML model for the smooth-wall simulations. ML model shows limitations in accounting for the wall roughness effect for yþ 2
30, which is attributed to the tentative modeling of the roughness effect in the buffer layer. No significant changes of results are observed by including the PG effect in TL model for full-scale computations. Resistance and powering computations are performed at full scale for self-propelled AH free to sink and trim using smoothand rough-wall WFs. In the rough-wall simulations Frs are predicted better than that in the smooth-wall simulations, when compared with full-scale data extrapolated from model scale measurements using ITTC ship-model correlation line including a correlation allowance. The boundary layer and wake profiles obtained from the full-scale rough-wall simulations using WF for the towed AH free to sink and trim are in good agreement with the full-scale EFD data. Seakeeping calculations are performed using WF for demonstration purpose at full-scale for both BH and AH. The ship motions are mainly governed by the incoming wave frequency and no significant Re or geometry effects are observed. Maneuvering calculation is performed for 5415 at full-scale using WF. Results show slightly more efficient rudder action, lower overshoots in the heading angle, and larger roll angles with slower damping compared with the model scale. Overall, the results are encouraging and demonstrate the versatility of a two-point, multilayer wall-function in accurately predicting model- and full-scale ship flows including resistance, DECEMBER 2009

• •

•

•

•

model for y2+
30, the correct formulation for downshift of the buffer-layer velocity profile has to be established. Aupoix (2007) recently proposed a general procedure to extend turbulence models to account for surface roughness that could be of interest for near-wall turbulence models. The author also provided a better relation of the downshift of log-law with roughness length that can be implemented along with current wall-function models. The benefits of including PG effects for more accurate prediction of flow separations need further investigation. It is valuable to further evaluate the current two-point approach by comparing with the one-point approach that allows solution of the pressure equation up to the wall for better mass conservation. WF simulations in maneuvering calculation showed that insufficient grid resolution near the transom corner may introduce error in the calculation of tangential and normal velocity direction. The sensitivity of the current WFs to different grid topologies will be further evaluated. A verification and validation study is required for modeland full-scale resistance and propulsion calculations for AH for which coarse grids were used in present study. WFs need to be further evaluated for more geometries and general applications to get more experience such as planning ships, ship-ship interaction, and so forth.

References AUPOIX, B. 2007 A general strategy to extend turbulence models to rough surfaces: Application to Smith’s k-L model, Journal of Fluids Engineering, 129, 1245–1254. BALAY, S., BUSCHELMAN, K., GROPP, W., KAUSHIK, D., KNEPLEY, M., CURFMAN, L., SMITH, B., AND ZHANG, H. 2002 PETSc User Manual, ANL-95/11, Revision 2.1.5, Argonne National Laboratory. BHUSHAN, S., XING, T., CARRICA, P., AND STERN, F. 2007 Model- and fullscale URANS/DES simulations for Athena R/V resistance, powering, and motions, Proceedings, Ninth International Conference on Numerical Ship Hydrodynamics, August 5–8, Ann Arbor, MI. BRADSHAW, P., AND HUANG, G. P. 1995 The law of the wall in turbulent flow, Proceedings of the Royal Society (London) Mathematical and Physical Sciences, 451, 1941 165–188. BULL, P., VERKUYL, J. B., RANOCCHIA, D., DI MASCIO, A., DATTOLA, R., MERLE, L., AND CORDIER, S. 2002 Prediction of high Reynolds number flow around naval vessels, Proceedings, 24th Symposium Naval Hydrodynamics, Fukuoka, Japan. CARRICA, P. M., WILSON, R. V., NOACK, R., XING, T., KANDASAMY, M., AND STERN, F. 2006a A dynamic overset, single-phase level set approach for viscous ship flows and large amplitude motions and maneuvering, Proceedings, 26th Symposium on Naval Hydrodynamics, September 17–22, Rome, Italy. CARRICA, P. M., WILSON, R. V., AND STERN, F. 2006b Unsteady RANS simulations of the ship forward speed diffraction problem, Computers & Fluids, 35, 6, 545–570. CARRICA, P. M., WILSON, R. V., AND STERN, F. 2007a An unsteady singlephase level set method for viscous free surface flows, International Journal for Numerical Methods in Fluids, 53, 2, 229–256. CARRICA, P. M., WILSON, R. V., NOACK, R., AND STERN, F. 2007b Ship motions using single-level set with dynamic overset grids, Computers & Fluids, 36, 9, 1415–1433. CARRICA, P. M., ISMAIL, F., HYMAN, M., BHUSHAN, S., AND STERN, F. 2008 Turn and zigzag maneuvers of a surface combatant using a URANS apJOURNAL OF SHIP RESEARCH

197

+

proach with dynamic overset grids, Proceedings, SIMMAN Workshop, April 14–16, Copenhagen, Denmark. CHEN, H. C., AND PATEL, V. C. 1988 Near wall turbulence models for complex flows including separation, AIAA Journal, 26, 6, 641–648. CHOI, J.-E., MIN, K.-S., CHUNG, S.-H., AND SEO, H.-W. 2003 Study on the scale effects on the flow characteristics around a full slow-speed ship, Proceedings, Eighth International Conference on Numerical Ship Hydrodynamics, Busan, Korea. CROOK, L. B. 1981 Powering Predictions for the R/V Athena (PG 94) Represented by Model 4950-1 with Design Propellers 4710 and 4711, David W. Taylor Naval Ship Research and Development Center, DTNSRDC/SPD-0833-05. DAY, W. G. Jr., REED, A. M., AND HURWITZ, R. B. 1980 Full-Scale Propeller Disk Wake Survey and Boundary Layer Velocity Profile Measurements on the 154-Foot Ship R/V Athena, David W. Taylor Naval Ship Research and Development Center, DTNSRDC/SPD-0833-01. ESCH T., AND MENTER, F.R. 2003 Heat transfer prediction based on twoequation turbulence models with advanced wall treatment, Proceedings, International Symposium on Turbulence, Heat and Mass Transfer, 4, Antalya, Turkey, 614–621. GORSKI, J. J. 2002 Present state of numerical ship hydrodynamics and validation experiments, Journal of Offshore Mechanics and Arctic Engineering, 124, 74–80. GORSKI, J. J., MILLER, R. W., AND COLEMAN, R. M. 2004 An investigation of propeller inflow for naval surface combatants, Proceedings, 25th Symposium Of Naval Hydrodynamics, St. John’s, NL, Canada. GROTJANS, H., AND MENTER, F. 1998 Wall functions for general CFD codes, Proceedings, Fourth European Comp. Fluid Dynamics Conference, Athens, Greece, 1112–1117. HANNINEN, S., AND MIKKOLA, T. 2006 Computation of ship-hull flows at model- and full-scale Reynolds numbers, Proceedings, Numerical Ship Hydrodynamics Seminar, Maritime Institute of Finland, Turku. HUANG, J., CARRICA, P. M., MOUSAVIRAAD, M., AND STERN, F. 2007 Semicoupled air/water immersed boundary approach in curvilinear dynamic overset grids with application to environmental effects in ship hydrodynamics, Proceedings, Ninth International Conference on Numerical Ship Hydrodynamics, August 5–8, Ann Arbor, MI. ISSA, R. I. 1985 Solution of the implicit discretised fluid flow equations by operator splitting, Journal of Computational Physics, 62, 40–65. JENKINS, D. 1984 Resistance Characteristics of the High Speed Transom Stern Ship R/V Athena in the Bare Hull Condition Represented by DTNSRDC Model 5365, Ship Performance Dept Research & Development Report, DTNSRDC-84/024. JIMENEZ, J. 2004 Turbulent flows over rough walls, Annual Review of Fluid Mechanics, 36, 173–196. KALITZIN, G., MEDIC, G., IACCARINO, G., AND DURBIN, P. 2005 Near-wall behavior of RANS turbulence models and implications for wall functions, Journal of Computational Physics, 204, 265–291. KIM, S.-E., AND CHOUDHURY, D. 1995 A near-wall treatment using wall functions sensitized to pressure gradient, ASME-FED Separated and Complex Flows, 217, 273–280. KNOBLOCH, K., AND FERNHOLZ, H.-H. 2002 Statistics, correlations, and scaling in a turbulent boundary layer at Red = 1.15 105, Proceedings, IUTAM Symposium on Reynolds Number Scaling in Turbulent Flow, September 11–13, Princeton, NJ. KNOPP, T., ALRUTZ, T., AND SCHWAMBORN, D. 2006 A grid and flow adaptive wall-function method for RANS turbulence modeling, Journal of Computational Physics, 220, 1, 19–40. LARSSON, L. 1997 CFD in ship design-prospects and limitation, 18th Georg Weinblaum Memorial Lecture, Ship Tec Research, 44, 3, 133–154. LAUNDER, B. E., AND SPALDING, D. B. 1974 The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering, 3, 269–289. LONGO, J., SHAO, J., IRVINE, M., AND STERN, F. 2007 Phase-averaged PIV for the nominal wake of a surface ship in regular head waves, Journal of Fluids Engineering, 129, 5, 524–540. MENTER, F. R. 1994 Two equation eddy viscosity turbulence models for engineering applications, AIAA Journal, 32, 8, 1598–1605. MILLER, R., GORSKI, J., XING, T., CARRICA, P., AND STERN, F. 2006 Resistance predictions of high speed mono and multi-hull ships with and without

198

DECEMBER 2009

water jet propulsors using URANS, Proceedings, 26th Symposium on Naval Hydrodynamics, September 17–22, Rome, Italy. NOACK, R. 2005 SUGGAR: a general capability for moving body overset grid assembly, AIAA paper 2005-5117, Proceedings, 17th AIAA Computational Fluid Dynamics Conference, Toronto, Ontario, Canada. OH, K. J., AND KANG, S. H. 1992 Full scale Reynolds number effects for the viscous flow around the ship stern, Computational Mechanics, 9, 85–94. PARK, I. L.-R., VAN, S.-H., KIM, J., AND AHN, H.-S. 2004 Two-phase viscous flow simulation around a commercial container ship hull form, Journal of Ships & Ocean Engineering, 37, 147–154. PATEL, V. C. 1998 Perspective: Flow at high Reynolds number and over rough surfaces—Achilles Heel of CFD, Journal of Fluids Engineering, 120, 434–444. SCHULTZ, M. P. 2002 The relationship between frictional resistance and roughness for surfaces smoothened by sanding, Journal of Fluids Engineering, 124, 492–499. SHIH, T. H., POVINELLI, L. A., AND LIU, N. S. 2003 Application of generalized wall function for complex turbulent flows, Journal of Turbulence, 4, 1–16. SIMMAN. 2008 Workshop on verification and validation of ship maneuvering simulation methods, April 14–16, Copenhagen, Denmark, http://www. simman2008.dk/. SO, R. M. C., AND LAI, Y. G. 1988 Low-Reynolds-number modeling of flows over a backward-facing step, Journal of Applied Mathematics and Physics, 39, 13–27. STARKE, B., RAVEN, H., AND PLOEG, A. 2007 Computation of transom-stern flows using a steady free-surface fitting RANS method, Proceedings, Ninth International conference on Numerical Ship Hydrodynamics, August 5–8, Ann Arbor, MI. STERN, F., KIM, H. T., PATEL, V. C., AND CHEN, H. C. 1988 A viscous-flow approach to the computation of propeller-hull interactions, JOURNAL OF SHIP RESEARCH, 32, 4, 246–262. STERN, F., CARRICA, P. M., KANDASAMY, M., GORSKI, J. J., O’DEA, M. H., MILLER, R., HENDRIX, D., KRING, D., MILEWSKI, W., HOFFMAN, R., AND GARY, C. 2006a Computational hydrodynamic tools for high-speed cargo transports, Proceedings, SNAME Maritime Technology Conference, Fort Lauderdale, FL. STERN, F., WILSON, R., AND SHAO, J. 2006b Quantitative approach to V&V of CFD simulations and certification of CFD codes, International Journal of Computational Fluid Dynamics, 50, 1335–1355. TAHARA, Y., KATSUUI, T., AND HIMEMNO, Y. 2002 Computation of ship Viscous flow at full scale Reynolds number, Journal of the Society of Naval Architects of Japan, 92, 89–101. TEMMERMAN, L., LESCHZINER, M. A., MELLEN, C. P., AND FROHLICH, J. 2003 Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions, International Journal of Heat and Fluid Flow, 24, 157–180. TZABIRAS, G. D. 1991 A numerical study of the turbulent flow around the stern of ship models, International Journal for Numerical Methods in Fluids, 13, 1179–1204. UTYUZHNIKOV, S. V. 2005 Generalized wall functions and their application for simulation of turbulent flows, International Journal for Numerical Methods in Fluids, 47, 1323–1328. WATSON, R. D., HALL, R. M., AND ANDERS, J. B. 2000 Review of skin friction measurements including recent high-Reynolds number results from NASA Langley NTF, Proceedings, AIAA Paper 2392, June 19–22, Denver, CO. WHITE, F. M. 2008 Fluid Mechanics, 6th ed., McGraw Hill, 362–363. WILCOX, D. C. 1989 Wall matching, A rational alternative to wall-functions, Proceedings, AIAA Paper 89-611, Reno, NV. WILCOX, D. C. 1993 Turbulence Modeling for CFD, DCW Industries, La Can˜ada, CA. WILSON, R. V., CARRICA, P. M., AND STERN, F. 2006 URANS simulations for a high-speed transom stern ship with breaking waves, International Journal of Computational Fluid Dynamics, 20, 2, 105–125. XING, T., AND STERN, F. 2008 Factors of safety for Richardson extrapolation for industrial applications, IIHR Technical Report No. 466. XING, T., CARRICA, P., AND STERN, F. 2008 Computational towing tank procedures for single run curves of resistance and propulsion, Journal of Fluids Engineering, 130, 2, 1–14.

JOURNAL OF SHIP RESEARCH