Journal of Ship Research, Vol. 53, No. 4, December …Journal of Ship Research, Vol. 53, No. 4,...

Journal of Ship Research Vol 53 No 4 December 2009 pp 179ndash198

Journal ofShip 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 com-puting model- and full-scale ship flows with wall roughness and pressure gradienteffects The wall-function model is validated for smooth flat-plate flows at Reynoldsnumbers up to 109 and it is applied to the Athena RV for resistance propulsion andseakeeping calculations and to fully appended DTMB 5415 for a maneuvering simula-tion Resistance predictions for Athena bare hull with skeg at the model scale comparewell with the near-wall turbulence model results and experimental fluid dynamics (EFD)data For full-scale simulations frictional resistance coefficient predictions usingsmooth wall are in good agreement with the International Towing Tank Conference(ITTC) line Rough-wall simulations show higher frictional and total resistance coeffi-cients where the former is found to be in good agreement with the ITTC correlationallowance Self-propelled simulations for the fully appended Athena performed at fullscale using rough-wall conditions compare well with full-scale data extrapolated frommodel-scale measurements using the ITTC ship-model correlation line including acorrelation allowance Full-scale computations are performed for the towed fullyappended Athena free to sink and trim and the boundary layer and wake profiles arecompared with full-scale EFD data Rough-wall results are found to be in better agree-ment with the EFD data than the smooth-wall results Seakeeping calculations areperformed for the demonstration purpose at both model- and full-scale Maneuveringcalculation shows slightly more efficient rudder action lower heading angle overshootsand 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 inthe development and validation of computational fluid dynamics(CFD) for ship hydrodynamics which has the potential of assist-ing in ship design (Larsson 1997 Gorski 2002 Stern et al 2006a)As documented in the previous symposiums on naval hydrody-namics most of the CFD and EFD studies have focused onmodel-scale ship flows that is Reynolds number Re 107 in-cluding resistance propulsion seakeeping and maneuvering (egCrook 1981 Miller et al 2006 Carrica et al 2006a Longo et al2007) In contrast full-scale ship flow computations (Re 109)

are limited (Gorski et al 2004 Hanninen ampMikkola 2006) This isthe result of the challenging issues with regard to both numericalmethods and turbulence modeling The overall objective of thisstudy is to extend the unsteady Reynolds averaged Navier-Stokes(URANS) solver CFDShip-Iowa (Carrica et al 2007a 2007b) tosimulate full-scale ship flows using wall functions for ship hydro-dynamics applications including resistance propulsion seakeep-ing and maneuvering Also of interest is the application ofwall-functions at model scale to reduce the number of grids pointsnear the wall

Numerical issues for full-scale simulations include severalaspects Near-wall turbulence models require high grid density nearthe wall usually unaffordable for ship geometries The extremelysmall grid spacing in the wall normal direction near the wall causeshuge aspect ratios of grid cells This significantly increases the

Manuscript received at SNAME headquarters January 28 2008 revised

manuscript received October 6 2008

DECEMBER 2009 0022-4502095304-0179$00000 JOURNAL OF SHIP RESEARCH 179

errors in mass and momentum flux calculations and may lead tosolution divergence Implementation of lower-order numericalschemes such as first order will likely overcome this problem butwith the penalty of significant loss of accuracy Development andvalidation of near-wall turbulence models for the Re of full-scaleship flows are lacking because EFD data are sparse even for canon-ical flows and difficulties of maintaining fixed environmental con-ditions to conduct practical full-scale ship measurements (Patel1998 Aupoix 2007)

The use of ldquowall functionsrdquo avoids the numerical limitationsof the near-wall turbulence models and significantly reduces thecomputational cost Wall functions are based on the near-wall be-havior of nonseparating two-dimensional turbulent boundary layersalso valid along the streamwise direction of three-dimensional(3D) turbulent boundary layers with small cross flow (Bradshaw ampHuang 1995) Thus the obvious limitations of the wall functions arein accurately predicting separated flows and 3D boundary layerswith significant cross flow For example in a backward facing stepflow the reattachment length is often underpredicted in wall-func-tion simulations Nevertheless near-wall turbulence models alsosuffer from the same deficiency as the model constants are derivedunder similar turbulent boundary layer assumptions (So amp Lai1988) Some of the issues addressed in the literature although notfully resolved are

bull Sensitivity to the spacing of the first grid point away fromthe wall (matching point herein)

bull Inclusion of pressure gradient effects for better prediction ofseparated flows

bull Modeling of wall roughnessbull Different implementation approaches for determination ofthe velocity direction at the matching point

The standard wall-function approach (Launder amp Spalding1974) is based on the stringent criteria that the matching pointlies in the log layer (one layer only) However different flowregimes make it difficult to place the matching point in the loglayer for example laminar-turbulent transition zone variation ofthe boundary layer thickness along the ship hull or when the shipis slowly accelerated from a static condition This limitation isaddressed by a two-layer model in which boundary conditionsfor the velocity and turbulent quantities are switched betweenthe sublayer and log-layer analytic profiles in the near-wall re-gion depending on the local y+ value of the matching point(Grotjans amp Menter 1998 Esch amp Menter 2003) A three-layermodel includes an additional formulation for the buffer layer(Temmerman et al 2003) but still suffers from the deficiencythat the velocity profile is not smooth across the near-wall regionThis deficiency was resolved by Shih et al (2003) who proposeda generalized multilayer model using curve fitting to provide acontinuous function to bridge the sublayers and log layers Kalit-zin et al (2005) developed a multilayer model along with look-uptables for evaluating the friction velocity The look-up tables areobtained from a separate zero-pressure gradient smooth flat-platesimulation using a near-wall turbulence model This approachleads to an accurate calculation of the friction velocity than theanalytic equations used in above models However the applica-bility of this approach for high Reynolds number flows wouldrequire examination of the look-up table The models discussedpreviously are based on Dirichlet-type boundary conditionsUtyuzhnikov (2005) proposed a differential form for the boundary

condition valid for the entire boundary layer Overall the multi-layer wall-function model provides the most flexibility in theplacement of the matching point

Evaluation of the pressure gradient effect on wall functions forseparated flows with mild pressure gradients was conducted bysome previous studies For a backward-facing step flow the two-layer models with pressure gradient effect predict the separationrecirculation and reattachment regions better than that withoutpressure gradient effect (Wilcox 1993 Kim amp Choudhury 1995)Knopp et al (2006) extended the Kalitzin et al (2005) approachfor nonequilibrium flows by including pressure gradient effectsTheir main conclusion is that in the regions of stagnationand strong pressure gradients a near-wall solution is the beststrategy for which they applied flow-based grid adaptation Inthe strong pressure gradient regions usually encountered in shipflows clipping of the pressure gradient magnitude to 75of the friction velocity is suggested to avoid numerical instabilityor divergence (Wilcox 1993) Thus the benefit of including pres-sure gradient effect for wall functions in separated flows isquestionable

The effect of surface roughness is more important for full-scalecomputations than for those at the model scale as it leads tosignificant increase in frictional and total resistances The mostcommonly used model for surface roughness is based on thedownshift of the log-layer profile (White 2008) Aupoix (2007)recently provided a formulation of the downshift of log-law that isin better agreement with EFD data which could be of futureinterest Several studies have investigated the effect of roughnesstype on the boundary layer profile (Jimenez 2004) Schultz (2002)performed model-scale experiments on several sanded andunsanded painted surfaces encountered in ship flows The studieshave shown that the roughness type does not show significanteffect in the transitional roughness regime (5 k+ 70) or thedownshift of log-layer profile Thus surface roughness modelingbased on the downward shift of the log-layer profile can be usedwith relative confidence for full-scale ship calculations whereroughness length mostly lies in the transitional regime (Patel1998 Tahara et al 2002) For one- and two-layer wall functionmodels modeling of roughness effect is straightforward How-ever for the multilayer model appropriate correlation is not avail-able for the buffer layer from either experimental or numericalstudies

Implementation of wall-function models requires evaluation ofthe friction velocity either analytically or using look-up tables(Kalitzin et al 2005) to provide boundary conditions for velocityand turbulence variables The one-point approach proposed byKim amp Choudhary (1995) uses the flow variables at the wallneighboring cells only This allows solutions of the momentumequations up to the matching point The one-point approach can beimplemented easily for finite-volume schemes but introduces ad-ditional complexities and challenges for finite-difference schemesAn alternative two-point approach was introduced by Chen ampPatel (1988) and extended by Tahara et al (2002) for ship flowsIt uses the tangential velocity magnitude and direction at thesecond grid point away from the wall to obtain velocity at thematching point Implementation of the two-point approach forfinite-difference schemes is straightforward The one-point ap-proach has advantages over the two-point approach as it does notrestrict the flow direction at the matching point to follow the flowdirection at the second point away from the wall

180 DECEMBER 2009 JOURNAL OF SHIP RESEARCH

Applicability of wall-functions for ship flows has beendemonstrated at model-scale by comparing predictions withnear-wall results and EFD data Park et al (2004) performedsimulations around the Korean Research Institute for Ships andOcean Engineering container ship (KCS) using the standardwall-function and the results were compared with EFD dataThey reported good agreement of the wave elevation patternand resistance coefficient predictions Tzabiras (1991) comparedthe standard wall-function and near-wall results for two shipmodels SSPA and HSVA The wall-function results comparedwell with the near-wall results for the velocity profile howeverdifferent results were obtained for skin friction coefficientsSeveral studies have used the wall-function approach in shipflows to investigate the Reynolds number effects A study con-ducted for a HSVA ship model (Oh amp Kang 1992 Re=109 and5 +10

6) using the standard wall function showed that the full-scale Re results in much reduced skin friction but has no effecton pressure coefficients in the thin boundary layer region In thethick boundary layer region around the stern and near the wakethe pressure coefficients for full scale are noticeably changed bythe reduction of viscous-inviscid interaction and have the trendof approaching the values of the inviscid region Similar conclu-sions were also drawn by Tahara et al (2002) using a two-layerwall-function model for the full-scale simulations of the Series60 ship model They also demonstrated the capability of thewall-function model in depicting surface roughness effect TheURANS study for a low-speed TDW VLCC (Choi et al 2003)using the standard wall-function showed that full-scale hasweaker strengths of bilge vortices which causes a smaller vor-tex and turbulence region and a smaller value of nominal wakefraction on the propeller plane However there is little scaleeffect on the limiting streamline and hull pressure except on thestern region Bull et al (2002) studied full-scale effects using atwo-layer model on two hull forms the Dutch frigate the ldquoDe-Ruyterrdquo and the NATO research vessel ldquoAlliancerdquo The resultscompare well with EFD data for the bare hull geometry andcapture the main flow features for the appended ships None ofthe aforementioned studies on ship flows has emphasized theneed for pressure gradient effect on wall-functions probablybecause of the issues already discussed

To achieve the overall objective two-layer (TL) (Esch ampMenter 2003) and multilayer (ML) (following Shih et al 2003)wall-function models with the ability to account for the wallroughness and pressure gradient effects are developed and imple-mented in CFDShip-Iowa The downshift of buffer layer is tenta-tively assumed to be the same as Whitersquos (2008) downshift ofthe log layer As CFDShip-Iowa is based on finite-differenceschemes wall-function models are implemented using the two-point approach (Tahara et al 2002) In the following sectioncomputational methods and the wall-function implementationapproach are discussed In section 3 the wall-function modelsare first validated for smooth flat-plate flows at high Re equiva-lent to full-scale ship flows In section 4 the experimental andsimulation conditions used for Athena RV and 5415 applicationsare summarized Resistance computations for Athena bare hullwith skeg at model- and full-scale with and without roughnessand pressure gradient effects using both TL and ML models arepresented in section 5 Self-propelled simulations and boundarylayer and wake profile comparison with EFD data for fullyappended Athena at full scale using both TL and ML models

with smooth- and rough-wall conditions are presented in sections5 and 6 respectively Seakeeping calculations at model- and full-scale for Athena and maneuvering simulations for full-scaleDTMB 5415 (5415) are performed using smooth-wall TL andML models for demonstration purposes in sections 8 and 9respectively Finally in section 10 conclusions and future worksare discussed

2 Computational method

The solver CFDShip-Iowa solves the URANS equations in theliquid phase of a free-surface flow The free surface is capturedusing a single-phase level set method

21 Equations of motion

The governing equations of motion are solved in either absoluteinertial earth-fixed or relative inertial coordinates for an arbitrarymoving but nondeforming control volume Xing et al (2008)showed that the solution of the equations in the absolute inertialearth-fixed (or relative inertial) coordinates has several advantagesover the noninertial ship fixed coordinate system such as simplic-ity in the specification of boundary conditions savings of compu-tational cost by reducing the solution domain size and allowingstraightforward implementation of ship motions The governingequations for the water phase in dimensionless form are

rsaquoUi

rsaquoxifrac14 0 eth1THORN

rsaquoUi

rsaquotthorn ethUj UGjTHORN rsaquoUi

rsaquoxjfrac14 rsaquop

rsaquoxjthorn 1

Re

rsaquo2Ui

rsaquoxjrsaquoxj rsaquo

rsaquoxjuiuj eth2THORN

where Ui = (UVW) are the Reynolds-averaged velocity compo-nents UGj is the local grid velocity in either the absolute inertialearth-fixed or relative inertial Cartesian coordinates xi = (xyz)p = pabsrU

20 + zFr2 + 2k3 is the dimensionless piezometric

pressure where pabs is the absolute pressure uiuj are the Reynoldsstresses Fr = U0

ffiffiffiffiffiffigL

pis the Froude number and k is the turbulent

kinetic energy (TKE) U0 is the free stream velocity L is the shiplength and Re is the Reynolds number based on L

22 Turbulence modeling

221 Blended k-vk-laquo and DES model Two-equation closureis used for the Reynolds stresses modeled as a linear function ofthe mean rate-of-strain tensor through an isotropic turbulent eddyviscosity (nt)

uiuj frac14 ntrsaquoUi

rsaquoxjthorn rsaquoUj

rsaquoxi

2

3dijk eth3THORN

where dij is the Kronecker delta The unknown turbulent eddyviscosity is evaluated from the TKE and the specific dissipationrate (v) Additional transport equations presented below are solvedfollowing Menterrsquos (1994) blended k-vk-e (BKW) approach

rsaquok

rsaquotthorn v sk rnteth THORN rk 1

Pkr2k thorn sk frac14 0 eth4aTHORN

rsaquov

rsaquotthorn v sv rnteth THORN rv 1

Pvr2vthorn sv frac14 0 eth4bTHORN

DECEMBER 2009 JOURNAL OF SHIP RESEARCH 181

The turbulent eddy viscosity and the effective Peclet numbers are

nt frac14 k=v Pk=v frac14 1

1=Rethorn sk=vnteth5THORN

and the source terms in k and v equations constitute the produc-tion and dissipation terms (refer to Carrica et al 2006b fordetails) The model constants say a are calculated from thestandard k-v (a1) and k-e (a2) values using a blending functionF1 (see Menter 1994 for the model constants values)

a frac14 F1a1 thorn 1 F1eth THORNa2 eth6THORNF1 is designed to be unity in the near-wall regions of boundarylayers and gradually switches to zero in the wake region to takeadvantage of the strengths of the k-v and k-e models respectively

222 Wall-function (WF) models In the TL model the velo-cities in the sublayer and log-layer regions are

U23

utfrac14

y thorn23 ythorn231167 sublayer

k1 lnethythorn23THORNthornBDB 113ythorn23Pthorn ythorn231167 log-layer

(

eth7THORNwhere subscripts 2 and 3 represent the first and second grid pointaway from the wall respectively The superscript + quantities are

nondimensionalized by the friction velocity (ut =ffiffiffiffiffiffiffiffiffiffitw=r

p) where

tw is the wall shear stress and Re The von Karman constant (k)and B are chosen to be 04 and 51 respectively (Knobloch ampFernholz 2002) The factor DB in equation (7) accounts for theeffect of wall roughness resulting in the downshift of the log-layer region (White 2008)

DB frac14 k1 lneth1thorn kthornTHORN 35 eth8aTHORNwhere k+ is the roughness parameter based on nondimensionalroughness length ks For naval applications the wall roughnesslies in the transitional roughness regime (Patel 1998) that is 5k+ 70

Pthorn frac14 nu 3t

rsaquop

rsaquox

eth8bTHORN

is a dimensionless parameter that is used to include the effect ofpressure gradient (PG) tangential to the wall in the WFs (Wilcox1989) The effect of PG is clipped such that ythorn2 P

thorn 34 asproposed by Wilcox (1989)

The TKE and v in the sublayer and log-layer regimes aredefined based on the analytic solution (Wilcox 1993)

k2 frac14 kjfrac143Dy2Dy3

323

ythorn2 1167

u2t03 1thorn 116ythorn2 P

thorn ythorn2 1167

8lt eth9THORN

v2 frac146n

0075Dy22 ythorn2 1167

ut03kDy2

1 03ythorn2 Pthorn

ythorn2 1167

(eth10THORN

where Dy is the distance normal to the wallA ML model is developed by using curve fitting (fourth-order

polynomial as adopted by Shih et al 2003) that blends the sub-

layer and log-layer velocity profiles in the TL model and providesthe buffer-layer profile

U23

utfrac14

y23thorn y23

thorn 5

a0 thorn a1ythorn

23 thorn a2ethy thorn23 THORN2thorn a3ethy23thornTHORN3thorn a4ethy23thornTHORN4 DB

5 y23thorn 30

1k lnethy23thornTHORN thorn B DB y23

thorn 30

8gtgtgtgtltgtgtgtgt

eth11THORNThe five unknown model coefficients in the buffer-layer region

are determined by satisfying continuity of the velocity and its firstderivative across the sublayer and buffer-layer and buffer-layerand log-layer intersections An additional equation is obtained byallowing the curve to pass through the analytic buffer-layer curveu+ = 5 ln (y+) 305 at y+ = 20 This yields the model constants

a0 frac14 1875736 a1 frac14 18158144 a2 frac14 0102066044

a3 frac14 000295224178 a4 frac14 33144178e 005

eth12THORNIn the absence of any numerical model or experimental data

showing the effect of wall roughness on the buffer layer down-shift of buffer layer is assumed to be same as that of the log layerin this paper as shown in equation (11)

As there is no analytic solution available for turbulence quan-tities in the buffer-layer regions approximated functions are usedThe TKE in the buffer-layer region is obtained by using Kalitzinrsquoset al (2005) approximation and for v the blending function pro-posed by Esch amp Menter (2003) is used

k2 frac14

kjfrac143

Dy2Dy3

323

ythorn2 5

dythorn

duthorn 1

v thorn2 5 ythorn2 30

u2t03 ythorn2 30

8gtgtgtgtgtgtgtltgtgtgtgtgtgtgt

eth13THORN

v2 frac14

6n0075Dy2

2

ythorn2 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6n

0075Dy22

2

thorn ut03kDy2

2s

5 ythorn2 30

ut03kDy2

ythorn2 30

8gtgtgtgtgtgtltgtgtgtgtgtgt

eth14THORN

223 Wall-function implementation The WFs are implemen-ted using the two-point approach by Tahara et al (2002) The TLmodel is implemented using the following steps (also refer to Fig 1)

1 The friction velocity is computed from the relative tangentialvelocity of the second point ( j = 3) away from the wall ( j = 1)either from the sublayer or log-layer equation (7) For thesublayer ut =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU3=ethReDy3THORN

p For the log layer the friction

velocity is obtained by iteratively solving the log-layer equa-tion using the Newton-Raphson method

2 The computed friction velocity is then used in equation (7) toobtain the magnitude of the tangential velocity at j = 2 withthe same direction of the tangential velocity at j = 3 Thenormal velocity at j = 2 is approximated using the linearinterpolation based on the normal velocity at j = 3 and wall


distances at j = 2 (Dy2) and j = 3 (Dy3) Coordinate transfor-mation is needed to give velocity boundary conditions in thephysical coordinates

3 The boundary conditions for the turbulence quantities k v arespecified at j = 2 using equations (9) and (10)

For the ML model a similar approach is used where the frictionvelocity computation is obtained from sublayer buffer-layer or log-layer regions as shown in equation (11) In the buffer-layer regionthe frictional velocity is obtained using the Newton-Raphson itera-tion For the turbulent quantities boundary conditions equations (13)and (14) are used

As the buffer layer is neglected in the TL model the imple-mentation of the two-point approach would result in the followingpossibilities (ythorn2 sublayer y

thorn3 sublayer y

thorn2 sublayer y

thorn3 log-

layer or ythorn2 log-layer ythorn3 log-layer) If either y

thorn2 or ythorn3 is placed

in the buffer layer (5 y+ 30) modeling errors will be largeThis limitation is overcome by the multilayer wall function asillustrated using smooth flat-plate simulations (discussed later)

In the current study the Re 107 to 109 for which the sublayeris thin and log-layer extent is large Thus both j = 2 and j = 3 canbe conveniently placed in the log-layer region for exampleTahara et al (2002) used ythorn2 to be 103 in Series 60 ship modelcalculations To resolve the boundary layer as best as we can andstill save sufficient grid points near the wall ythorn2 30 is appliedfor most simulations in the current study However in the low-pressure regions of the ship such as the stern the local ythorn2 islower than rest of the hull and lies in the buffer-layer region TheML model is thus expected to perform better than the TL modelfor the overall calculations

23 Free-surface modeling

The location of the free surface is given by the zero value ofthe level set function (f) positive in water and negative in air

Since the free surface is a material surface the level set functionfollows a simple advection equation For stability purposes asmall artificial diffusion term is added to the equation Negligibleshear stress in the air phase is assumed that provides the jumpcondition at the free surface As a good approximation for airwater interfaces the pressure in the air is assumed equal to theatmospheric pressure The velocity and turbulent quantities k andv are extended from the airwater interface to air by solving anequation similar to f over the whole air domain Readers arereferred to Carrica et al (2007a) for details

24 Propeller model

A simplified body force model for the propeller is used toprescribe axisymmetric body force with axial and tangential com-ponents (Stern et al 1988) The propeller model requires thrusttorque and advance coefficients as input and provides the torqueand thrust forces These forces appear as a body force term in themomentum equation for the fluid inside the propeller diskThe location of the propeller is defined in the static condition ofthe ship and moves according to the ship motion

25 Six degrees of freedom (6DOF) module

The total force and moment vectors are computed in the abso-lute inertial earth-fixed coordinates from the nonstatic pressurehydrostatic pressure (buoyancy) and frictional forces acting onthe ship surfaces and propeller thrust and torque In this study itis assumed that the center of rotation is coincident with the centerof gravity The forces and moments are then projected into thenoninertial ship-fixed coordinates These forces and moments areused to evaluate the surge sway and heave velocities and rollpitch and yaw angular velocities assuming rigid body motionThe equations are solved using a predictorcorrector implicitsolver The velocities are then transformed back to the absolute

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


inertial earth-fixed coordinates to predict the evolution of thelocation and attitude of the ship The time marching is done usingthe Euler second-order backward differences The readers arereferred to Carrica et al (2006a) and Xing et al (2008) for moredetails of the implementation

26 Numerical methods and high-performance computing

The governing equations are discretized using finite differenceschemes with body-fitted curvilinear grids The convection termsare discretized using second-order upwind scheme and the diffu-sion terms are discretized using second-order central differencescheme A pressure-implicit and splitting of operators (PISO) (Issa1985) algorithm is used to obtain a pressure equation and satisfycontinuity The pressure Poisson equation is solved using the PETSctoolkit (Balay et al 2002) The software SUGGAR (Noack 2005)runs as a separate process from the flow solver to get the interpola-tion coefficients required for the dynamic overset grids MessagePassing Interface (MPI) based domain decomposition is usedwhere each decomposed block is mapped to a single processor Formore technical details readers are referred to Carrica et al (2007a)

3 Smooth flat plate simulations

The TL and ML models are first validated for smooth flat-platesimulations for a range of Re (106 to 109) Results are compared withBKW solutions analytic log-layer profiles and EFD data at large Re(Watson et al 2000) The grids used for BKW and WF simulationsconsist of 201 +91 +5 and 201 +71 +5 grid points in the stream-wise wall-normal and spanwise directions respectively

At first simulations are performed to study the effect of ythorn2 andythorn3 on the prediction of the boundary layer profiles at Re =8 +10

5 As shown in Fig 2 both TL and ML results are in closeagreement with BKW results when both ythorn2 and ythorn3 lie in the log-layer or the sublayer region (not shown in figure) However theML results are better than those of TL especially for the turbulenteddy viscosity predictions when ythorn2 lies in the buffer-layer regionFlat-plate boundary layer profiles obtained from the WF simula-tions for a range of Re (106 to 109) using ythorn2 30 were comparedwith the analytic log-layer profiles (Bhushan et al 2007) Thecomparison shows good agreement for u+ k+ v+ and n+ profilesThe variation of the skin friction coefficient along the plate forRe = 109 is shown in Fig 3 The results agree well with theKarman-Schoenherr equation which is suitable for high Reflows (Watson et al 2000) and the EFD data for ythorn2 = 30 to 600Results are found to be slightly dependent on the ythorn2 for lowerRex that is toward the leading edge of the plate The aboveresults demonstrate that the WF is capable of simulating high Reflows equivalent to full-scale ship Re

4 Application to Athena and 5415

The Athena geometry used herein for resistance propulsionand seakeeping calculations was previously used at the 2005ONR wave-breaking workshop to assess the capability of CFDmethods in predicting bow and transom wave breaking The fully

appended surface combatant 5415 was used in the SIMMAN(2008) workshop for verification and validation of ship maneu-vering simulations methods This study extends previous model-scale BKW simulations using CFDShip-Iowa for Athena (Wilsonet al 2006 Miller et al 2006 Xing et al 2008) and 5415 (Carricaet al 2008) to full scale

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)

Fig 3 Variation of the skin friction coefficient along the flat plate com-

pared with experimental data of Watson et al (2000) and Karman-

Schoenherr equationKarman-Schoenherr equation cf = (0558cd)(0558 +

ffiffiffiffifficd

p) where 0242ffiffiffiffiffi

cdp

= log10(cdRex) = log10(2Reu)


41 Experimental data and simulation conditions

The EFD data for Athena bare hull with skeg (BH) are obtainedfrom the towing tank experiments conducted on a 1825-scale shipmodel (Jenkins 1984) Measurements of the total resistance waveresistance bow and stern sinkage and wave elevation along the hullat several stations were made for Frs ranging from 028 to 100Miller et al (2006) performed resistance calculations for fixed sink-age and trim BH at model scale using BKW for six Frs Resistancesinkage and trim calculations performed by Xing et al (2008) atmodel scale using BKW included a single run full Fr curve and twosteady-state computations for Fr = 048 and 08 Simulations per-formed here for BH using WF are (1) resistance calculations forfixed sinkage and trim at model scale (Fr = 028 048 and 08) andfull scale (Re = 2 +10

9 Fr = 048) using the smooth-wall TLmodel(2) verification study at full scale (Re = 2 +10

9 Fr = 048) for theresistance coefficients sinkage and trim using the smooth-wall TLmodel (3) resistance sinkage and trim predictions at model- andfull-scale (Fr = 048 and 08) using both ML and TL models assum-ing smooth- and rough-wall conditions Effect of PG is evaluatedfor full-scale simulation using the smooth-wall TL model and (4)sensitivity study of the ML and TL model resistance sinkage andtrim predictions on ythorn2 (ythorn2 = 30 100 and 300) at full-scale forsmooth- and rough-wall (k+ = 28) conditions

The EFD data for the self-propelled fully appended Athena(AH) (Crook 1981) include resistance and powering characteris-tics sinkage and trim at 17 different Frs ranging from 0336 to0839 open water curves for thrust and torque coefficients for thepropeller model 4710 with the corresponding propeller revolu-tion per seconds (RPS) thrust deduction and wake factors TheEFD data were presented at full-scale extrapolated using theITTC ship-model correlation line including a correlation allow-ance CA = 65 +10

4 Xing et al (2008) performed self-pro-pelled simulations for AH free to sink and trim at model scaleusing BKW Their simulations included full Fr powering curvesand steady-state computations at Fr = 0432 0575 and 0839Herein ship speeds and consequently Frs are predicted at full scalefor AH free to sink and trim using operational propeller conditionsat Fr = 0432 0575 and 0839 in the EFD The effect of wallroughness is demonstrated for the Fr = 0432 and 0575 cases usingroughness length of 100 micron with both TL and ML models

The full-scale boundary layer and wake profiles were measuredon AH by Day et al (1980) The ship was self-propelled by theoperating propeller on the port side The course of the ship wasmaintained with a small port side rudder angle between 2 and 5On the starboard side the rudder was set at 0 and the propellerblades were removed Measurements on the starboard side in-cluded (1) boundary layer profiles at four rake locations and (2)wake profiles including axial tangential and radial velocities onthe forward propeller plane at three radial locations 0417 0583and 075R where R is the radius of the propeller and on thepropeller plane at four radial locations 0456 0633 0781 and0963R The measurement locations are shown in Fig 4 The trimangle was reported to be less than 3 whereas the sinkage was notdocumented The wall roughness length was reported to be be-tween 50 and 130 mm To approximate the EFD conditions on thestarboard side the ship was towed at the speed corresponding toRe = 42 +10

8 and Fr = 0362 for which the most comprehensive

EFD data are available The simulations are performed using bothsmooth- and rough-wall conditions Three roughness lengths areconsidered 50 (k+ = 15) 100 (k+ = 32) and 130 (k+ = 43) mmcovering the reported roughness range

Seakeeping EFD data are not available for Athena thus simu-lations are performed for demonstration purposes using thesmooth-wall TL model for Fr = 03 An incoming regular headwave is specified with nondimensional wave length 12 and am-plitude 001 The encounter period of the wave is 0488LU0BKW and WF calculations are performed for towed BH at modelscale (Re = 372 +10

7) Full-scale (Re = 2 +109) calculations are

performed using TL model for towed BH and AHFor the SIMMAN (2008) workshop EFD data were acquired for

captive and free-sailing (6DOF) model tests for tanker containership and surface combatants hull forms The free-sailing modelEFD data were not provided before the workshop thus the compar-isons were blind Carrica et al (2008) performed steady turn andzigzag maneuvers for self-propelled fully appended 5415 Steadyturn computations were performed at model scale using BKW for20 and 35 rudder deflections for Fr = 025 and 041 in calm andregular waves and for constant torque and constant RPS 2020zigzag maneuvers were performed using BKW for constant RPSfor Fr = 025 and 041 at model scale (Re = 104 +10

7) An addi-tional simulation was performed for the Fr = 041 case at full scale(Re = 22 +10

9) using the smooth-wall ML model The zigzag

Fig 4 Towed fully appended Athena EFD measurement locations (Day

et al 1980) a Wake profile measurement planes (AndashB propeller plane

CndashD forward propeller plane) b Boundary layer measurement locations

CA frac14 eth105k1=3s 064THORN middot 103


maneuver is controlled by the rudder angle which changes from20 starboard to 20 port The rudder angle is changed at the rate of9 degs (in full scale) to the other side when the heading angle is20 which is referred to as rudder check point Here results of theFr = 041 model- and full-scale 2020 zigzag maneuvers are pre-sented for demonstration purposes

Table 1 summarizes the geometry ythorn2 simulation conditionsand comparisonvalidation data used for various WF simulationsperformed in this paper

42 Domains grids and boundary conditions

Figures 5 and 6 show the grid topology used for the Athena and5415 geometries The simulations performed using Athena geo-metries are for half domain only taking advantage of the symme-try of the problem about the center plane y = 0 The AH consistsof a skeg rudder stabilizer propeller shaft and struts The fullyappended 5415 consists of skeg split bilge keels stabilizers shaftsand struts rudder seats and rudders Body-fitted ldquodouble-Ordquo typegrids are generated for the BH Overset body-fitted ldquoOrdquo typegrids are generated for the ship appendages except the rudderstabilizer and shaft caps which are open topology A Cartesianbackground grid with clustered grid near the free surface to re-solve the wave flow pattern is used for specifying boundary con-

ditions away from the ship hull Model-scale BH simulations areperformed using a grid consisting of 360K points A grid consistingof 17 million (M) points is found to be appropriate for the full-scaleWF simulations from the verification study (performed later in thepaper) In the model-scale simulations ythorn2 30 is used whereas forfull-scale calculations ythorn2 = 30 to 300 The AH grid consists of22M points split into 24 blocks (Xing et al 2008) For the full-scale WF simulations the grid design is such that ythorn2 = 30 to 60The grid used for the 5415 simulation consists of 7M grid pointsdecomposed into 72 blocks (Carrica et al 2008) The ythorn2 valuesfor the WF simulations are shown in Table 1

Towed and seakeeping simulations are performed using relative

inertial coordinates whereas self-propelled simulations are in abso-

lute inertial earth-fixed coordinates The ship-hull skeg and ap-

pendages 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 2In 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 ythorn2

Wall-Function Model

Wall Condition Results and Compare

Fixed sinkage

and trim

13 +107 028 30ndash40 Resistance coefficients BKW results

223 +107 048 30 TL smooth-wall (Xing et al 2008) and EFD

372 +107 08 30ndash40 (Jenkins 1984) data

2 +108 048 3267

TL smooth-wall Resistance ITTC line8 +108 048 11835

2 +109 048 27715

Towed Athena

bare hull with 223 +107

048

30ndash40

ML and TL

smooth-wall

Resistance coefficients sinkage and trim

BKW results (Xing et al 2008) and EFD

skeg Predict sinkage 08 TL smooth-wall (Jenkins 1984) data at model-scale

and trim

2 +109

048

08

30 100 300

6- grids

30

ML and TL smooth

and rough-wall

TL smooth-wall

TL smooth-wall

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

Pitch and heave

in waves

372 +107

2 +109 03

30ndash40

30ndash40TL smooth-wall BKW solution at model-scale

Self-propelled fully

appended Athena

Predict sinkage

and trim

366 +108

488 +108

0432

0575

65

85

TL smooth and

rough-wall

ML and TL smooth

and rough-wall

Resistance power and motion BKW

results (Xing et al 2008) and EFD

(Crook 1981) data at model scale

extrapolated to full-scale

715 +108 0839 120 TL smooth-wall Compare TL and ML results and

wall roughness effect

Towed fully

appended Athena

Predict sinkage

and trim

42 +108 0362 50 TL smooth and

rough-wall

Boundary layer and wake survey full-scale

EFD (Day et al 1980)

Pitch and heave

in waves

2 +109 03 30ndash40 TL smooth-wall Bare hull with skeg in waves


appended 5415

6DOF 22 +109 041 100 ML smooth-wall BKW solution at model scale


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 con-

stant RPS obtained by the controller in the model- and full-scale

calculations is 1950 and 1878 in full-scale dimensions respec-

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 towedAthena with skeg

51 Fixed sinkage and trim

The model- and full-scale results obtained using smooth-wallTL models are presented in Fig 7 The total resistance coefficients(Ctotx) at model scale are within 5 of the EFD data for Frs = 028and 048 but is overestimated by as much as 675 for Fr = 08Model- and full-scale (Re = 107 to 109) frictional resistance coeffi-cients (Cfx) follows the ITTC

line However at model-scale WFpredictions are larger than BKW results by 5 to 7 Figure 8 ashows that the full-scale boundary layer is thinner than that inmodel scale The free-surface elevation pattern is not significantlyaffected by Re as seen in Fig 8 b A closer inspection of thetransom wave elevation shows that the wave slope at full scale isslightly lower than at model scale (Fig 8 c) which is consistentwith the findings of Starke et al (2007)

52 Verification study for resistance sinkage and trim

Solution verification study is performed following the quantita-tive methodology and procedures proposed by Stern et al (2006b)(ST) and recent modifications proposed by Xing amp Stern (2008)(XST) for situations when estimated order of accuracy (PG) islarger than the theoretical order of accuracy (PGth) and correctionfactor 1CG 2 CG 2 solutions are too far from the asymp-totic range and also regarded as divergent in the XST method Forthe verification study six grids are designed with a systematicrefinement ratio of rG = 214 summarized in Table 3 Six sets ofverification studies are possible four with rG = 214 (123 234345 and 456) and two with rG = 212(135 and 246) Resultshave been plotted in Figs 9 a and c and the converged results areshown in Table 4 Convergence condition is defined by RG whichis the ratio of solution changes for medium-fine and coarse-me-dium solutions Correction factor 1 CG indicates how far thesolution is from asymptotic range where CG =1 Figures 9 b and dshow the relative change of the solution (eN) between two succes-sive grids along with iterative errors (UI) obtained using fivenonlinear iterations on each grid for resistance coefficients andmotions respectively UI are of the same order of magnitude forall the grids which suggests that they are determined primarily bythe iterative method and independent of grid resolution Theresults are discussed in the light of recent study by Xing et al(2008) (XET) for BH at model scale (Re = 223 +10

7 Fr = 048)using BKW where an additional grid consisting of 81M gridpoints was considered

Ctotx monotonically converges on grids (246) and (234)oscillatorially converges on grids (135) oscillatorially divergeson grids (123) and (456) and monotonically diverges on grids(345) The pressure resistance coefficient (Cpx) monotonicallyconverges on grids (246) and (345) oscillatorially converges

Fig 6 Grid topology for a fully appended Athena and b fully appended

5415

Fig 5 Boundary conditions for the simulation domain represented for

Athena bare hull with skeg grid topology

ITTC Cf = 0075(log Re 2)2


on grids (135) and (234) oscillatorially diverges for (456)and monotonically diverges on grids (123) Cfx monotonicallyconverges on grids (123) oscillatory converges on grids(135) (246) and (234) oscillatorially diverges on grids(456) and monotonically diverges on (345) Sinkage mono-tonically converges on grids (234) oscillatory converges ongrids (123) (345) and (456) oscillatorially diverges on grids(135) and monotonically diverges on grids (246) Trimmonotonically converges on grids (123) oscillatorially con-verges on grids (345) and (456) oscillatorially diverges ongrids (246) and (135) and monotonically diverges on grids(234) It must be noted that the monotonically converged solu-tions involve the three finest grids 1 2 or 3 and most of the

diverged solutions involve coarsest grid 6 XET also obtaineddiverged solutions involving this grid which is likely caused byinsufficient grid resolution

Ctotx shows large values of 1CG 245 for grids (246) and74 for grids (234) The grid uncertainty (UG) is 155 on grids(246) and 0132 on grids (234) using the ST method UG is553 on grids (135) Grids (246) are closest to the asymptoticrange for Cpx for which 1CG is 02 1 CG for grids (345) is279 UG for Cpx is below 4 on grids (246) for both theST and XST methods 1359 on grids (345) based on theST method and 11 and 324 on grids (135) and (234)respectively 1 CG for Cfx is 081 and UG is 291 and278 on grids (123) 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

Table 2 Boundary conditions for all the variables

Boundary f p k v u n w

Towed f = z rsaquoprsaquon frac14 0 kfs frac14 107 vfs frac14 9 Uinf = 1 Vinf = 0 Winf = 0

Inlet Self-

propelled

f = z rsaquoprsaquon frac14 0 kfs frac14 107 vfs frac14 9 Uinf = 0 Vinf = 0 Winf = 0

Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)

kfs frac14 107 vfs frac14 9 Uinf = Eq (40)

(Carrica et al

2007a)

Vinf = 0 Winf = Eq (41)

(Carrica et al

2007a)

Exit rsaquofrsaquon frac14 0 rsaquop

rsaquon frac14 0 rsaquokrsaquon frac14 0 rsaquov

rsaquon frac14 0 rsaquo2Ursaquon2 frac14 0 rsaquo2V

rsaquon2 frac14 0 rsaquo2Wrsaquon2 frac14 0

Far field No 1 rsaquofrsaquon frac14 0 0 rsaquok

rsaquon frac14 0 rsaquovrsaquon frac14 0 rsaquoU

rsaquon frac14 0 rsaquoVrsaquon frac14 0 rsaquoW

rsaquon frac14 0

Far field No 2 rsaquofrsaquon frac14 0 rsaquop


rsaquon frac14 0 Uinf Vinf Winf

Symmetry rsaquofrsaquon frac14 0 rsaquop


rsaquon frac14 0 rsaquoUrsaquon frac14 0 0 rsaquoW

rsaquon frac14 0

No-slip v Carrica et al

(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)

rsaquofrsaquon frac14 0 rsaquop

rsaquon frac14 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


tively For other grids on which oscillatory convergence is ob-served UG is below 35 A large range of 279 1 CG

02 neglecting the largest value for resistance coefficientssuggests that all solutions are still far from the asymptoticrange XET also observed significant variation in142 1CG 093 however the range of variation wassmaller than the present case The variations of Cfx in the WFsimulations are within 4 for all the grids whereas in the BKWsimulations by XET significant grid dependence up to 8 isobserved eN of Cpx and Ctotx linearly increase when grid isrefined from 6 to 5 which suggests that the coarsest grid is toocoarse then linearly decrease when grid is refined from 5 to 2Further refinement to grid 1 leads to a rebound of eN which iscaused by the problem of separating UI from UG as they are ofthe same order of magnitude eN for Cpx in the WF simulationsare about 7 which is significantly larger than 2 in XETOverall UI around 2 in the present simulations are higher thanthat reported by XET around 06 Thus to achieve the asymp-totic range more nonlinear iterations andor implementation ofmore accurate and efficient iterative methods are required

Grid study (234) is closest to asymptotic range for sinkage as1 CG is 016 UG is 1548 on grids (234) about 4 on grids(123) and large values of 744 and 1164 are observed ongrids (345) and (456) respectively The solutions of trim are

far from the asymptotic range as evident from the large value of1 CG around 39 on grids (123) UG is 256 on grids (123)using the ST method and less than 6 on grids (345) and(456) Compared with resistance coefficients motions are diffi-cult to converge eN for sinkage decreases almost linearly whengrid is refined from 6 to 1 eN for trim also shows a lineardecrease with grid refinement from 6 to 3 A sudden rise in eN isobserved for grids 2 and 3 when UI and UG are of same order ofmagnitude Maximum UI for both sinkage and trim are obtainedbetween grids 2 and 3 which are 27 and 34 respectivelywhich are larger compared to 02 in XET

Of all the converged grid studies the lowest grid uncertaintyfor the resistance coefficients and motions are obtained on grids(234) and (123) respectively The XST method predicts morereasonable estimate for UG compared with ST method for Cpx ongrids (123) where PG PGth Grid studies (234) and (246)are closest to asymptotic range for sinkage and Cpx respectivelyThese calculations show that to reach the asymptotic range furthergrid refinement is required To maintain an affordable computa-tional cost with a reasonable accuracy grid No 3 is used for full-scale calculations

The full-scale results are consistent with the XET results as inboth the cases (a) the converged and diverged solutions involvefinest and coarsest grids respectively (b) large variations in

Table 3 Grids used for verification study for Athena bare hull with skeg (Re = 2 +109 Fr = 048)

Grid number 6 5 4 3 2 1

Ratio 1 214 212 234 2 254

Ship 111 +29 +56 132 +34 +66 157 +41 +79 187 +49 +94 222 +58 +112 264 +69 +133Background 111 +29 +56 132 +34 +66 157 +41 +79 187 +49 +94 222 +58 +112 264 +69 +133Total 360528 592416 1017046 1722644 2884224 4845456

ythorn2 94 78 62 50 40 30

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 = 001 L for Fr =048


1 CG are observed suggesting that further grid refinement isrequired (c) a significant increase in eN is observed when UI andUG are of the same order of magnitude There are specific differ-ences such as (a) The variations of Cfx in the WF simulations are50 of that in near-wall simulations in XET which is expected asboundary layer in the WF is treated using the log-layer formula-tion (b) The variations of Cpx in the WF simulations are signifi-cantly larger compared with BKW solutions in XET This couldbe due to the zero pressure gradient boundary condition at thewall used in the two-point approach (c) Overall higher levels ofgrid uncertainties are observed in WF simulations compared to

BKW solution in XET The verification study presented herecould be significantly affected by (a) contamination of UG dueto higher levels of UI (b) sensitivity of the solution on y+ asdiscussed in section 54 and (c) the pressure boundary conditionat the wall used in the WF

53 Predicted sinkage and trim

In Table 5 and Fig 7 smooth- and rough-wall TL and MLresults at model and full scale are presented In model-scalesimulations TL and ML models overpredict Cfx by 8 and 5

Fig 9 Verification for resistance coefficients and motions for Athena bare hull with skeg (Re = 2 +109 Fr = 048) 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

Table 4(a) Verification study for resistance coefficients and motions of Athena bare hull with skeg at full-scale (Re = 2 + 109 Fr = 048)

Monotonically converged solutions UG is S1 S2 or S3

Parameter Grids

Refinement

Ratio (rG) RG PG 1 CG

UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780

Sinkage 234ffiffiffi24

p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash


compared with BKW respectively Cpx is within 3 of the BKWresults for both the WF models Ctotx is within 6 of the BKWresults and 25 of the EFD data for both the WF models TheWFs overestimate sinkage for the higher speed when comparedwith the EFD data Similar trend is observed for the BKW resultsNotice that the absolute error is small but the low value ofsinkage increases the relative error The trim values obtainedusing WFs are within 5 of the BKW results and 32 of theEFD data for Fr = 08 Larger differences around 10 are ob-served between the WF results and EFD data for Fr = 048Overall the ML results are within 2 of the TL results

Cfx values obtained from full-scale (Re=2 +109) simulations

are found to be in good agreement with the ITTC line whererelative errors are less than 5 The differences in the model-

and full-scale Cpx are less than 65 consistent with the previousconclusions by Oh amp Kang (1992) Sinkage and trim comparewell with the model-scale WF results where differences arewithin 4 For this case again the TL and ML results are within2 Results obtained using TL model with PG effect are within1 of the TL results without PG effect This demonstrates thatthe effect of PG in WF models is not important for full-scaleship flows and is not included for the rest of simulations per-formed in this paper

An additional simulation is performed for Fr = 048 to demon-strate the effect of wall roughness (k+ = 28) For the TL modelCfx increases by 30 over the smooth-wall results which isconsistent with the ITTC correlation allowance for this roughnesslength The corresponding increase for the ML model is higherabout 41 which is discussed in the next section Cpx is unaf-fected by the wall-roughness effect as the differences are within01 Ctotx increases by 10 and 15 for the TL and ML mod-els respectively Sinkage and trim for both TL and ML modelsare within 15 of the smooth-wall results

54 Sensitivity of resistance sinkage and trim to y2+

As shown in Figs 10 a and c for the smooth-wall TL model theCtotx Cpx and trim vary by as much as 15 when ythorn2 is increasedfrom 30 to 300 Respective variations in the Cfx and sinkage arewithin 5 The variations in resistance coefficients and shipmotions for the smooth-wall ML model are within 5

Figures 10 b and c show that the resistance coefficients pre-dicted by the rough-wall TL model are less sensitive to ythorn2 com-pared with rough-wall ML model The relative changes in TLmodel are only 5 whereas ML model shows variations of asmuch as 10 The wall-roughness increases Cfx by 28 whichis in good agreement with the ITTC correlation allowance of30 ML model overpredicts Cfx for ythorn2 = 30 by as much as10 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-scaleError (E) is based on EFD data (D)

Fr Re Sinkage Trim Ctotx Cfx

+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216

TL with PG effect 20 +109 E 000305 (1062) 0664 (539) 000382 (078) 0001436 (028) 000217 (+046)

ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)

Results are based on static areas+E is based on BKW results

Table 4(b) Verification study for resistance coefficients andmotions of Athena bare hull with skeg at full-scale (Re = 2 + 10

9 Fr =048) Oscillatory converged solutions UG is S1 S2 S3 or S4

Parameter Grids

Refinement

Ratio (rG) RG UG ()

Ctotx 135ffiffiffi22

p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


roughness effect in the buffer layer Sinkage and trim vary onlyby 3 for the range of ythorn2 considered

6 Self-propelled fully appended Athena

As shown in Table 6 and Fig 11 the smooth-wall WF over-predicts Frs by 2 to 3 caused by the underprediction of CtotxCfx for the AH is 10 to 15 higher than the ITTC line asshown in Fig 7 because of the added frictional resistance fromthe appendages Sinkage is within 3 of the EFD value forlower speed Fr = 0432 The differences are larger for Fr =0575 and 0839 similar trend is also observed in the BKW

model-scale results It must be noted that the absolute error islow but relative error is high due to lower absolute value of thesinkage The behavior is reversed for the trim which has largererror for Fr = 0432 than those for Fr = 0575 and 0839 Forhigher Frs results are within 35 of the EFD The ML and TLresults are in close agreement with each other with differenceswithin 2

Rough-wall simulations are performed for Fr = 0432 and 0575using a roughness length ks = 100 mm corresponding to k+ = 25to 35 for lower and higher speeds The rough-wall simulations leadto better prediction of the Frs that are within 1 of the EFD dataThe good agreement is attributed to the increase in Cfx by60 +10

4 for the TL and 55 +104 for ML model which are in

close agreement with the correlation allowance CA = 65 +104

used in EFD data extrapolation (Crook 1981) Similar to the BHcase no appreciable effect of wall roughness is observed for sink-age and trim values for both ML and TL models

7 Towed fully appended Athena boundary layerand wake profiles

The wall roughness effect increases Cfx by 21 316 and395 over the smooth-wall for k+ = 15 32 and 43 respectivelyThis is in good agreement with the ITTC correlation allowanceCpx variations are within 3 for all the surface conditionsThe corresponding increases in Ctotx are 63 105 and 123 Sink-age and trim values are 000233 and 1844 respectively andshow variation within 1 on the wall roughness Trim is consistentwith the EFD value which was reported to be less than 3

The smooth-wall simulation predicts thinner boundary layercompared with the EFD data The boundary layer thicknessincreases with increasing roughness resulting in much better com-parison than the smooth-wall results The best results are obtainedfor k+=32 In Fig 12 the boundary layer velocity profilesobtained from the smooth- and rough-wall (k+ = 32) simulationsare compared with the EFD data The rough-wall results comparebetter with the EFD data than the smooth-wall results for rakes1 and 2 whereas the nature is reversed for rake 4 At rake 2 thediscrepancy between the EFD and numerical results is large Thiscould be the result of either the uncertainty in the EFD data as thevelocity outside the boundary layer is reported to be 092 opposedto the expected value of 10 or due to grid resolution issues asprobe 2 lies in the overset region of the hull and propeller hubgrids Further investigation is required using finer grid resolutionsandor comparison with additional EFD data

In Fig 13 results at two of the radial locations (0417 and075 R) on the forward propeller plane (CndashD in Fig 4 a) arecompared with EFD data No significant differences are observedbetween the smooth- and rough-wall simulations for tangentialand radial velocities The axial velocity predictions using rough-wall condition are better than that using the smooth-wall Resultsshown in Fig 14 are at the propeller plane (AndashB in Fig 4 a) forradial locations 0781 and 0963 R The tangential and radialvelocity components are predicted well in the rough-wall simula-tion when compared with the EFD data whereas the smooth-wallresults underpredict them significantly The rough-wall resultsalso capture the sharp decline in the axial velocity close to0 and 360 Similar results were obtained at other radial loca-tions for which figures are not shown

Fig 10 Sensitivity of results on ythorn2 for Athena bare hull with skeg (Re =

2 + 109 and Fr = 048) using TL and ML models a resistance coeffi-

cients for smooth-wall simulations b resistance coefficients for rough-

wall simulations and c sinkage and trim for both smooth- and rough-

wall conditions


8 Seakeeping for bare hull with skeg and fullyappended Athena

Figure 15 shows the motions for the seakeeping case Theheave has a phase lag of 20 compared with the pitch At modelscale WF predicts the peak of pitch 6 higher than the BKWwhereas no significant differences are observed for the heaveThe full-scale ship motions vary within 5 of the model-scaleWF results The transfer functions for the pitch at model scale are08 and 085 using the BKW and WF respectively Thecorresponding value is 081 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 loz Rough-wall simulation (k+ = 25 to

35) at full-scale using wall-function lines (and filled circles) EFD

(Crook 1981)

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 Cfx

EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3

E shows the percentage change from the smooth-wall values

Fig 12 Boundary layer velocity profiles obtained using smooth- and rough-

wall conditions at four rake locations shown in Fig 4 b for fully appended

Athena (Re = 42 +108 Fr = 03621) are compared with EFD data (Day et al

1980) The coordinate zL is the distance normal to the wall


Fig 13 Wake profile (axial tangential and radial velocity components) at forward propeller plane (CndashD in Fig 4 a) for fully appended Athena (Re =

42 +108 Fr = 03621) left 0417 R and right 075 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 (AndashB in Fig 4 a) for fully appended Athena (Re =

42 +108 Fr = 03621) left 0456 R and right 0963 R Black line smooth-wall gray line rough wall k+ = 32 EFD (Day et al 1980)


of the heave is 065 for both model and full scale Figure 16shows that the peak of Cfx correlates with the trough of theheave whereas the Ctotx peaks in phase with the pitch A bimodalpattern is observed in the WF simulations at Cfx minimathat occurs after the peaks of the pitch and heave Similar bimodalpattern in the Ctotxminima is caused by the 180 phase difference inCpx and Cfx troughs The WF predictions of Cfx at model scale are5 to 10 lower than the BKW Cfx in the full-scale calculation isabout 12 lower than the model-scale WF which is in good agree-ment with the ITTC line There is no significant scale effect on Cpx

and Ctotx for which the relative changes are within 2Figure 17 compares the full-scale AH results with those of

the full-scale BH results In the AH results pitch lags by 12

compared with the BH case whereas heave is in the samephase The amplitude of pitch is about 4 higher than the BHcase whereas the differences observed for heave amplitude arewithin 2 The transfer functions for the pitch and heave are084 and 066 respectively Figure 17 b shows high-frequencyoscillations in the trough of the Ctotx for both the BH and AHgeometries which coincides with the trough of the pitchSource of these fluctuations are the pressure fluctuations andcould be avoided by better convergence of the implicit motionssolver (Carrica et al 2007b) The emphasis of this example isto demonstrate the capability of WFs to simulate ship motions

Fig 15 Evolution of the pitch and heave for Athena bare hull with

skeg (Fr = 03) in presence of incoming regular head waves wave

length = 12 L amplitude = 001 L and wave encounter frequency =

0488 LU0

Fig 17 Time histories of a pitch and heave and b total resistance coefficients for bare hull with skeg and fully appended Athena (Fr = 03) in

presence of regular head waves

Fig 16 Time histories of a frictional and b total resistance coefficients for Athena bare hull with skeg (Fr = 03) in presence of regular head waves


in the presence of waves thus this aspect is not further investi-gated Ctotx of the AH is higher by about 10 to 15 com-pared with the BH case

9 2020 zigzag maneuver for self-propelled 5415

Figure 18 a shows the histories of heading and rudder anglesfor the model- and full-scale calculations In the full scale therudder check point is reached faster than in model scale Theperiods between rudder checks are 38 and 36 s for model and fullscale respectively This indicates slightly more efficient rudderaction for full scale The overshoot on the heading is 58 for themodel-scale and 51 for the full-scale There are no significantdifferences in the yaw rate for the model and full scale as seen inFig 18 b The yaw rate right before the rudder check point is 18degs close to the asymptotic value of 183 degs reported byCarrica et al (2008) for the turning circle case under similarconditions Figure 18 c shows that the full scale has larger rollangle peaks and smaller damping of roll oscillations than the

model scale This is caused by the smaller effective viscosity inthe full-scale computation The peak value of the roll angleis 153 for the model scale and 175 for the full scaleFigure 18 d shows the time histories of pitch and heave A dy-namic response to the rudder direction change can be observed forboth pitch and heave The ship goes nose down and moves up justafter the rudder check point and tends to recover to an asymptoticvalue Pitch and heave show oscillations similar to the roll wherethe damping is lower for full scale The asymptotic value of pitchis estimated to be 094 for model scale and 098 for full scaleThe minimum for the pitch is observed to be 024 for the modelscale and 029 for the full scale Heave values do not show muchvariation in model- and full-scale simulations The steady value isestimated to be 00059 and maximum is 00035 for both themodel- and full-scale simulations

There is no significant Re effect on free-surface elevationHowever the transom rooster tail shows slightly higher elevationin the full-scale computations compared with the model scaleVelocity contours on the cross sections immediately upstream

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 =

041) at a cross section upstream of the shaft struts


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

10 Conclusions and future work

This study develops and implements TL and ML models withwall roughness and pressure gradient effects using a two-pointapproach in the URANS solver CFDShip-Iowa The TL and MLmodels are first validated for smooth flat-plate simulations atlarge Re equivalent to full-scale ship Re Results show that theTL model is accurate when both ythorn2 and ythorn3 are placed in eithersublayer or log layer whereas ML has no such limitations

The WF models are further validated for BH simulations atboth model and full scale with fixed or predicted sinkage andtrim Resistance sinkage and trim predictions at model scaleagree well with the BKW results and EFD data Quantitativeverification of resistance coefficients and motions at full scaleshows that the grids considered are still far from the asymptoticrange A sufficiently fine grid is identified and used for full-scale simulations In the WF simulations the iterative errors andrelative changes between two successive grids of Cpx are highercompared with the BKW results of Xing et al (2008) at modelscale For full-scale simulations Cfx predictions are in goodagreement with the ITTC line Rough-wall simulations showhigher Cfx and Ctotx where the former agrees well with theITTC correlation allowance As expected sinkage and trim arenot significantly affected by the Re or the wall roughness Infull-scale simulations boundary layer is thinner and the slope ofthe transom wave is slightly lower compared with the modelscale It is shown that the TL model is more sensitive toythorn2 300 than the ML model for the smooth-wall simulationsML model shows limitations in accounting for the wall roughnesseffect for ythorn2 30 which is attributed to the tentative modeling ofthe roughness effect in the buffer layer No significant changes ofresults are observed by including the PG effect in TL model forfull-scale computations

Resistance and powering computations are performed at fullscale for self-propelled AH free to sink and trim using smooth-and rough-wall WFs In the rough-wall simulations Frs are pre-dicted better than that in the smooth-wall simulations when com-pared with full-scale data extrapolated from model scalemeasurements using ITTC ship-model correlation line includinga correlation allowance The boundary layer and wake profilesobtained from the full-scale rough-wall simulations using WF forthe towed AH free to sink and trim are in good agreement withthe full-scale EFD data Seakeeping calculations are performedusing WF for demonstration purpose at full-scale for both BH andAH The ship motions are mainly governed by the incoming wavefrequency and no significant Re or geometry effects are observedManeuvering calculation is performed for 5415 at full-scale usingWF Results show slightly more efficient rudder action lowerovershoots in the heading angle and larger roll angles withslower damping compared with the model scale

Overall the results are encouraging and demonstrate the versa-tility of a two-point multilayer wall-function in accurately pre-dicting model- and full-scale ship flows including resistance

propulsion seakeeping and maneuvering However there areissues and questions that need to be resolved in the future

bull To relax the sensitivity of rough-wall simulations using MLmodel for y2

+30 the correct formulation for downshift ofthe buffer-layer velocity profile has to be established

bull Aupoix (2007) recently proposed a general procedure to ex-tend turbulence models to account for surface roughness thatcould be of interest for near-wall turbulence models Theauthor also provided a better relation of the downshift oflog-law with roughness length that can be implementedalong with current wall-function models

bull The benefits of including PG effects for more accurate pre-diction of flow separations need further investigation

bull It is valuable to further evaluate the current two-point ap-proach by comparing with the one-point approach thatallows solution of the pressure equation up to the wall forbetter mass conservation

bull WF simulations in maneuvering calculation showed that in-sufficient grid resolution near the transom corner may intro-duce error in the calculation of tangential and normalvelocity direction The sensitivity of the current WFs todifferent grid topologies will be further evaluated

bull A verification and validation study is required for model-and full-scale resistance and propulsion calculations for AHfor which coarse grids were used in present study

bull WFs need to be further evaluated for more geometries andgeneral applications to get more experience such as planningships ship-ship interaction and so forth

References

AUPOIX B 2007 A general strategy to extend turbulence models to roughsurfaces Application to Smithrsquos k-L model Journal of Fluids Engineering129 1245ndash1254

BALAY S BUSCHELMAN K GROPP W KAUSHIK D KNEPLEY M CURFMANL SMITH B AND ZHANG H 2002 PETSc User Manual ANL-9511Revision 215 Argonne National Laboratory

BHUSHAN S XING T CARRICA P AND STERN F 2007 Model- and full-scale URANSDES simulations for Athena RV resistance powering andmotions Proceedings Ninth International Conference on Numerical ShipHydrodynamics August 5ndash8 Ann Arbor MI

BRADSHAW P AND HUANG G P 1995 The law of the wall in turbulentflow Proceedings of the Royal Society (London) Mathematical and Physi-cal Sciences 451 1941 165ndash188

BULL P VERKUYL J B RANOCCHIA D DI MASCIO A DATTOLA R MERLEL AND CORDIER S 2002 Prediction of high Reynolds number flowaround naval vessels Proceedings 24th Symposium Naval Hydrodynam-ics 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 approachfor viscous ship flows and large amplitude motions and maneuveringProceedings 26th Symposium on Naval Hydrodynamics September17ndash22 Rome Italy

CARRICA P M WILSON R V AND STERN F 2006b Unsteady RANSsimulations of the ship forward speed diffraction problem Computers ampFluids 35 6 545ndash570

CARRICA P M WILSON R V AND STERN F 2007a An unsteady single-phase level set method for viscous free surface flows International Journalfor Numerical Methods in Fluids 53 2 229ndash256

CARRICA P M WILSON R V NOACK R AND STERN F 2007b Shipmotions using single-level set with dynamic overset grids Computers ampFluids 36 9 1415ndash1433

CARRICA P M ISMAIL F HYMAN M BHUSHAN S AND STERN F 2008Turn and zigzag maneuvers of a surface combatant using a URANS ap-


proach with dynamic overset grids Proceedings SIMMAN WorkshopApril 14ndash16 Copenhagen Denmark

CHEN H C AND PATEL V C 1988 Near wall turbulence models forcomplex flows including separation AIAA Journal 26 6 641ndash648

CHOI J-E MIN K-S CHUNG S-H AND SEO H-W 2003 Study on thescale effects on the flow characteristics around a full slow-speed shipProceedings Eighth International Conference on Numerical Ship Hydrody-namics Busan Korea

CROOK L B 1981 Powering Predictions for the RV Athena (PG 94)Represented by Model 4950-1 with Design Propellers 4710 and 4711David W Taylor Naval Ship Research and Development CenterDTNSRDCSPD-0833-05

DAY W G Jr REED A M AND HURWITZ R B 1980 Full-Scale Propel-ler Disk Wake Survey and Boundary Layer Velocity Profile Measurementson the 154-Foot Ship RV Athena David W Taylor Naval Ship Researchand Development Center DTNSRDCSPD-0833-01

ESCH T AND MENTER FR 2003 Heat transfer prediction based on two-equation turbulence models with advanced wall treatment Proceedings In-ternational Symposium on Turbulence Heat and Mass Transfer 4 AntalyaTurkey 614ndash621

GORSKI J J 2002 Present state of numerical ship hydrodynamics andvalidation experiments Journal of Offshore Mechanics and Arctic Engi-neering 124 74ndash80

GORSKI J J MILLER R W AND COLEMAN R M 2004 An investigation ofpropeller inflow for naval surface combatants Proceedings 25th Sympo-sium Of Naval Hydrodynamics St Johnrsquos NL Canada

GROTJANS H AND MENTER F 1998 Wall functions for general CFD codesProceedings Fourth European Comp Fluid Dynamics Conference AthensGreece 1112ndash1117

HANNINEN S AND MIKKOLA T 2006 Computation of ship-hull flows atmodel- and full-scale Reynolds numbers Proceedings Numerical ShipHydrodynamics Seminar Maritime Institute of Finland Turku

HUANG J CARRICA P M MOUSAVIRAAD M AND STERN F 2007 Semi-coupled airwater immersed boundary approach in curvilinear dynamicoverset grids with application to environmental effects in ship hydrodynam-ics Proceedings Ninth International Conference on Numerical ShipHydrodynamics August 5ndash8 Ann Arbor MI

ISSA R I 1985 Solution of the implicit discretised fluid flow equations byoperator splitting Journal of Computational Physics 62 40ndash65

JENKINS D 1984 Resistance Characteristics of the High Speed TransomStern Ship RV Athena in the Bare Hull Condition Represented byDTNSRDC Model 5365 Ship Performance Dept Research amp DevelopmentReport DTNSRDC-84024

JIMENEZ J 2004 Turbulent flows over rough walls Annual Review of FluidMechanics 36 173ndash196

KALITZIN G MEDIC G IACCARINO G AND DURBIN P 2005 Near-wallbehavior of RANS turbulence models and implications for wall functionsJournal of Computational Physics 204 265ndash291

KIM S-E AND CHOUDHURY D 1995 A near-wall treatment using wallfunctions sensitized to pressure gradient ASME-FED Separated and Com-plex Flows 217 273ndash280

KNOBLOCH K AND FERNHOLZ H-H 2002 Statistics correlations and scal-ing in a turbulent boundary layer at Red = 115 +10

5 Proceedings IUTAMSymposium on Reynolds Number Scaling in Turbulent Flow September11ndash13 Princeton NJ

KNOPP T ALRUTZ T AND SCHWAMBORN D 2006 A grid and flow adap-tive wall-function method for RANS turbulence modeling Journal ofComputational Physics 220 1 19ndash40

LARSSON L 1997 CFD in ship design-prospects and limitation 18th GeorgWeinblaum Memorial Lecture Ship Tec Research 44 3 133ndash154

LAUNDER B E AND SPALDING D B 1974 The numerical computation ofturbulent flows Computer Methods in Applied Mechanics and Engineering3 269ndash289

LONGO J SHAO J IRVINE M AND STERN F 2007 Phase-averaged PIV forthe nominal wake of a surface ship in regular head waves Journal of FluidsEngineering 129 5 524ndash540

MENTER F R 1994 Two equation eddy viscosity turbulence models forengineering applications AIAA Journal 32 8 1598ndash1605

MILLER R GORSKI J XING T CARRICA P AND STERN F 2006 Resis-tance predictions of high speed mono and multi-hull ships with and without

water jet propulsors using URANS Proceedings 26th Symposium on Na-val Hydrodynamics September 17ndash22 Rome Italy

NOACK R 2005 SUGGAR a general capability for moving body oversetgrid assembly AIAA paper 2005-5117 Proceedings 17th AIAA Computa-tional Fluid Dynamics Conference Toronto Ontario Canada

OH K J AND KANG S H 1992 Full scale Reynolds number effects for theviscous flow around the ship stern Computational Mechanics 9 85ndash94

PARK I L-R VAN S-H KIM J AND AHN H-S 2004 Two-phase vis-cous flow simulation around a commercial container ship hull form Jour-nal of Ships amp Ocean Engineering 37 147ndash154

PATEL V C 1998 Perspective Flow at high Reynolds number and overrough surfacesmdashAchilles Heel of CFD Journal of Fluids Engineering 120434ndash444

SCHULTZ M P 2002 The relationship between frictional resistance androughness for surfaces smoothened by sanding Journal of Fluids Engineer-ing 124 492ndash499

SHIH T H POVINELLI L A AND LIU N S 2003 Application of generalizedwall function for complex turbulent flows Journal of Turbulence 4 1ndash16

SIMMAN 2008 Workshop on verification and validation of ship maneuver-ing simulation methods April 14ndash16 Copenhagen Denmark httpwwwsimman2008dk

SO R M C AND LAI Y G 1988 Low-Reynolds-number modeling offlows over a backward-facing step Journal of Applied Mathematics andPhysics 39 13ndash27

STARKE B RAVEN H AND PLOEG A 2007 Computation of transom-sternflows using a steady free-surface fitting RANS method Proceedings NinthInternational conference on Numerical Ship Hydrodynamics August 5ndash8Ann Arbor MI

STERN F KIM H T PATEL V C AND CHEN H C 1988 A viscous-flowapproach to the computation of propeller-hull interactions JOURNAL OF SHIPRESEARCH 32 4 246ndash262

STERN F CARRICA P M KANDASAMY M GORSKI J J OrsquoDEA M H MILLERR HENDRIX D KRING D MILEWSKI W HOFFMAN R AND GARYC 2006a Computational hydrodynamic tools for high-speed cargo trans-ports Proceedings SNAME Maritime Technology Conference FortLauderdale FL

STERN F WILSON R AND SHAO J 2006b Quantitative approach to VampVof CFD simulations and certification of CFD codes International Journalof Computational Fluid Dynamics 50 1335ndash1355

TAHARA Y KATSUUI T AND HIMEMNO Y 2002 Computation of shipViscous flow at full scale Reynolds number Journal of the Society of NavalArchitects of Japan 92 89ndash101

TEMMERMAN L LESCHZINER M A MELLEN C P AND FROHLICH J 2003Investigation of wall-function approximations and subgrid-scale modelsin large eddy simulation of separated flow in a channel with streamwiseperiodic constrictions International Journal of Heat and Fluid Flow 24157ndash180

TZABIRAS G D 1991 A numerical study of the turbulent flow around thestern of ship models International Journal for Numerical Methods inFluids 13 1179ndash1204

UTYUZHNIKOV S V 2005 Generalized wall functions and their applicationfor simulation of turbulent flows International Journal for NumericalMethods in Fluids 47 1323ndash1328

WATSON R D HALL R M AND ANDERS J B 2000 Review of skinfriction measurements including recent high-Reynolds number resultsfrom NASA Langley NTF Proceedings AIAA Paper 2392 June 19ndash22Denver CO

WHITE F M 2008 Fluid Mechanics 6th ed McGraw Hill 362ndash363WILCOX D C 1989 Wall matching A rational alternative to wall-func-tions Proceedings AIAA Paper 89-611 Reno NV

WILCOX D C 1993 Turbulence Modeling for CFD DCW Industries LaCanada CA

WILSON R V CARRICA P M AND STERN F 2006 URANS simulationsfor a high-speed transom stern ship with breaking waves InternationalJournal of Computational Fluid Dynamics 20 2 105ndash125

XING T AND STERN F 2008 Factors of safety for Richardson extrapola-tion for industrial applications IIHR Technical Report No 466

XING T CARRICA P AND STERN F 2008 Computational towing tankprocedures for single run curves of resistance and propulsion Journal ofFluids Engineering 130 2 1ndash14


errors in mass and momentum flux calculations and may lead tosolution divergence Implementation of lower-order numericalschemes such as first order will likely overcome this problem butwith the penalty of significant loss of accuracy Development andvalidation of near-wall turbulence models for the Re of full-scaleship flows are lacking because EFD data are sparse even for canon-ical flows and difficulties of maintaining fixed environmental con-ditions to conduct practical full-scale ship measurements (Patel1998 Aupoix 2007)

The use of ldquowall functionsrdquo avoids the numerical limitationsof the near-wall turbulence models and significantly reduces thecomputational cost Wall functions are based on the near-wall be-havior of nonseparating two-dimensional turbulent boundary layersalso valid along the streamwise direction of three-dimensional(3D) turbulent boundary layers with small cross flow (Bradshaw ampHuang 1995) Thus the obvious limitations of the wall functions arein accurately predicting separated flows and 3D boundary layerswith significant cross flow For example in a backward facing stepflow the reattachment length is often underpredicted in wall-func-tion simulations Nevertheless near-wall turbulence models alsosuffer from the same deficiency as the model constants are derivedunder similar turbulent boundary layer assumptions (So amp Lai1988) Some of the issues addressed in the literature although notfully resolved are

bull Sensitivity to the spacing of the first grid point away fromthe wall (matching point herein)

bull Inclusion of pressure gradient effects for better prediction ofseparated flows

bull Modeling of wall roughnessbull Different implementation approaches for determination ofthe velocity direction at the matching point

The standard wall-function approach (Launder amp Spalding1974) is based on the stringent criteria that the matching pointlies in the log layer (one layer only) However different flowregimes make it difficult to place the matching point in the loglayer for example laminar-turbulent transition zone variation ofthe boundary layer thickness along the ship hull or when the shipis slowly accelerated from a static condition This limitation isaddressed by a two-layer model in which boundary conditionsfor the velocity and turbulent quantities are switched betweenthe sublayer and log-layer analytic profiles in the near-wall re-gion depending on the local y+ value of the matching point(Grotjans amp Menter 1998 Esch amp Menter 2003) A three-layermodel includes an additional formulation for the buffer layer(Temmerman et al 2003) but still suffers from the deficiencythat the velocity profile is not smooth across the near-wall regionThis deficiency was resolved by Shih et al (2003) who proposeda generalized multilayer model using curve fitting to provide acontinuous function to bridge the sublayers and log layers Kalit-zin et al (2005) developed a multilayer model along with look-uptables for evaluating the friction velocity The look-up tables areobtained from a separate zero-pressure gradient smooth flat-platesimulation using a near-wall turbulence model This approachleads to an accurate calculation of the friction velocity than theanalytic equations used in above models However the applica-bility of this approach for high Reynolds number flows wouldrequire examination of the look-up table The models discussedpreviously are based on Dirichlet-type boundary conditionsUtyuzhnikov (2005) proposed a differential form for the boundary

condition valid for the entire boundary layer Overall the multi-layer wall-function model provides the most flexibility in theplacement of the matching point

Evaluation of the pressure gradient effect on wall functions forseparated flows with mild pressure gradients was conducted bysome previous studies For a backward-facing step flow the two-layer models with pressure gradient effect predict the separationrecirculation and reattachment regions better than that withoutpressure gradient effect (Wilcox 1993 Kim amp Choudhury 1995)Knopp et al (2006) extended the Kalitzin et al (2005) approachfor nonequilibrium flows by including pressure gradient effectsTheir main conclusion is that in the regions of stagnationand strong pressure gradients a near-wall solution is the beststrategy for which they applied flow-based grid adaptation Inthe strong pressure gradient regions usually encountered in shipflows clipping of the pressure gradient magnitude to 75of the friction velocity is suggested to avoid numerical instabilityor divergence (Wilcox 1993) Thus the benefit of including pres-sure gradient effect for wall functions in separated flows isquestionable

The effect of surface roughness is more important for full-scalecomputations than for those at the model scale as it leads tosignificant increase in frictional and total resistances The mostcommonly used model for surface roughness is based on thedownshift of the log-layer profile (White 2008) Aupoix (2007)recently provided a formulation of the downshift of log-law that isin better agreement with EFD data which could be of futureinterest Several studies have investigated the effect of roughnesstype on the boundary layer profile (Jimenez 2004) Schultz (2002)performed model-scale experiments on several sanded andunsanded painted surfaces encountered in ship flows The studieshave shown that the roughness type does not show significanteffect in the transitional roughness regime (5 k+ 70) or thedownshift of log-layer profile Thus surface roughness modelingbased on the downward shift of the log-layer profile can be usedwith relative confidence for full-scale ship calculations whereroughness length mostly lies in the transitional regime (Patel1998 Tahara et al 2002) For one- and two-layer wall functionmodels modeling of roughness effect is straightforward How-ever for the multilayer model appropriate correlation is not avail-able for the buffer layer from either experimental or numericalstudies

Implementation of wall-function models requires evaluation ofthe friction velocity either analytically or using look-up tables(Kalitzin et al 2005) to provide boundary conditions for velocityand turbulence variables The one-point approach proposed byKim amp Choudhary (1995) uses the flow variables at the wallneighboring cells only This allows solutions of the momentumequations up to the matching point The one-point approach can beimplemented easily for finite-volume schemes but introduces ad-ditional complexities and challenges for finite-difference schemesAn alternative two-point approach was introduced by Chen ampPatel (1988) and extended by Tahara et al (2002) for ship flowsIt uses the tangential velocity magnitude and direction at thesecond grid point away from the wall to obtain velocity at thematching point Implementation of the two-point approach forfinite-difference schemes is straightforward The one-point ap-proach has advantages over the two-point approach as it does notrestrict the flow direction at the matching point to follow the flowdirection at the second point away from the wall










rsaquoUi


rsaquoUi



rsaquoxjthorn 1

Re

rsaquo2Ui













rsaquoxi

2

3dijk eth3THORN


rsaquok



rsaquov










U23

utfrac14



(



p) where



Pthorn frac14 nu 3t

rsaquop

rsaquox

eth8bTHORN





323

ythorn2 1167


thorn ythorn2 1167

8lt eth9THORN

v2 frac146n

0075Dy22 ythorn2 1167

ut03kDy2

1 03ythorn2 Pthorn

ythorn2 1167

(eth10THORN




U23

utfrac14

y23thorn y23

thorn 5

a0 thorn a1ythorn


5 y23thorn 30


thorn 30

8gtgtgtgtltgtgtgtgt




a3 frac14 000295224178 a4 frac14 33144178e 005




k2 frac14

kjfrac143

Dy2Dy3

323

ythorn2 5

dythorn

duthorn 1


u2t03 ythorn2 30


eth13THORN

v2 frac14

6n0075Dy2

2


0075Dy22

2

thorn ut03kDy2

2s

5 ythorn2 30

ut03kDy2

ythorn2 30


eth14THORN












thorn3 sublayer y

thorn2 sublayer y

thorn3 log-








24 Propeller model






















ffiffiffiffifficd


cdp









































Wall-Function Model


Fixed sinkage

and trim




2 +108 048 3267


2 +109 048 27715

Towed Athena


048

30ndash40

ML and TL

smooth-wall




and trim

2 +109

048

08

30 100 300

6- grids

30

ML and TL smooth

and rough-wall

TL smooth-wall

TL smooth-wall


smooth-wall



models


Pitch and heave

in waves

372 +107

2 +109 03

30ndash40



appended Athena

Predict sinkage

and trim

366 +108

488 +108

0432

0575

65

85

TL smooth and

rough-wall

ML and TL smooth

and rough-wall







Towed fully

appended Athena

Predict sinkage

and trim

42 +108 0362 50 TL smooth and

rough-wall



Pitch and heave

in waves



appended 5415






















5415













Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References





































































rsaquoUi


rsaquoUi



rsaquoxjthorn 1

Re

rsaquo2Ui













rsaquoxi

2

3dijk eth3THORN


rsaquok



rsaquov










U23

utfrac14



(



p) where



Pthorn frac14 nu 3t

rsaquop

rsaquox

eth8bTHORN





323

ythorn2 1167


thorn ythorn2 1167

8lt eth9THORN

v2 frac146n

0075Dy22 ythorn2 1167

ut03kDy2

1 03ythorn2 Pthorn

ythorn2 1167

(eth10THORN




U23

utfrac14

y23thorn y23

thorn 5

a0 thorn a1ythorn


5 y23thorn 30


thorn 30

8gtgtgtgtltgtgtgtgt




a3 frac14 000295224178 a4 frac14 33144178e 005




k2 frac14

kjfrac143

Dy2Dy3

323

ythorn2 5

dythorn

duthorn 1


u2t03 ythorn2 30


eth13THORN

v2 frac14

6n0075Dy2

2


0075Dy22

2

thorn ut03kDy2

2s

5 ythorn2 30

ut03kDy2

ythorn2 30


eth14THORN












thorn3 sublayer y

thorn2 sublayer y

thorn3 log-








24 Propeller model






















ffiffiffiffifficd


cdp









































Wall-Function Model


Fixed sinkage

and trim




2 +108 048 3267


2 +109 048 27715

Towed Athena


048

30ndash40

ML and TL

smooth-wall




and trim

2 +109

048

08

30 100 300

6- grids

30

ML and TL smooth

and rough-wall

TL smooth-wall

TL smooth-wall


smooth-wall



models


Pitch and heave

in waves

372 +107

2 +109 03

30ndash40



appended Athena

Predict sinkage

and trim

366 +108

488 +108

0432

0575

65

85

TL smooth and

rough-wall

ML and TL smooth

and rough-wall







Towed fully

appended Athena

Predict sinkage

and trim

42 +108 0362 50 TL smooth and

rough-wall



Pitch and heave

in waves



appended 5415






















5415













Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References



































































U23

utfrac14



(



p) where



Pthorn frac14 nu 3t

rsaquop

rsaquox

eth8bTHORN





323

ythorn2 1167


thorn ythorn2 1167

8lt eth9THORN

v2 frac146n

0075Dy22 ythorn2 1167

ut03kDy2

1 03ythorn2 Pthorn

ythorn2 1167

(eth10THORN




U23

utfrac14

y23thorn y23

thorn 5

a0 thorn a1ythorn


5 y23thorn 30


thorn 30

8gtgtgtgtltgtgtgtgt




a3 frac14 000295224178 a4 frac14 33144178e 005




k2 frac14

kjfrac143

Dy2Dy3

323

ythorn2 5

dythorn

duthorn 1


u2t03 ythorn2 30


eth13THORN

v2 frac14

6n0075Dy2

2


0075Dy22

2

thorn ut03kDy2

2s

5 ythorn2 30

ut03kDy2

ythorn2 30


eth14THORN












thorn3 sublayer y

thorn2 sublayer y

thorn3 log-








24 Propeller model






















ffiffiffiffifficd


cdp









































Wall-Function Model


Fixed sinkage

and trim




2 +108 048 3267


2 +109 048 27715

Towed Athena


048

30ndash40

ML and TL

smooth-wall




and trim

2 +109

048

08

30 100 300

6- grids

30

ML and TL smooth

and rough-wall

TL smooth-wall

TL smooth-wall


smooth-wall



models


Pitch and heave

in waves

372 +107

2 +109 03

30ndash40



appended Athena

Predict sinkage

and trim

366 +108

488 +108

0432

0575

65

85

TL smooth and

rough-wall

ML and TL smooth

and rough-wall







Towed fully

appended Athena

Predict sinkage

and trim

42 +108 0362 50 TL smooth and

rough-wall



Pitch and heave

in waves



appended 5415






















5415













Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References

































































thorn3 sublayer y

thorn2 sublayer y

thorn3 log-








24 Propeller model






















ffiffiffiffifficd


cdp









































Wall-Function Model


Fixed sinkage

and trim




2 +108 048 3267


2 +109 048 27715

Towed Athena


048

30ndash40

ML and TL

smooth-wall




and trim

2 +109

048

08

30 100 300

6- grids

30

ML and TL smooth

and rough-wall

TL smooth-wall

TL smooth-wall


smooth-wall



models


Pitch and heave

in waves

372 +107

2 +109 03

30ndash40



appended Athena

Predict sinkage

and trim

366 +108

488 +108

0432

0575

65

85

TL smooth and

rough-wall

ML and TL smooth

and rough-wall







Towed fully

appended Athena

Predict sinkage

and trim

42 +108 0362 50 TL smooth and

rough-wall



Pitch and heave

in waves



appended 5415






















5415













Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References













































































ffiffiffiffifficd


cdp









































Wall-Function Model


Fixed sinkage

and trim




2 +108 048 3267


2 +109 048 27715

Towed Athena


048

30ndash40

ML and TL

smooth-wall




and trim

2 +109

048

08

30 100 300

6- grids

30

ML and TL smooth

and rough-wall

TL smooth-wall

TL smooth-wall


smooth-wall



models


Pitch and heave

in waves

372 +107

2 +109 03

30ndash40



appended Athena

Predict sinkage

and trim

366 +108

488 +108

0432

0575

65

85

TL smooth and

rough-wall

ML and TL smooth

and rough-wall







Towed fully

appended Athena

Predict sinkage

and trim

42 +108 0362 50 TL smooth and

rough-wall



Pitch and heave

in waves



appended 5415






















5415













Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References



































































































Wall-Function Model


Fixed sinkage

and trim




2 +108 048 3267


2 +109 048 27715

Towed Athena


048

30ndash40

ML and TL

smooth-wall




and trim

2 +109

048

08

30 100 300

6- grids

30

ML and TL smooth

and rough-wall

TL smooth-wall

TL smooth-wall


smooth-wall



models


Pitch and heave

in waves

372 +107

2 +109 03

30ndash40



appended Athena

Predict sinkage

and trim

366 +108

488 +108

0432

0575

65

85

TL smooth and

rough-wall

ML and TL smooth

and rough-wall







Towed fully

appended Athena

Predict sinkage

and trim

42 +108 0362 50 TL smooth and

rough-wall



Pitch and heave

in waves



appended 5415






















5415













Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References
















































































5415













Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References





































































Inlet Self-

propelled


Seakeeping Eq (43)

(Carrica et al

2007a)

Eq (42)

(Carrica et al

2007a)


(Carrica et al

2007a)


(Carrica et al

2007a)








rsaquon frac14 0







rsaquon frac14 0


(2007a)

Eq (37)

0 6n0075Dy2

2

uship nship wship

Wall-function

( j = 2)



ML Eq (13)

TL Eq (10)

ML Eq (14)













Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References





































































Ratio 1 214 212 234 2 254


ythorn2 94 78 62 50 40 30













Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References






































































Parameter Grids

Refinement


UG ()

Stern et al

(2006a)

Xing and Stern

(2008)

Ctotx

246ffiffiffi22

p0224 431 245 155 ndash

234ffiffiffi24

p00313 201 7384 0132 ndash

Cpx

246ffiffiffi22

p0555 17 02 398 398

345ffiffiffi24

p0389 545 279 1359 ndash

Cfx 123ffiffiffi24

p0571 323 081 291 2780


p0743 172 0164 1548 1548

Trim 123ffiffiffi24

p0057 165 3871 256 ndash












+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References







































































+ Cpx+

EFD 048 223 +107 000341 0710 000575 ndash ndash

BKW 223 +107 E 000320 (616) 0676 (479) 000578 (0522) 0002480 000223

WF

TL223 +10

7 E000310 (882) 0641 (971) 0005873 (2139) 0002673 (778) 0002288 (260)

ML 000305 (1056) 0638 (1014) 0005800 (0870) 0002585 (423) 0002284 (242)

TL 20 +109 E 000307 (997) 0667 (606) 000385 000144 000216


ML 20 +109 E 000307 (999) 0668 (592) 000390 (+130) 000147 (+208) 000218 (+009)

TL k+ = 28 20 +109 E 000308 (968) 0673 (521) 000431 (+1195) 000185 (+2847) 000217 (+005)

ML k+ = 28 20 +109 E 000306 (1026) 0657 (746) 000454 (+1641) 000208 (+4150) 000216 (+000)

EFD 08 372 +107 000194 0997 000430 ndash ndash

BKW 372 +107 E 000145 (+25) 0934 (63) 000411 (465) 000220 000176

WF

TL 372 +107 E 000120 (+38) 1029 (+32) 000437 (+163) 000236 (+727) 000178 (114)

TL 20 +109 E 000128 (+34) 1061 (+64) 000341 000141 000187 (+625)




Parameter Grids

Refinement

Ratio (rG) RG UG ()


p 077 553

Cpx

135ffiffiffi22

p 080 1059

234ffiffiffi24

p 028 324

Cfx

135 ffiffiffi22

p 097 139

246 030 342

234ffiffiffi24

p 055 274

Sinkage

123 ffiffiffi24

p 035 409

345 075 744

456 068 1164

Trim345 ffiffiffi

24p 059 357

456 067 547


















wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References













































































wall conditions









(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References




































































(Crook 1981)



EFD ndash 161 +107 0432 000360 0726 ndash

BKW (E) 1537 161 +107 0423 (208) 000381 (58) 0740 (+19) ndash

WF

TL (E)1518 367 +10

8 0441 (+208) 000368 (22) 0813 (12) 195 +103

TL k+ = 26 (E) 0433 (+023) 000370 (278) 0812 (+118) 256 +103 (+31)

EFD ndash 215 +107 0575 000335 146 ndash

BKW (E) 2008 215 +107 0567 (14) 000305 (+9) 154 (+55) ndash

WF

TL (E)

1985 488 +108

0589 (+244) 000279 (+179) 143 (210) 19 +103

TL k+=35 (E) 0580 (+087) 000279 (+179) 142 (273) 25 +103 (+32)

ML (E) 0590 (+261) 000273 (+185) 146 (002) 185 +103

ML k+ = 35 (E) 0581 (+104) 000274 (+182) 145 (007) 241 +103

(+30)

EFD ndash 313 +107 0839 80 +10

4 1595 ndash

BKW (E) 2772 313 +107 0830 (11) 30 +10

4 (+63) 175 (+97) ndash

WF TL (E) 2743 715 +108 0856 (+202) 20 +10

4(+75) 165 (+35) 176 +10

3



















0488 LU0











5415



















References









































































0488 LU0











5415



















References



































































5415



















References












































































References





























































Journal of Ship Research, Vol. 53, No. 4, December …Journal of Ship Research, Vol. 53, No. 4,...

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Transcript of Journal of Ship Research, Vol. 53, No. 4, December …Journal of Ship Research, Vol. 53, No. 4,...