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Twenty-Third Symposium on Naval Hydrodynamics (2001)
Naval Studies Board (NSB)

Page
441
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Page
441
Front Matter (R1-R19)
Modern Seakeeping Computations for Ships (1-45)
Forces, Moment and Wave Pattern for Naval Combatant in Regular Head Waves (46-65)
New Green-Function Method to Predict Wave-Induced Ship Motions and Loads (66-81)
Validation of Time-Domain Prediction of Motion, Sea Load, and Hull Pressure of a Frigate in Regular Waves (82-97)
Ship Motions and Loads in Large Waves (98-111)
Prediction of Vertical-Plane Wave Loading and Ship Responses in High Seas (112-125)
Basic Studies of Water on Deck (126-142)
Second Order Waves Generated by Ship Motions (143-156)
Prediction of Nonlinear Motions of High-Speed Vessels in Oblique Waves (157-170)
Optimizing Turbulence Generation for Controlling Pressure Recovery in Submarine Launchways (171-180)
Hull Design by CAD/CFD Simulation (181-190)
Steady-State Hydrodynamics of High-Speed Vessels with a Transom Stern (191-205)
Practical CFD Applications to Design of a Wave Cancellation Multihull Ship (206-222)
Simulation of Ship Maneuvers Using Recursive Neural Networks (223-242)
Flow- and Wave-Field Optimization of Surface Combatants Using CFD-Based Optimization Methods (243-261)
Marine Propulsor Noise Investigations in the Hydroacoustic Water Tunnel 'G.T.H.' (262-283)
Propulsor Design Using Clebsch Formulation (284-300)
Unsteady Flow Quantities on Two-Dimensional Foils: Experimental and Numerical Results (301-313)
Hydrofoil Turbulent Boundary Layer Separation at High Reynolds Numbers (314-329)
Pressure Fluctuation on Finite Flat Plate Above Wing in Sinusoidal Gust (330-341)
Control of the Turbulent Wake of an Appended Streamlined Body (342-354)
Investigation of Global and Local Flow Details by a Fully Three-Dimensional Seakeeping Method (355-367)
Prediction of Wave Pressure and Loads on Actual Ships by the Enhanced Unified Theory (368-384)
Frequency Domain Numerical and Experimental Investigation of Forward Speed Radiation by Ships (385-401)
International Collaboration on Benchmark CFD Validation Data for Surface Combatant DTMB Model 5415 (402-422)
Validation of High Reynolds Number, Unsteady Multi-Phase CFD Modeling for Naval Applications (423-440)
Free Surface Viscous Flow Computation Around A Transom Stern Ship by Chimera Overlapping Scheme (441-456)
Anti-Roll Tank Simulations With A Volume of Fluid (VOF) Based Navier-Stokes Solver (457-473)
Validation of Tab Assisted Control Surface Computation (474-484)
Experimental and Numerical Investigation of the Flow Around the Appendices of a Whitbread 60 Sailing Yacht (485-492)
Propeller Wake Analysis by Means of PIV (493-510)
Experimental and Numerical Investigation of the Unsteady Flow Around a Propeller (511-526)
Simulation of Incompressible Viscous Flow Around a Ducted Propeller Using a RANS Equation Solver (527-539)
On Submerged Stagnation Points and Bow Vortices Generation (540-552)
Numerical Prediction of Scale Effects in Ship Stern Flows with Eddy-Viscosity Turbulence Models (553-568)
The Experimental and Numerical Study of Flow Structure and Water Noise Caused by Roughness of a Body (569-578)
Large-Eddy Simulations of Turbulent Wake Flows (579-598)
Instability of Partial Cavitation: A Numerical/Experimental Approach (599-615)
An Unsteady Three-Dimensional Euler Solver Coupled with a Cavitating Propeller Analysis Method (616-638)
On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise (639-653)
An Experimental Investigation of Cavitation Inception and Development of Partial Sheet Cavaties on Two-Dimensional Hydrofoils (654-669)
Modeling 3D Unsteady Sheet Cavities Using a Coupled UnRANS-BEM code (670-686)
Ship Wake Detectability in the Ocean Turbulent Environment (687-703)
An Experimental and Computational Study of the Effects of Propulsion on the Free-Surface Flow Astern of Model 5415 (704-712)
Breaking Waves in the Ocean and Around Ships (713-745)
Numerical and Experimental Study of the Wave Breaking Generated by a Submerged Hydrofoil (746-761)
The Numerical Simulation of Ship Waves Using Cartesian Grid Methods (762-779)
Radiation Loads on a Cylinder Oscillating in Pycnocline (780-791)
Wave Resistance Computations - A Comparison of Different Approaches (792-804)
Computations of Nonlinear Turbulent Free Surface Flows Using the Parallel Uncle Code (805-819)
Submarine Maneuverability Assessment Using Computational Fluid Dynamic Tools (820-832)
Simulation of UUV Recovery Hydrodynamics (833-847)
Reynolds-Averaged Modeling of High-Froude-Number Free Surface Jets (848-862)
On Roll Hydrodynamics of Cylinders Fitted with Bilge Keels (863-880)
Combining Accuracy and Effciency with Robustness in Ship Stern Flow Computation (882-896)
An Unstructured Multielement Solution Algorithm for Complex Geometry Hydrodynamic Simulations (897-909)
Ship Stern Flow Calculations on Overlapping Composite Grids (910-926)
Study on the Prediction of Flow Characteristics Around a Ship Hull (927-940)
Analysis of Turbulence Free-Surface Flow Around Hulls in Shallow Water Channel by a Level-Set Method (941-956)
A Design Tool for High Speed Ferries Washes (957-967)
Flow Around Ships Sailing in Shallow Water - Experimental and Numerical Results (968-982)
Ship Stability Study in the Coastal Region: New Coastal Wave Model Coupled with a Dynamic Stability Model (983-992)
Waves and Forces Caused by Oscillation of a Floating Body Determined Through a Unified Nonlinear Shallow-Water Theory (993-1005)

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Free Surface Viscous Flow Computation Around A Transom Stern Ship By Chimera Overlapping Scheme CW Lin, S. Percival (Naval Surface Warfare Center, Carderock Division, USA) ABSTRACT The focus of this p Her is to investigate the c mm ility of a numerical scheme Hat inco porates a free so face bommdffy treatment md a smkage/tam calculation into a viscous flow computation mound a su face ship Utilizmg d is scheme computations me pe to med on two selected h msom tern ships The ff ility to compute file wet, partially wet, or d y t msom is developed, including a new g id topology for h msom .-. - 3 Computational results on two t msom-stff ships are consistent with available model experimental data The sh me of the wave profile along the hull is generally predicted well, while the magnitude of wave elevation still needs to be improved The wave pattern is also well copulated, except that wave cre ts md troughs me not ff Shop ff the model measurements The computed total resister ce is in good con elation wish model measured values Aldhough two different types of grid pacing md di tribution techniques me investigated, a father tudy is needed to dete mule a more effective gad system for free su face viscous flow computations INTRODUCTION Computing file flow around a so face ship moving steadily in calm water hff beff a challengmg research task for yews The complexity of file flow physics ff und a smface hip hff generally required tw sepmate numerical approaches, one to compute the wwe elevation on the fi ee smfff e md a second to compute file viscous boundary layer mound ship hull The basehne equation to compute file fiee so face elevation is usually based on potential flow Leo y The numerical scheme c m use either a complicated Green fmmction or use file p mel method with simple Rmkme sources The governing equations to compute file viscous bommdffy layer on a ship hull su face me based on the Nwie~Stokes equations The mlm en cat medhods to compute viscous flow h we been developed fiom a simpli led boundary iayff mprox im at ion to a vat i ety of different lo mu l at i ons of N wier-Stokes equations in this p me' a combmed numerical techmique based on computing the free su face viscous flow ff und a realistic ship hull is developed it is based on sol ing the Reynolds Averaged Navier-Stokes (R NS) equations using a computational scheme that also calculates file water su face elevation generated by a moving hip hull on the fi ee su face boundary A other challengmg task is to be ff le to compute flow mound a su face ship wish mucous types of ship geomet y in addition to the necessary appendages required to operate a hip, designers have developed a variety of hull h Yes to meet file fmmctional requirements of a ship, eifLer for commercial benefits or miEtffy advff tages Trot som .-. -I ships me the focus of flus paper T msom .-.- ships have beff used for ! ems, especially for high- speed ship designs The hyd odynamic pfffommmce of this type of hip hff beff investigated m ruious model test facilities However, the flow ff und file t msom ten is ve y complicated md di hcult to compute due the discontinuity of file hull geomet y It is e pecially challengmg to compute file free so face elevation mommd file trmsom ten The issue of wet d yh msom condition is amajorconcem for a h msom-stenh ship design in flus paper, a numerical technique is developed to hff die this complicated flow phff amend The numerical technique mcludes a new gad topology for t msom .-. -I flow calculations in this way the wet d y t msom condition c m be computed A special topic of tree so face flow computation is to calculate file Linkage md tl im of the vessel The qu mtities of sinkas e md trim depend on the sh me of underwater geomet y md the ship peed Model test results have how a difference m measured resist mce between hoed models md free to sink/trim models Since file si kage md tam d Ha me not know in advance, file free su face flow

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computaion ha to caculae them a pat of the flow solutions The nmm~sica scheme in fLis paper includes this capability The vaidaion discussed m fLis paper is the compaison of computaiona results wifh a set of expenmenta model dma For fiee su fa e viscous flow caculaions, two series of experim:~ta meaurements a~e chosen, DTMB Model 5365 (R Ath:~a) md 5415 (a su fa e comba mt hull selected by ONR for CFD w~lidaion) The compaisons include tota resistmce coeffcients, smkage/tam, wave prohles, md wave paten~ m the fiee su fa e The numerica treaments for computmg fi ee smfa e elevaion md smkage/tam a~e discussed m detail The computaiona g id topology used to hmdle t msom tem geomet y is descabed, a well a gad spa mg, grid di tabution, md computaiona grid domain A limited grid st dy on flow computaions is briefly discussed NUMERICAL SCHEME Nmmerica caculaion of viscous flow aound a su fa e hip mu t mclude tw major physica mechmisms: computaion of the water elevaion a the fiee su fa e md computaion of fLe viscous bounday layer a und fLe ship hull Until tly, fLe fiee su fa e wave elevaion generaed by a moving ship ha been h mdled by the potentia flow approximaion, which asumes no iscosity The caculaion of fLe iscous bounday layer is pe fo med by fLe doublebody approximaion, which igmores fLe effect of the fiee su fa e Ba ed on this kmd of decoupled nmm~sica treament, the free su fa e viscous flow a ommd a moving hip c m be affordably solved The interation between viscous bounday lay~s a d fiee su fa e wavemaking is a sumed to be minima As for pra tica :~gineering ca cula ions, this a sumption rem ains va id to suppo t ship design effo ts However, due to recent advmces in CFD numenca schemes md computer CPU power, fLis decouplmg of the mmmerica medhodology may not be necessay One cm compute the iscous bounday layer md the free smfae wave elevaion together in m affordable ma ner This is a hieved by solving the R NS equa ions wifh ca ef I trea ment of free su fa e bommda y The fundam:~ta RANS equaions in fLe free su fa e flow computaions remain the same a those used for caculaions of doublebody flow, with fLe exception of the pressure te m which ha to mclude staic head fiom fiee su fa e bounday The numerica heament of double-body RANS computaions cm be gen~saly adopted in fLe free sufae R NS caculaions wifLout my modiEcaions The mmmerica scheme for fLis double- body R NS computaion in FREE9S ha been m developm:~t for yea a d will not be discussed m this paper Lin et a 1995, Lm et a 2000) The mam numenca effo t h~se is to h mdle the movement of the fi ee su fa e bounda y Sev~sa numenca methods have been developed to compute this hee su fa e bounday, md they cm generaly be clasibed m th ee ways: su fa e htting, su fa e t~ckmg, md su fa e capt rmg The effo ts of al these methods a~e to compute fiee su fa e elevaion by saisfymg bodh kmemaic md dynamic bounday conditions required in the free su fa e The kinemaic bounday condition forces water paticles on the fiee su fa e to remam m the bommday su fa e al the time The dynamic condition sai hes constmt amosphenc pressure on the fiee su fa e bounday F EE9S ha adopted fLe su fa e htting medhod developed by Famer (Fam~s et a 1993) A volume gad comprismg fLe computaiona domam is developed simila to fha for a double body caculaion, with the difference tha it extends abow fLe still wat~slme The Imes of this initia g id fha a~e no ma to the fiee su fa e a~e htted wifh B- plme cmves, which will be used to t~ck the fiee su fa e elevaion durmg computaions At fLe beginnmg of fLe computaion, the grid is moved to the still water line by redi tributmg the grid points aong the B-splme cm es The RANS equaions a~e fLen solved for this cunent g id Ba ed on fLe new computed flow w~riables, the wat~s su fa e elevaions in the free su fa e bounday ae updaed to saisfy both fLe kinemaic md fLe dynamic bounday conditions For fLe fully nonlinea condition, the fiee su fa e must move with the flow bounday md fLe bounday conditions must be applied on fLis disto ted fiee su fa e The water elevaion on the free su fa e is computed by integmtmg fLe fiee su fa e kinemaic equaion, which is denved by treamg the fiee su fa e a a maena su fa e requned by fiee su fa e kinemaic bounday condition Through fLe interior of fLe fiee sufae computaiona domam, al d~sivaiws a~e computed usmg the second order cenha differ:me On fLe bounday a second order centra stencil is used aong the bounday tmgent md a fust order one-sided diff OCR for page 443
ff possible it hff beff found Hat nmmfficai dissipation m file free so face computation hff signi ~cmt effect on file do per On of file wwe system Excessive dissipation may have better numerical conve gence, but gives a w ong phase mgle of the wave system Once the tree so face update is completed, the pressure is adjusted on file free su face The updated fiee su face serves ff new boundary values for file bulk R NS flow computations The coupling is ester li bed by computing a bulk flow solution md Glen using file bulk flow ff a boundary condition for the fi ee nxtfn e computation The tree su face elevation is updated md its present values are used ff a boundary condition for the pressure on file bulk flow This therefore completes a numerical iteration of free su face RANS flow calculation SINKAGE AND TRIM COMPUTATION A nxtsce ship moving at a con tant peed will End equilibrium m si kage md trim due to file balm ce of dy amic forces generated by the motion Model tests me pe to med in the same m mner; file model is fiee to sink md tam, md file values of smkage md tam me measured at different speeds The si kage md him chmges the underwater hme of the hull and the resulting hyd odynamic pe to mmce it is therefore impo tmt to include file smkage md h im in file computations The smkage md him cm be computed simultmeously wish the fiee so face R NS flow computations At each iteration, the forces md moments acting on the underwater geomet y me used to compute file ch retie in si kage md him th ough a simple hyd tatic calculation based on the hip's wate piece geomet y Using the mitial watffpime instead of calculating file cannot watemlme swes a sub tmtial amount of computations The assumption is the watemlme mea will not chmge sub tmtially due to si kage md trim md therefore this imposes a limitation The smkage md him is implemented by moving the hull (md the local grid) relative to file global gad, other ah m t ying to move the me m free su face reference All the fi ee nxtfx e elevations then need to be recalculated to fulfill both kinematic md dyn m ic b ommdffy conditions m the new fi ee surface locations The cm ent venion of F EELS c m activate the calculation of smkage md trim at file begmning or =.- a ce tam number of fiee so face RANS computations The results show in fLis paper me obtained by activating file smkage md tam fiom the bedimming of the computation md pe fomming one smkage md him calculation for es fix 10 iterations of free su face RANS computations HULLFORM GEOMETRY The tw hul fomms in this p Off were chosen for pled t msom .-.- sh me md file availed ility of model te t data Model 5365 R/V Athena) is a high- speed, h msom .'. -I dispiacemfft ship Model 5415 is a so face combat mt hull selected by OUR for CFD validation Dimensions for file hullfomms me show m table I md bodyplm plots of tenons 0 to 20 se show in figures I md 2 The only appendages included in either file compm.rons or file model tests me show in figures I md 2 Model 5365 hff a centerline skeg md model 5415 hff m integrated skeg md sonar dome Figmre I DTMB Model 5365

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Table 1. Hullform dimensions 1 5365 1 5415 Length (m) 5.960 5.724 Beam (m) 0.836 0.764 Draft (m) 0.183 0.248 CHIMERA OVERLAPPING GRID Grid Topology Free surface viscous flow computations involve two major physical parameters, Reynolds number and Froude number. To accurately compute viscous flow phenomena due to the effect of Reynolds number, fine grid resolution is needed in the boundary layer. Conversely, to resolve the free surface wave systems due to Froude number effect requires relatively fine grid resolution outside the boundary layer. The chimera approach has the capability to handle the necessary grid refinement to I've,. ~~:~\'~' of/' 'A ; / meet both requirements without using an impractically large number of points. The computational domain is divided into free surface and double body grid components, which are connected to each other with an interpolation interface. By decoupling the topology of the two grids, enough flexibility is gained to generate a suitable grid for a viscous free surface calculation. In addition, free surface flow computation around a transom stern is a challenge. Very large pressure and velocity gradients exist at the transom, requiring fine grid resolution and high grid quality. Furthermore, the free surface elevation moves up- and-down with the transition from wet to dry transom. Getting the computational grid to move with the free surface around the discontinuous edge of the transom is a topological challenge. The decoupling of the two grid components enables a suitable grid block topology to be developed. 'aft; ~ ~~] ~ Figure 3. Perspective view of overall grid topology .\ i.< \.. y \. / · \. \, ~ .. . / ~ / , , , ,

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A reverse "~" 1opology, ~ co~in~bon of ~n ~llo~nce ~ be m~de 10 provide room ~r 1be tree "H" 1opology ~11be bo~ ~nd ~n "~" 1opology ~11be sur~ce grid 10 redi~ribute ~self ~s ~ moves ~ ~nd stern, ~s used ~r 1be grid ~1 1be tree sur~ce do~ 1be ~1L In order 10 si~li~ 1be do~le bo~ bound~. Tbe "H" 1opology si~liEes seHing 1be grid, 1be son~r dome on model 5415 ~ modeled ~ilb -~re~m bound~ condition ~nd 1be "~" 1opology ~ ~ 1b~d grid system. A 1r~nsverse cu1 of ~11 1bree grids e~cien1 ~nd provides ~r ~ rel~bvely uni~rm, ~1 s1~ion 1 ~ sbo~ in Egure 4. rec1~ngul~r grid ~round 1be stern ~ere grid qu~li~ is Tbe mos1 co_on ~pro~cb 10 "ridding mos1 crhic~L In 1be 1r~nsverse pl~ne 1be grid s~eeps R~nsom ~ern ship b~s been 10 ex1ode ~ block hom do~ hom 1be tree sur~ce ~round 10 1be pl~ne of . . 1be b~nsom ~ce ~R 10 1be outer bound. Amongs1 1be s~e1~ 1n "~" ~shlon. Fo~d of 1be stern 1be . . . . , m~ 1r~de-o~ of 1bls ~pro~cb, 1bere ~re ~o m~or CO-u1~08~1 dOm~lD 1S spl~ equ~lly be~een 1ne . d~dv~n1~ges ~r ttee sur~ce co~u1~bons. tree sur~ce ~nd do~le bo~ grid co~onents ~long . Conb~ing 1be bound~ l~er sp~cing ~R o~ 1be 45-degree line. In 1be ~R region 1bls dlvldlug 45- . . . . . . . 1r~nsom results 1n extreme v~rl~1lons 1n cell sl~e ~nd degree bue 1S ro1~1ed ~ou1 ~ vedlc~1 Q\1S Q1 1be stem . dls~lbutlon. Tbe second m~or dls~dv~n1~ge 1S n order 10 ~0~ 1bC "~" 10pOlOgy. A perspective . . . 1be b~nsom block b~s no~ere 10 move 1n 1be c~se of vle~ of 1be over~H 1opology ~ sbo~ 1n Egure 3. In 1r~nsom Tbe "~" 1opology in co~unction ~ilb bis ~nd succeeding grid Egures only eve~ 2nd poin1 . . . chimer~ grids solves botb of 1bese problem~ Tbe cell sbo~ ~r cl~n~. Abboupb 1b~ Egure depicts 1be . sl~es ~re ~lrly unl~rm ~nd 1be dlstrlbutlons ~re and ~r model 5415 ~ 20 ~o~ 1be 1opology ~r ~1 . . . . ' smootb. Flgure 5 1S ~ blo~-- of 1be stern ~re~ in 1be grids 1n 1bls p~er 1S ldOD11C~l. Figure 3~ sbo~ing jus1 1be tree sur~ce block. Tbe grid hnes 1b~1 move norm~110 1be tree sur~ce R~vel do~ 1be 1r~nsom ~nd ~rd under 1be ~lt 1~s ~ ~ ~ ~e ^e ~ ~ ~ s~ ~ ~d 2~/ ~/ ~ / ~ - ~ - - ~ - - - - ~ - - - - - -!- - beneE1 of 1bis 1opology is 1b~1, bec~se of its rel~1ive ~,~ ~ ~ ~ ~ ~S,~,~: B~ ~ ~^~^~ ~ ~ ~ \~ ~ ~ ~ ~ ~ ~/ ~ ( [~1eatofDom~la ~ad Crid D1stribu110a, S1ze Figure 4. A 1r~nsverse cu1 of 1be grids ~1 S1~1ion 1 To co~ute ~ccur~1ely 1be ~ve system gener~1ed ~round ~ moving ship, especi~Ny 1be . . . diverging ~ve system requires ~ sui1~le dom~in Tbe grid on 1be ~11 sur~ce 1S Spll1 be~een ' . . ~nd grid reEnemen1 in 1be tree sur~ce. ~orm~lly in 1be ~o grid systems ~1 ~pro~lm~1ely b~lf 1be ~e11ed ' . . do~le bo~ ~S 1be di~nce 101be outer bound g~1b lengib Ide~lly 1be do~le bo~ grid ~ould ' of 1be co~u1~bon~l dom~in is b~sed on 1be ~11 enco-~ss mos1 of 1be ~H sur~ce. Ho~ever, some

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length For fiee so face computations, lager Froude numbers me m longed wavelengths m the wave system, which requite lager computational domains to cover several wave components of file generated wave system it wa decided to set the dist mce to file outed bound at 2 chaacteri tic wavelengths Lw = 2vFr: L) in m attempt to define the grid resolution it wa decided to set the ma imum dimension of my cell wifLin one wwelength of the hull to be no la tiff thmLw/20 md as cell my here m be no la=er~hm Lw /10 20 points hould be enough to satisfactorily define a wave md 10 pomts hould give a rough deEmition at the outer bound The most difficult dimension to achieve flus criterion is no mat to the hull, where the boundary layer requires e tremely mall spacing Tw approaches were tried using different dishibution fmmctions for file fi ee smfa e block The hat gad for Model 5365, hereafter retorted to a G id 1, used a smgle tw -sided hyperbolic tmgent dishibution (TANH) h om the hull to the outer bound The mitial spacing no mat to file wal wa si ed to y+ value of 10 The growdh rate of the grid pacing wa a con t mt 10% for file Bust wavelength away fiom file hull md then law, mptoticaly approaches 0% at file outer bound, where the gad spacing is Lw /10 The effect of this di tabution is to concentrate a large Faction of the total the points inside file boundary layer A single tw -sided Monotonic Rational Quadratic Spline distribution fiom the hull to file outer bound wa used for subsequent gads of Model 5365, refened to a G id 2, md for bodh 5415 grids The initial spacing nommal to the wall wa sized to y+ value of I The mitial grow h rate of the grid off file wall is exLemely large md this results in a more unifo mly spaced g id, with fat fewer points Aside the bounda y layer For file double body blocks in all gads a TANH distribution wa used no mat to the hull Gnd I wa intended to be relatively con e a d file dimension on file fiee so face nommal to file hull wa chosen to be 61 The mproximate total number of grid points wa 500,000 Grid 2 md bodh 5415 grids were si ed to 111 pomts md reamed in file other dimensions a well The mproximate total number of grid points wa 1 3 million for each gad EXPERIMENTAL DATA Model 5365 wa tested m the bare hull condition ~ ah the centerline skeg in place Model tests were condu ted at tw different facilities, one at David Taylor Model BE m (Jenkins 1964), md file other at the National Maritime Instit te (Gadd md Russell 1961) The model emeriments were conducted m two modes: ~ ah file model fiee to sink md trim, md with the model rigidly resh amed at file static d afl Total resistmce of file model wa mea wed with file floating girds Hat is attached to the c or age The bow md stem Linkage of the model were mea Red with two displacement transduced that were mounted at the bow md -. n of the model The wave elevation along file side of the model wa reco ded mamally, using a grew e pencil to mark file water height at 21 locations Subsequently, file di tmce between the marked heights md file cam water so face wa measured, taking into account file Linkage md him of the model b addition, a set of mea moments of the .-. -I wave elevations wa made on Model 5365 in the hoed condition a a Froude number of 0 46 The .-. -I wave heights were obtamed using thm rods fha were ma ually adjusted until then tips just touched the water smfa es, once the model reached a steady stale condition The accuracy of file mea red total resi tmce wa repo ted to be +1 5% No sy tematic unce tamty malysis wa done a file time of flus experiment The experimental data for 5415 wa collected during several tests pmning mmy yea (Radcliffe 1999) The original bare hull resistmce tests were done in 1962 Around 1990 mea rements of the fiee smfae wwe heights aommd file model were obtamed using tereophotoghvmmehic techniques Tw Ha selblad met tic still camera were mounted on a ceding mounted hat re above file Carriage I bat in As the model wa towed doss file bat m, sequences of photog mhs were taken, sy chronized by trobe lights The water smfa e wa "seeded" using computer punch cat d chips fha .: e di tributed on file so face before each m A array of calibraed target gad pomts pa endowed file free su face m the fat held These grid points were also imaged in the stereo pans a d served a control points Ming the malysis of file photographs The resultmt -. .o photog mhs wise subsequently "measured" by the Naiona Ocem Sm e! over a 76 2 mm X 76 2 mm grid The da a w ff obtained a model speeds of 4 01 md 6 02 kmots, cohespondrg to Fh ude numbers of 0 26 md 0 41 Subrace wave height me inurements wed made on the bow md .-. - waves r 1996-7 The mea Hem wed obtamed on file ha e hull model off d We berg towed r the Carriage 3 Towing Baler For these mea rements, a dy amic wave height probe called a "whisker probe" We used At each da a collection point, x, !, md z Adzes were recorded

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via a voltage readmg from the probe md traverse encoded The him conditions for file tw peed were repo tedly file same a the 1982 test Wave profile mea moments Song file length of the hull were obtamed on file bare hull model during emeriments run in 1997 The model wa fiee to sink md tam a en h peed The wave profile truces were d a on file model md reco ded wifh relerence to the diva waterlme The tam conditions for file tw speeds were repo tedly the same a file 1982 te t The whisker probe wave data wa combmed wifh file stereophotogrmbic data by the whop to obtam a more complete pict re of file wave patem Due to male unkmow s file accmacv of flus data c mnot be qu mtital ively established RESULTS AND ANALYSIS Model 5365 Flow computations using Grid I ale pe to med for Fn = 0 35, 0 4g md 0 65 m hxed condition it is obvious this Gnd I is too smal for Fn = 0 65 a d magina for Fn = 0 48 Flg ire. 6-8 how the wave patems using G id I for these th ee Froude mlmbeh m a hxed condition These wave prlems aso show file la held bounday condition is well implemented a file Mar boundary of file computations domain Computed wwe profiles Song the hull for these fEree Froude numbers se compa cd with expenmenta mea ured da a show m Flg n e. 9-11 Flow computations ale pe to med on Gnd 2 for Fn = 0 '8 0 35, 0 4g md 0 65 in both hxed md sink/trim modes Wave patems on these fEree grids for fom Froude numbers a e show m Figures 12-15 for hxed condition md Figures 16-19 for smk/trim condition Since no mea nemem is available, these wave pa -. ~ e how to exam me then rela ionship wifh a wide retie of different Froude mlmbeh quaitaively Compamg with wave pat:ns fiom Gnd I caculaions, only minor differences are found except that file wave patem in G id I is trunca cd by the outer bounda y Only m mor doff erences a e found on the wave patems between hxed md sinktmm conditions The computed wave profiles ale compa cd wifh the expenmenta mea urements m both hxed md silLWtrim modes, which ale how re pectively in Figures 20-27 Generals the wave profiles new bow area a e well predicted compa ing wifh both sets of expenmenta data The bough of wave profile a the mid ad po tion of the hull is predicted consi tent with Gadd's expert enta data md under predi ted compa mg with Jenkins's da a A compa i on of the mea red md computed .-. -I wave elevations in the hxed condition a Fn = 0 48 is show m Flgn e. 28-29 The computed elevations fiom Gnd 2 how better conelaion with expenmenta data ah m G id I Since the meatt ed data is archer limited, file comparison here is only qu mtitaively -and ted Forces md moments are calclllaed Song wifh computations of flow va iables m the process of flow computations They a e obtamed by integmting the shed stress md pressure on file wa I smfa e The total resist mce of a smfa e ship is flus computed a the x-componem of total mtegraed force a ting on the ship wetted hull su fa e Table 2 how file computed total resista cc compa cd wifh available mea ured da a for Model 5365 computation For file hxed condition, bodh calclllalons on G id I a d Gnd 2 ale mostly over-predi ted compa Id with Jenkms's measurements, while file result from G id 2 computation is closes to file model meatt ed data However, the prediction treed is consi tent wifh expenmenta t end b Table 3 file computed results for sink tam condition conelae well wifh the model test data with file exception of Fn = 0 65 in Tables 4 md 5, file computed Linkage md him forModel 5365 computation using Gnd 2 are listed Song wifh file model measured data it is found that both sinkage md trim cumputatiuts have consistent trends wifh model experimental data An mcreaed positive tam (bowup) is obtained trom Fn = 0 28 to 0 65, but file magnitude is over-predicted The Linkage caculaions are well conelaed in magnitude wifh experimental data a low peed, but over-predict a both higher Froude ma hers Model 5415 For model 5415 caculaions, two computaiona gads were generated for tw different ship speeds, 20 md 30 knots Only the caculaion on the hxed-a-fLe-mea ured-smk/trim condition is pe to med. The computed wave prtems for both ship speeds are show m Figmres 30-31 Song wifh file meatt ed wave contours b genera, computed wwe prtems a e consistent with the mea ured patems for both bow md stem wwe wstems The pha e mgle of both wave wstems is well conelaed The mamit de of wave elevation is however under-predicted a d file wave crests md troughs are not a sham a mea Ned data This may he due to file problem of having not enough grid Solution r those Alan of cre t/tmugh tom awns The wave pmEles Song file hull ale

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shown in Figures 32-33 for both speeds respectively. They are well correlated with experimental data, except being under-predicted in the bow wave elevation. The trend of wave elevation profile is well predicted. The total resistance for both ship speeds are shown in Table 6 and correlate well with the model measurements. CONCLUSION Computations of free surface viscous flow around two transom stern ships are performed to investigate the capability of a numeric scheme developed for surface ship hydrodynamic performance prediction. The inclusion of free surface boundary treatment into the viscous flow computation is first described. The technique to handle wet/dry transom computation is developed, which includes a new grid topology for transom stern flow calculations. The capability to compute sinkage and trim for a moving surface ship is important, which is one of the focuses in the paper. Computational results on two transom-stern ships are consistent with available model experimental data. The shape of the wave profile along the hull is generally predicted well, while the magnitude of wave elevation still needs to be improved. The wave pattern is also well correlated, except that wave crests and troughs are not as sharp as the model measurements. The computed total resistance is in good correlation with model-measured values. Although two different types of grid spacing and distribution techniques are investigated, a further study is needed to determine a more effective grid system for free surface viscous flow computations. In addition, more systematic comparisons between numerical flow computations and model experimental measurements are needed so that the applications of numerical flow tools into ship design effort can finally be accomplished. REFERENCES Lin, C.W., Percival, S., Gotimer, E.H., "Viscous Drag Calculations for Ship Hull Geometry", Ninth International Conference on Numerical Methods in Laminar and Turbulent Flow, Atlanta, 1995. Lin, C.W., Percival, S., Fisher, L., "Validation of Computational Forces and Moments on an Appended Body", International Maritime Association of Mediterranean IX Congress, Italy, 2000. Farmer, J., Martinelli, L., Jameson, A., "A Fast Multigrid Method for solving the Nonlinear Ship Wave Problem with a Free Surface," 6th International Conference on Numerical Ship Hydrodynamics, Iowa, 1993. Jenkins, D.S., "Resistance Characteristics of the High Speed Transom Stern Ship R/V Athena in the Bare Hull Condition, Represented by DTNSRDC Model 5365," DTNSRDC-84/024, June 1984, David W. Taylor Naval Ship Research and Development Center, Bethesda, MD. Gadd, G.E., & Russell, M.J., "Measurements of the Components of Resistance of a Model of R.V. 'Athena'," NMI R1 19, October 1981, National Maritime Institute. Ratcliffe, T.J., (1999) "Model 5415," i, (1 May 2000). Table 2. Comparison of Resistance Coefficient for Model 5365 at Fixed Condition Froude # 0.28 0.35 0.48 0.65 Jenkins 4.774 4.239 4.437 4.219 CT X 1000 Grid 1 Grid 2 5.102 4.795 5.085 3.784 n/a 5.428 5.987 5.727 Table 3. Comparison of Resistance Coefficient for Model 5365 at Sink/Trim Condition Froude # CT X 1000 Jenkins | Grid2 0.28 0.35 0.48 0.65 5.531 5.030 5.774 4.924 5.432 5.020 5.516 3.962 Table 4. Comparison of Sinkage for Model 5365 Froude # ATm/L x 100 Jenkins | Grid2 0.28 0.35 0.48 0.65 0.105 0.200 0.245 0.095 0.135 0.209 0.389 0.25 1 Table 5. Comparison of Trim for Model 5365 Froude # | (ATf- ~Ta)/Lx 1OO I Jenkins | Grid2 0.28 0.35 0.48 0.65 0.060 0.095 1.240 1.760 0.224 0.392 1.249 2.250 Table 6. Comparison of Resistance for Model 5415 Froude # CT X 1000 Ratcliffe Computed 0.28 0.41 4.14 4.541 7.01 7.218

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Wave Height/(L*Fr2) -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 1 ~ ~> J To 1 x/L 2 Fig 6. Wave Pattern for Fr # = 0.35 Wave Height/(L*Fr2) -0.05 -0.03 -0.01 0.01 0.03 0.05 ~ ~ \ 1 1 1 1 X/L ~ Fig 7. Wave Pattern for Fr # = 0.48 Wave Height/(L*Fr2) -0.04 -0.02 0.00 0.02 0.04 1 \ ! ~ 1 x/L 2 Fig 8. Wave Pattern for Fr # = 0.65 0.15 1 ~ 0.1 LL —0.05 . _ I O > ~ -0.05 OR 1'-' Fixed Condition · Jenkins ------- Grid 1 — ~ — ~ , ~ , 0 0.2 0.4 0.6 0.8 Distance from FP, x/L Fig 9. Comparison of Wave Profile for Fr # = 0.35 0.15 1 ~ 0.1 LL —0.05 .= I O > ~ -0.05 Fixed Condition · Jenkins 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 10. Comparison of Wave Profile for Fr # = 0.48 0.15 ~ 0.1 LL —0.05 . _ I O > ~ -0.05 Fixed Condition · Jenkins ------- Grid 1 I_ 0 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 1 1. Comparison of Wave Profile for Fr # = 0.65

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Wave Height/(L*Fr2) 0.15 1 -0 08 -0 04 0 00 0 04 0 08 ~ 0.1 LL —0.05 I O > -0.05 Do 1 x/L 2 Fig 12. Wave Pattern for Fr # = 0.28 Wave Height/(L*Fr2) \- · · ~ Fixed Condition · Jenkins Grid 2 1 0 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 20. Comparison of Wave Profile for Fr # = 0.28 0.15 1 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 t~ 0.1 —0.05 . _ I O > -0.05 ../ ~ O. I` 1 x/L 2 Fig 13. Wave Pattern for Fr # = 0.35 Wave Height/~*Fr2) 21 - 1 -0.05 -0.03 -0.01 0.01 0.03 0.05 ~ of\ ,., ..... ...... ~ O. ~ 1 x/L 2 Fig 14. Wave Pattern for Fr # = 0.48 I'm- - ~ ~ ~ ~ ~ ~ I ~ ~ ~ I ~ ~ ~ I ~ ~ ~ I 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 21. Comparison of Wave Profile for Fr # = 0.35 0.15 ~ 0.1 LL —0.05 . _ I O > ~ -0.05 Fixed Condition · Jenkins — Grid2 · - - — i... it. Fixed Condition · Jenkins Grid 2 ·,m, 1 0 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 22. Comparison of Wave Profile for Fr # = 0.48

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Wave Height/(L*Fr2) -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 2 o( ) 1 2 x/L 3 4 5 Fig 15. Wave Pattern for Fr # = 0.65 Wave Height/(L*Fr2) 0.15 1 ~ 0.1 LL —0.05 . _ I O > ~ -0.05 Fixed Condition · Jenkins Grid 2 0 0.2 0.4 0.6 0.8 Distance from FP, x/L Fig 23. Comparison of Wave Profile for Fr # = 0.65 0.15 1 -n nP -n n4 n nn n n4 n nP ~ 0.1 LL —0.05 I O > ~ -0.05 ~ . . . . ,. 1 x/L 2 Fig 16. Wave Pattern for Fr # = 0.28 Wave Height/(L*Fr2) -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 ,:;~ x/L Fig 17. Wave Pattern for Fr # = 0.35 . Sink&Trim Condition Jenkins Gadd&Russell — Grid2 ~ . \ . · · · - . · ·- ,r' ~~ · . ~ — 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 24. Comparison of Wave Profile for Fr # = 0.28 0.15 ~ 0.1 LL —0.05 . _ I O > ~ -0.05 Sink&Trim Condition Jenkins ~ Gadd&Russell /) Grid2 I \ A\— 1 ~ . l 0 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 25. Comparison of Wave Profile for Fr # = 0.35

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Wave Height/(L*Fr~) 0 1 AL 2 3 Fig 18. Wave Pattern for Fr # = 0.48 Wave Height/(L*Fr2) 0.15 0.1 LO —0.05 . _ I O > ~ -0.05 Sink&Trim Conditiol. Jenkins Gadd&Russell Grid 2 . _ - ~ . \~ · :,__ ~ ~~ i 0 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 26. Comparison of Wave Profile for Fr # = 0.48 0.15 1 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 ~ 0.1 \- ~ ~ 0.05 .= I O > ~ -0.05 n 0.06 I, O 0 03 Sink&Trim Condition Jenkins Gadd&Russell — Grid2 . ~ ma_ ·~ ~. ...... * ~ t ~ ~ · ~ ~ 0 1 2 x/L 3 4 5 Fig 19. Wave Pattern for Fr # = 0.65 ~ . ~ !·····- _~ 0.2 0.4 0.6 0.8 1 Distance from FP, x/L Fig 27. Comparison of Wave Profile for Fr # = 0.65 L*FK) ~040 0.030 0.020 0.010 0.000 -0.010 -0.020 -0.030 -0.040 1 , ~ ., mu. 1 -0.050 1 1.05 C/L 1.1 1.15 Fig 28. Comparison of Wave Pattern for Fr # = 0.48 I, u 0 03 a'\ ' ' J 0 004 -0.004 1oo36 -0.044 0.067_ _ - ~~ >~> ~ -0.052 1 1.05 C/L 1.1 1.15 Fig 29. Comparison of Wave Pattern for Fr # = 0.48

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0.6 0.04 0.00 -0.04 I:: ~ -0.6 0.6 n -0.6 Wave Height _ z/(L*Fr2) 0.12 0.08 ~ Ratcliffe 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0.5 1 X/L 1.5 Fig 30. Comparison of Wave Pattern for Fr # = 0.28 Wave Height _ z/(L*Fr2) ~ ComDuted 1 1 2 0.12 aim ~~ r---- ~ /~) Jut ~ .. ~ .... ~ . ~ O 00 ~ ~ . ~ . Ratcliffe 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0.5 1 X/L 1.5 Fig 31. Comparison of Wave Pattern for Fr # = 0.41 1 1 2

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03q 5415 Fate ffe 28 Computed 0 2- i\ 0 1- \- 0 2 0 4 0 3 0 8 1 D stance from FP x/L Fig 32 Comp of Wave Profile forFr#= 0 28 u \ 01- ~ 03~ 5415 e Fate ffe41 — Computed 0 2- 0 1- W. . . ~ o ~ 1 0 1- . . = 0 02 04 03 08 1 D stance from FP x/L Fig 33 Comp of Wave Profile forFr#= 0 41

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DISCUSSION Gzbor Kzrafizfh Naval Surface Wzrfz~e C nter, Czrderock Div:, USA I The mthors must be congratulated for devising the ingenious griddmg medhod Fiat enables the combmed free surface viscous flow computation This methodology was also used by He mthors to perfomm free surface viscous flow calculations to study the flow behind z stern flap md help co fimm the scale effect Fiat occurs between model size md ship size barafiath et al 1999) in that particular case, the use of the Chimera gridding was z great asset with regard to the efhcient mmmerical geometry definition of She stern tarp 2 in ener31, She cunent commercial practice associated with z ship hull fomm development is to use either the free su face potential flow computation or the double body RANS calculation for viscous flow The combined free surface viscous flow calculation, either She one presented by the mthors or one of the f w of her emerging he surface viscous flow codes, is generally not used becmse of the newness of She codes md he mse of the associated extent co t of She calculation Could She mfhor's comment on the turn around time, grid preparation md calculation effo t that is reqmred for the combined fiee surface viscous flow calculation? 3 Given that Here is z somewhat greater cost for this new calculation method, what do we gain? Could we see z comparison of the wave height prediction relative to the equivalent tree surface potential flow code prediction Ed also z comparison of She boundary layer fhickmess rektive to the prediction from She double body RANS code? 4 in Figure 30 md 31, we see z comparison of the predicted fiee surface viscous wave field to the model measurements using the whisker probes As characteristic of mmy similar comparisons, the predictions are very smooth in nature whereas the model measurements have z great deal of high frequency content that is not captured m the prediction My observation of mmy model tests conducted in very still calm water is that the flow field around She model is in reality m unsteady flow with z significmt temporal flow variation tendency near She tr msom md that the high frequency content in She wave field is real Could the mfhor's comment on She resolution of this problem of z steady state prediction for phffmmenz that has some temporal variztmn? Kzrafiath G. Cusanelli,D S. tnd Lm, C, W. "Stern Wedges md Stern Flaps for mproving Powering U. S. No y E perience" T msactions of the Society of Naval A chitects md Mzrme E gineers, 1999, Bzltimore,Mzrylmd AUTHOR'S REPLY DISCUSSION: Y. Tzharz Oskskz Frefectme University, Jzp m (I ) As the mfhors mentioned in the paper, Most recent high- peed fine ships as w 11 as Model 5415 have t msom stem in order to obtain wide waterplane area to secures fficient stability H wever,th widetrmsoms tend to Increase disturbance on t msom wave fields, md that results in increase of hull resistance The present mfhors md of hers Kawasaki et al, 1996; Tzharz et al, 1997) had carried out investigation on tr m, m flow md wave fields using computational md experimental models in the work, it appeared that tr msom wave field c m be classffied as the following 3 types: (A) wish dead water zone right after stern end; B I with no dead water zone, but wave Mel m g in near wake region; md (C) with neither dead water zone nor wave breaking in near wake region, i e, free surface is smooth y continuous from She stern end in your paper, the above are simply retorted to d ylwet conditions My question is how accurately your mmmerical method c m predict She above (A) th ough (C) for ship models considered m your work (2) For She zbove-mentioned type (A) tr msom wave condition, She signffic mt bubble end Pi merit is usually observed in the measmements The effects must be included in tree-cm t Ice boundary conditions m order to accurately predict the wave field Cunently, inclusion of the effects may not be focused in your work; how ver, I would like to Mow if you have prospect or suggestion for feasible mmmerical treatment to include the effects REFERENCES: Iwasaki, Y. Tzharz, Y. Okuno, T. Himeno, Y. md Yam mo, T. "Studies on Re ktionship between Water Surface behind Stern md Stern E d Fomm of Fine Ships," J. of Society of Nils Pi Architects of Jzpm, Vol. 130,1996,pp 13-20[Jzpanese]

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Taharc, Y. md Iwasaki, Y. "A St d. of Tr msom- Stem Free-Su face Flows by 2-D Computatiom~l md E perimentclModels,"J F.cnssi Society of Ncs-cl A chitects, No 227, 1997, pp 7-19 [Jsp mew]; also, Proceedings of She 2nd Co terence for New Ship & Marine Techmology into 21 st C ntury, Hong Kong, June 1995, pp 53-92 [E glish] AUTHOR'S REPLY None received

Representative terms from entire chapter:

surface viscous