Twenty-Third Symposium on Naval Hydrodynamics (2001) / Chapter Skim
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The Numerical Simulation of Ship Waves Using Cartesian Grid Methods
The Numerical Simulation of Ship Waves Using Cartesian Grid Methods
Pages 762-779
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From page 762... ...
to mod I he fiee-surface interface A body-fmce tech iq~x is used to impose She hull bo mdary condition The predictions of bodh m mericcl tech iq~xs are compared to whisk r-probe mecsurements of She DDG 5415 The level-set tech iq~x is also used to investigate the breakup of c two-dimffmsiorul spmy sheet 1 Introtlttcbott At moderate to high sped, She turbulent flow along the hull of c ship md behind the stem is characterized by complex physical processes which involve breaking waves, air entrainment, fiee-surface turbulence, md She formation of prey Irsdmonal m meri al mproahes to these problems, which use bo mdary-fltted g ids, me di flcult md time-consummg to impleme t Also, es wakes steepen, bo mdary-fltted g ids will Ixek d wn Bless cd hoc h ectments me implemented to preve t fhe wa~s from g tting too steep At fhe wxy lest, c bridge is req ired betw en potenticl-fl w medhods, which mod I limited physics, md more c mplex bo mdary-fltted g id methods, which mcorpomte mme physics, clbeit wifh g ect effo t md with limitations on the wave steepness Cartesi m-g id methods me c m~tural choice be mse fhey all w mme compl:x physics thm potential-fl w medhods, md, unlik bo mdaryfltted medhods, cartesi m-g id methods ~equi~e minimal effo t wifh no limitation on the wave teepness Although cartesim-g id methods (CGM are p~esently mccpable of resolving fhe hull bo mdary-kyer, CGM c m model wa~ b~eckmg, fre-smface tmbulence, cir ff~t~aimment, spmy-sheet fommation, md complex interactions betwen the ship hull md the free surface, such es t msom-stern fl ws md t mblehome bows The cartesi m-g id methods fnat are descobed m fhis pcper use the pmelized ge meby fnat is used by potenticlflow medhods to cutomatically con truct c represent~ tion of the hull The hulMepresentation is fhen immersed inside c cartesim g id that used to tmck fhe interface No cdditiorul g iddi g beyond what is clredy used by potenticl-fl w medhods is ~equi cd We note that mother variction of this cpprocch is to use cartesim-g id medhods to hack the fiee-surface interface md body-fltted g ids to m odel the ship hull For the cclcoktion of ship waves, VOF md levelset methods have cc tam cdvmtages md disadvmtages VOF uses the vol me fraction (F) to h cck fhe mterface F = 0 corre ponds to gas md F = I corre ponds to liquid For intermedicte values, betw en ero md one, there :xists m interface betw en the gas md the liquid The interface betw en the gas md the liquid is sharp for c pure VOF medhod L vel-set methods use c levelset f mction (f)
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From page 763... ...
The levelset interfae-hscking slgorid m uses a new isoturfae sch me to calculate She zm3 levels t Then th minimal distance betw en She cartesim pomts ad the zero levels t is calculated in a narr w bad The minimal di tance is made positive m the water ad negative in the sit This sigmed dista e to the free surface is used to reimtiali:D: She thickmess of She interfae The two m merical reproaches ah used to simulate the flow pro Ed the DDG 5415 The CLS technique is still mder development, so only preliminary results ah presented The level-set technique inchdes upg odes to the m meri 91 technique that is det Ned m [8] Those upg Ides include a new body-fcrce formulation dint is mollified, a new h initialization procedure, md a new flmite-vol me hestment of the convective temms The original m merical procedure is not mollified md does not use remitislizati m in addition, She original cenbaldifference formulation of She convective terms is not as robust as the new to stment using a flux integ al formulstion We -i st revi w the governing equation md then we die uss She m meri 91 spproahes Finally, w present some pa lim inary m merical results which illustrste various festmes of the m merical slgorid ms The application of level-set methods to She be tkup of spray sheets is also illu orated 2 Field Equations As in Dommermuth et al, (1998)
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provides z Poisson equation for the dynamic pressm e ~ I SPa go\ P be\ the velocity onto z solenoidal field (11) whereZ is z somceterm Equation II isusedto moiect 3 Enforcement of Body Bounditry Conditions Two different cartesi m-g id methods are used to simulate She flow arommd the DDG 5415 The -i st tech iqm~ imposes She no-fl x boundary condition on She body using z flmitewolmme tech iqm~ The second tech iqm~
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d~ \+~/2,~,u r +~ F\+r/2,~,u r presents the lef fae of z comput~ tional element; similar deflmitions zpply to r\ r/2 ~ u r\,>+~/2,u. In or der t o discr te Iy e f orce 6he b o mdary c ondit ions (16)
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The zdjustment f mction reduces th gen mtion of nonphysical high-frequen y waves As constn ted, the velocities of 6he pomts withm 6he body are forced to vmo For z body 6~t is fixed in z fiee sheam, 6his corre ponds to imposmg no-slip bo mdary conditions 4 Interface Tracking Two methods me presented m our work for computing ship fl ws Both medhods use z "fiont-captmmg" typ procedme for representmg 6he flee surface sep~ rating the zi md water The fl st techmique is based on 6he Coupled vol me-of-fluid md level set medhod (CLS) md 6he second techmique is based on 6he level set method LS)
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From page 767... ...
follow d by She "x-swep" (done explicitly) he scalar flux s\+~/2,> is computed differently de pending on whether ~ represents She level set function f or She vol me fraction F For She case when ~ represents the level set function f w have the foll wing representation for s\+~/2,> (U\+~/2,)
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, w have ~ ~d V V: In other words, i formation propagates nommcl to 6he geomet y su fae For contat mgles dfffe~ent fi om 90 deg ees, 6he folI wing procedu e is tsken to flmd ue~ d: Vf Vf V: nw~ll = Vi,, n~ = n x nw~ n x nw~ll nI x nw~n n2 = nI x nw~n c = n n2 Remarks: ~ n~,, GOfl1 8) n2 ffc < 0 ~ d I n~,, GOt c~n2 ., = ~ ~,,+GOI Wr ff c > 0 I nw~+cot ~ s~n2 ~ nw~z ffc = 0 o ~ 3d, the contat line (CL)
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From page 769... ...
13 (35) Simibrexp~essionsholdalong6hef th mdj thmdices Repeated zpplications of weighted avemges provide z nnr w b md that en ompasses 6he :D:ro level-set Thenarrowbmdcorre pondstoth ~egion T\,~,u < I The sig ed di tance f mction D is expressed in terms of th level-set f mcti m md the mimmal distance: D = sign(f)
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6 Preliminarv Results In section 6 1, w prese t prelimi By computations of -I w past z DDG 5415 ship ~ section 62, w present preliminary computations of She breakup Of Z two-dimensiomtl prayshet 6.1 Ship Wave Results As z demonstration of She level-set md She coupled level-set md volume-of-fluid fcrmnlations, we predict the flee-su face di tu bance near She b w of She DDG 5415 moving with forward speed The experiments w re performed It She David Taylor Model Basin DTMB) , md me available viz the world wide web at http ://wwwSO dt ma y m ii/5415/ This is the same flow that Dommermuth, et al, (1 998)
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From page 771... ...
md She fine resolution uses 512 x 128 x 128 g id points The length, widdh, md height of She computational domain are L = 2, W = 0 5, md 11 = 0 5, respectively The water depth is d = 0 25 Th g id Pacing is constant zlong all th ee cane i m axes ~ the next phi He of our research, we will implement g id shetchmg, which will all w g eater water depths to be simulated The d u ttions of the CLS simulations are t = 0 75 Unlike She level-set results, th CLS re itsf ts extend the fiee-surfae i terfae into She h al using She tech iq~xs o timed earlier m our paper The fre-smfae elevation was measured at DTMR wasmmexpeotedlf~ctdow for fi e days of m mtenmce~m before fi is ~ Per was due using z whisk r probe Tw nty-one h traverse cuts were pe formed near She bow, extendmg fiom ~ = 0 to ~ = 0 178 in dimensionless mits The whisker probe measures the highest pomt of She free surface in regi ms where here is wave t Gel i g, She whisker probe m easures the t op of She be eaking wave Seventeen h msverse cuts w re performed m th stern, extending fr m = 1 01 to ~ = 1 22 Figures 3 md 4 compare measurements at She bow md stern to the m me f ii tl predictions The b w mew smements include profile md whisker-probe measurements Comparisons to the b w data are performed at four tations: ~ = 0 0444, ~ = 0 0622, ~ = 0 0800, md ~ = 0 0978 The circular symbol denotes profile measur merits The solid black lines denote the outline of the hull md the whisker-probe measurements The solid blue Ime is medi m CLS md the dashed blue Ime is coarse CLS Th solid ff d Ime is medi m level-set md the dashed red Ime is coarse level-set in general, th CLS technique captures th mpid rise up the side of the hull The level-set technique does less w 11 in this regard ~ the outer-fiow region the CLS coarse results are slightly better th m the CLS fine res dts This may be zth fluted to the shallow depth that is used in th CLS The level-set results appear to converge bettef in the outer-fi w r gi m, but She re mlts of fihe fine simulation f re requi ed for confirmation Figure 4 sh ws the enti e flow f mumd fihe ship for th medium resolution level-set simulation The ste m whisk r-probe mea smements are Merle id for fihe pUfposes of comp If if on Although the m mericaI results are not sfatiorrtfy, the shape of fihe stem contours sh w gee f a I a g cement with Dora tory measurements How ver, the f mplitude of fihe m mericaZ remits f f e signifi mtly low r thm the measurements Note that fihe stern is pa ftia lly d y m the m mericaZ simulations The cull if f of fihe hull is visible m the m merica I simulations because the If l-set fumcti m intersects fihe hull 6.2 Spray Sheet Results The Na ier-Stokes eqmtriom in c mbination with f If l f t formulation are used to so dy th breakup of two-dimensiornl sheet of wate' The sheet is ZO = -mm thick The length of fihe sheet is 24mm The top md bosom of fihe sheet f f e bounded by f i' The initial mem-velocity of the Water is f 0 = 3m/s The initiaI mms f rbuZent velocity of fihe Water is i = 1 2m/s The fir is initially quiescent Based on th sheet fihickr ss To) md fihe me m velocity (f 0)
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i, where r is th radius of curvature of The tip 7 Conclusion In this paper, w have outlined the key m mericcl clgorithms for simulating free-smfae flows m cartesim g ids usi g level-set md coupled level-set md vol meof-fluid techniques Preliminary m mericcl Results have been show for ship waves md spay sheets he ship _ He results indicate Nat cartesi m-g id methods are car pable of resolving the flow aro md c ship if the g id resolution is s fflcient Near the bow md rem, w estimate that the g id spacing along ail f ee cartesim axes should be ~ = 0 0005 abased on ship length) m order to resolve breaking waves On c parallel computer, it is possible to cpproah this level of g id resol Lion, but adaptive g idding may also be hequi ed to fully Resolve the entire flow aro md c ship [15]
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From page 773... ...
Comp t Phys, 148:81 124,1999 [16] M Sussm madE G Puck tt Acoupled level set md vol me of fluid medhod for computmg 3d md axi mmetric incompressible two-phcse fl ws J
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-_:R f F=~' ~ ~ I: ~ I^~4 ~ _ _ W_: ,....... Figure 5: 2d spray sheet.
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if ~4 -~ Figure 5: 2d spray sheet continued.
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AUTHOR'S REPLY We are m the process of dev loping z body-fitted method The method will be described It the next .-.~npp nun DISCUSSION K Hend ickson Massachusetts institute of Tech olo :, USA The mthors of d is paper show m zggressiv use of z mmmerical method which has, to date only, been used for smaller en m ermg problems The fact that Hey are ah mpting it for this type of problem says much about their patiep e Ed ambition The blending of the volume of fluid and lev I set methods is quite creativ Ed shows promising re pits I believ that the Cartesi m Grid Medhod is wonderfully useful in fast m my of She gridding difficulties hav been remov d Ed or Educed to panel method that has been dealt with in detail in the literature I wish th m both luck as they push the method further Quesdom I At d is stage in dev lopment of the two methods, it seems fast the coupled lev l-set/volume of fluid technique (CLS) is more zccurate/robust m t~eatmg the hull boundary conditions mainly 1 e.
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such es the one proposed by the discusser, are effective away fr m the interface 4 The CLS method is cutout twice es expensive es the US method How ver, She computational costs associated with both methods are less th m ten percent of the Po isson so l- er 5 Irlre face tracking methods me capable of ccpturmg physics that SIRENS md 2D+T will never be capable of modeling Although i te face tracki g methods are more computationally expensive th m unRANS md 2D+T, f is will become less of m issue es computers become faster Ten years from today, inte face tracking will be the medhod of choice for mod hng breckmg waves md the near-field flow around real combat mts 6 The treatment of She hull boundary condition is not accurate enough This issue is currently being add essed by using c body-fitted grid with c level-set treatment of the flee-su face elevation in the near field of She hull in the outer-flow region, the im r solution is mat bed to c combirLttion of spectral methods md panel methods This matching procedure reduces the mmmber of grid pomts md the amount time that is required to generate 3D grids
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