Twenty-Third Symposium on Naval Hydrodynamics (2001) / Chapter Skim
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Waves and Forces Caused by Oscillation of a Floating Body Determined Through a Unified Nonlinear Shallow-Water Theory
Waves and Forces Caused by Oscillation of a Floating Body Determined Through a Unified Nonlinear Shallow-Water Theory
Pages 993-1005
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From page 993... ...
md Sch oter (1995) he par tmeter reqtti cd for defiming fhe ve ticcl level m cgam be rpff ified by comparing the di persion relations betwefft the re mlting Imearized eqtutions md fhe lit ar wave theo y By sttitable selfftion of fhe cssocicted par tmetff valt es fhe m odff cd Bottssmesq's eqtutions are valid for c rctio of wave leng h to water-depfh d wn to c valt e of 2, practicclly to fhe deep-water region In comptttation of waves get mted by ships, c signific mt conh ibtttion was fhe mclltsion of fhe ship's i fittet e on fhe tmbient fiow As repo tedby Ji mg (2000b)
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From page 994... ...
To verify 6 is miffed Theory Ed to examine our m mericcl method, we investigate the wan generation of c vertically oscillating cl cohr cylinder m shall w WCtff For mall cmplit des of body motion, the calculated wave profiles Ed wave forces will be compared with Hose from She Imear wave theory The nonlinear effects occurring et urger motion cmplit de will also be discussed THEORETICAL FORMULATION Gen Description Considering wave generctionby c surfae-piercingbody oscilktmg m shallow water of varymg water depth /1ii.1~. Of I Ed using prime ~ es superscript to denote all dimensional q entities, c Cartesi m coordinate system O~'lf,' is used to describe She velocity Ed pressure field, where She plane O
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From page 995... ...
—~' [(~;—/'i"[ = 0 (Ig) with the dep6h-averagedhori ontcl velocity components W = id{- (-'l = .
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From page 996... ...
(22) The relation betweff fhe depth~emged horizontal velocity ll(.~ '~./1 md fhe wave elevation <~.t 'l./1 c m be obtamed by integ cting the horizonbl m oment m equations (I O)
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From page 997... ...
ensures mass conservation th ough 6he waterline NUMERICAL IMPLEMENTATION Inthepresent tudyw cpplythe mffednonlmear6heory to mvestigate the waves ge ffcted by c smfae-piercmg circular cylmder oscilhting ve tica lly in shallow water of constat water-depth /~ = /` G md 6he cssocicted forces For c problem having axicl mmetry ctout 6he verticcl axis _, the dimensiorul equations, now omittmg the primes ~ for simplicity, m c cylmd iccl coordim~te systffm corresponding to equations (36-39) reduce to: (!
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From page 998... ...
the ~oove implemented scheme is generally sbble but lecds often to m mericcl oscillations which c m be mppressed by me ms of global md local flltering Staggerffd Grid The cpplication of 6he ctove Cr mk-Nicolson scheme for the governmg equations m the im r fleld is not shaightforword Due to 6he decouplmg of 6he pressure /96 on the w tted body-smfcce from the dep6h-averaged contmnity equation, c speci~l trectment is requi cd Fortunctely, 6he dff ouplmg effect is mathematicclly ormlogous to that k ow f om Enler's equations Thus, the well~stablished concept of c sbggered g id c m be cpplied to overcome this diflflculty A discretization based on 6his concept is impiffmented for fhe governmg equations m fhe im r fleld Becimse of fhe lineority of fhe vertically integ oted contmnity eq mion (46) , it is discreti:osd using c fully implicit schffme: 1 , —, I, , ~ ,1 _ ,,: ~ il]
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From page 999... ...
s t ~ = I .2.3. , ~ 1r—I it i s worth m ent i onmg 6~t the tffms ii ssoci~ted wi6h ~ s e discretized usmg the C lmk Nicolson scheme lmd 6he pressure tffm usmg s fully im plicit scheme N mffical test r ms htive sh wn 6~t 6his treli tment mppressed the pressure oscilktions clmsed by s complete C lmk Nicolson schffme h the Ih tter cs se lm mderrellixation procedure wi6h s flictor of lipproxi mlitely 0 6 hti d to be s dditiomdly pe fommed Scheme for Coupling Conditions At the mtersection ~ = ~ 1l,6he ve tica lly integ s ted outer contimmity equation (44)
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From page 1000... ...
Fig 7 compares the th ee time hi tories of he re ponse force also nom Shed by he cmplit de of motion Furthermore, es no difference in the nom Prized wave elevation or response force c m be noticed for oscillation cmplit des smeller 6 m 0 01 m, linear theo y maybe safely used up to .: < 0.01 m m he present co figuration CONCLUSIONS A mified nonlinear shtll3w water then y 03mprismg two t ts of Bcussinesq t pe equaticus was introduced to detffm me wows Ed forces o me d by cscilktions of ~ fixating body m shall w-water A m merited medhod base d on he Cr mk-Nio 3 is on sohem e 0 3mb me d wi6h fully implicit t h me was successfully implemented to solve he governing equaticus in ~ taggered g id Fm small motion tmplit des ~ good ag cement between the miffed nonlinear chewy Ed linear wee then y was cbserved for wave elevation Ed re pause force within the r mge of validity of he classical B cussinesq's equations H wever, the pret nt miffed then y did display nonlmesr effects for larger motion tmplit des Mme f mdamenhlly,w own w03 fidently~pplythenew miffed nonlmesr htlbw Water chewy to simulate wee hod interactions REFERENCES Chen,X-N at Swarms, SD 1994: N3nlinesr6heory of ~ mehio motion of ~ slender ship in ~ shallow charnel Prop of he 206h Symp 3nNsval HydLcdytmios, CA, USA, pp 386-407 Etekm,RC,Qim,ZM &Websusen,JV 1997: Upstre mm scliton ge Ration by ~ slender, vertical stn t mdship: B3ussmesqequstions Prop cfthe -that
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From page 1001... ...
Fig.1 Comparison of numerical and analytical solutions for the depth-averaged radial velocity generated by a vertically oscillating vertical circular cylinder
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14 Fig. 4 Comparison of time histories of the response force determined from the unified nonlinear shallow-water theory with that from the linear wave theory for the subject cylinder
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From page 1003... ...
5 Comparison of normalized wave-amplitudes determined from the unified nonlinear shallow-water theory with those from the linear wave theory for a vertically oscillating horizontal circular cylinder 0.500.250.00-0.2 -0.50-0.7 -1.00a = 0.1 m -- -- - a=O.Olm a= 0.001 m l , 10 ~ ~ 40 r m 50 \\/ ~ V Id Fig. 6 Comparison of normalized wave-profiles generated by a vertically oscillating vertical circular cylinder with a forcing period 2.3 s 40 300200100~ O-100-200- 300 a= 0.1 m -- -- a= 0.01 m a= 0.001 m 0 2 4 6 8 10 12 14 t [s]
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From page 1004... ...
compares our c mptotic time history with Nat of WAMIT Agam, the cg e merit is rem.ukrhle for th cmplit de U fortunately, the phase shift betw en She two different calculations persists Ed Thus c m not be explained by possible k k of convergence, neither m pace nor in time discretizatioa Mmeover, our cdditiorurl simulations show that the coupling conditions betw en the im r Ed the outer flow field have some i flume on the phase shift We ogre Nat c uniform Cartesi m g id will not workw 11 for c ship-lke geomeby Cune fly, w me implementing om computer program m c curvilinear grid Ed w hope to present the new results et the next Symposium on NavalHyd odynamics
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From page 1005... ...
Fig. 8: Time histories of the response force (b)
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