Twenty-Third Symposium on Naval Hydrodynamics (2001) / Chapter Skim
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Numerical Prediction of Scale Effects in Ship Stern Flows with Eddy-Viscosity Turbulence Models
Numerical Prediction of Scale Effects in Ship Stern Flows with Eddy-Viscosity Turbulence Models
Pages 553-568
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From page 553... ...
But if one cims et selecting c tmbulence closure that is m merically robust et model and f 11 sccle R y olds n mbers, eddy-viscosity models are still fhe only ~ecsoruible choice ~ fhis p mer we present c m mericcl inve tigation mto fhe prediction of sccle effects with eddy-viscosity tmbulence models, mcludmg cigeb~aic, one~quation md two-equation models Two mcin gocis are conside~ed: e Ir~stigate which turbulence m odels are m mericcily robu t from model up to f 11 sccle R y olds m mbers, without fhe need of fmfher t ming e Evalucte fhe i fluence of fhe R y olds n mber on fhe diffe~ences betw en sohtions obtamed with dffferent eddy-viscosity tmbulence models With cchie ing these gocis, we expect to inxecse the co fldence m fhe use of CF et full sccle R y olds n mbers, usmg turbulence models that have been origincily d veloped for thin shear hyers et model sccle Rey olds m mbers The pcper is orgmised in fhe following way: section 2 gives c brief description of fhe tmbulence models md fheir m mericcl impleme htion The ~esults of cpplication to fhe fl w aro md the Myste y t mker ctR y oldsn mbersfiommodelsccleuptofullsccle Rn me presented md discussed in section 3 Section 4 summarizes fhe conchsions of fhe pcper
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From page 554... ...
, w re aIso tested How v r, remlts of 6bese models will notbe mcluded indbe presentpaper Tbe Baldwm & Barth model sh wed ~ v ry poor bebaviour m 6be mitial te t r ms, while the SST v sion of Menter's k_ m does r t perform berer 6 m 6be other k_ m models te ted It is poss~ble to improv 6be quality of the pr dictions of ship tern flows wibb the or -equation md two-equation tmbuler e models usmg ~ simple corr ction to the production term of 6be hansport equa tions, [16] However, in 6bis paper w will gdopt 6be tmdardv rsionsofthemodels 2.1 Algebraic Models Tbe two algcbrdic models are w 11-kmown md based m ~ two-hyer d flmition of the eddy viscosity, v~, where 6be eddy viscosity is obtamed from 6be m mim m of its values m 6be two 1ayers ~ 6be i merkyer, bobb models use 6be mixmg-length approach wibb the Vm D lest dgmping f mction m 6be r ar-wall rgim ~ 6be Cebeci & Smithmodel, the eddy viscosity in 6be outer region is obtair d from (v,)
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From page 555... ...
deriv s 6he following h m port equstion fi om 6he k— emodel: Dt clDlvt:3+V ((v+ t jVv) —c:E The eddy-viscosity is giv nby v~ = D: v~ D = v~+v v~+v Ele = sE~t mh(~)
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From page 556... ...
The stmdard k—d constmts me cp = 0.09, Cl = 1.44, C: = 1.92, t = I md t = 1.3 The k y fectme of 6his two-kyer m odel is the determinati m oftheboumdarybetwe nthe im~er md outer kyers which is often deflmed by c criteri m based on y However, withy it is diflflcult to establish c criterion which is insensitiv to 6he Rey olds m mber in ou cpprocch, 6he i mer-lcyer region is deflmed by the following criteric: f~ < 0.99 Ay < 50 . The fl t criterion would be 6he nstmal choice to border 6he im r-hyer ~egion How v r, in the iterctiv detemmination of the eddy viscosity fleld it may lecd to excessiv Iy large regions, which provoke m mericalconvergeneproblems Therefore,w hav cdded 6he second criterion which originates fi om the kmowledge on flct pkte boumdary kyers, where the fullytmbuient region starts et y ~ 30 - 50 This cpprocch does not guarmtee that 6he f~ is close to I et 6he edge of im~er-kye' Therefore, m 6he outer-hyer 6he d h m port equation is solv d but f~ is still obtain d from (21)
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From page 557... ...
7he h msport equations of q md 5 are deriv dfr m 6he h mspo t equations of k md ~ with the rehtions: Dq = I Dk Z) F 2qNf D ~Ds _ ~Dq Ot ~ ~T q Ot ~ 6he p~esent impl menbtion of the method, hien's low Rey olds v rsion of 6he k—~ model was cdopted to obbin 6he h msport equations of q md 5 7he eddy-viscosity is giv nby v~ =C~f~25 (34)
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From page 558... ...
mddhe valows of vt obtamed from th Cebeci & Smith clgeb~aic model ~ 6he outer region, Yr > 0.156, m Hermite cubic mterpohti m is used to obtain 6he tu bulent qu mtities, assuming wro derivativ s et the exteukd boumdary At 6he ship surfae, k, q md 5 are wro in Chien's formohtion of the k—d model, d is clso wro et the wall However, the wall boumdary condition for o asks for c few mme words According to [12] , o behav s in 6he vicmity of 6he wall es: whe~e o = for Yr < 2.5 j (43)
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From page 559... ...
v A Cartesi m coordinate system is inh oduced with 6he z axis zlong the mdisturbed sheam, the z axis v rtical positiv pomtmg upwards md y completing z right-h md system The origin of the coordinate system is located on the forward p rpendicular zt 6he ship mmet y phne on the k cl Ime All the variables pr sented cc made non-dimensional using U~ md l zs 6he v locity md length r fer n e cales The c mputational domain covers only 6he flow fleld n ar 6he tem The inlet md outlet phne are z c mst mt pkmes The inlet pkme is located zt z = 0.5l md 6he outlet phne zt z = I.25l Th extemal b mdaryis mellipticalcylinder,giv nby: y ~ /z—0.056lN (0,149) +1~ 0,140 J = The r mainmg bo mdaries are 6he fiee surice, pkme z = 0.056l, 6he symmeby phne of the ship, y = 0, md the hull surface The vol me g ids w r cr zted wi6h z proprietary elliptic PDE g id gen ztor, based on 6he GRAPE zpproach [23]
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From page 560... ...
X 1 O CD X 1 O Wf uW) m:~ mir 4 L.Vt~me' X 10 (V~>m~; X 1O BCt 1 944 0911 0 609 0 343 -0 075 0 265 I 582 BC2 1 612 0 878 0 565 0 365 -0 076 0 257 I 504 Tetle 1: Comparison of solutioms obtamed with Mbuter's k—P model usi g differ nt implementations of the P wall boumdary condition w hav calcuiated the flpw et f = Sxl06 with Mbuter's v rsion of the k—P model with fhe two options considered: BCt, which obteins P from fhe th or ticel valu for y < 2.5 md BC2, which is based on m od hoc deflmition of c flmite valu et fhe wall Tetle I presents the fl e selected flpw qumtities md fhe me m md maximum valu s of v~ obtamed with BCt md BC2 The dffferen s obtain dbetwe n fhe two solutions are certamly not neglig~ble As on might expect, fhe fi iction resist mce coefflcient, CDf, e hibits the largest d fference How v r, the limitmg treamlin s of both prlcuiations, which are depicted m flgme I, are simi~ BCZ, (T~)
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From page 561... ...
O ( ) ~ T whe~e Sw is th wetted surface of the ship included m 6he computatiom~l domam, which is ass med to be half of the total w tted surface The p~edictions of a11 6he turbulence models e h~bit the correct hend with sewval turbulence models at different Rey olds n mbers 6he mcrease of the Rey olds m mber, but there is clear diffe~ence m slope As ~ fur6her relevmt result, w have plotted the ae m wake fracti m, Wf, as afunction of the Rey olds ~ mber m figme 4 It is interesting to note 6~t m both igmes 3 md 4 the~e is good ag eement betweff~ the iA model md the two k—m models, K\Af md K\AfM The calculated limitmg sheamlines at R = 5 x I o6, b'= 108 mdRn=2 x lO9are illu trated mfigmes 5 o 7 fcr th models CS, ~iA, MT, K\Af, KWM md KE As in the previous re mits, 6he dffferences betw en he predictions of the various tmbulence models tend o diminish with 6he inmease of Rn Once mme, the
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From page 562... ...
OMT ~ KE TL ~QW x O KWM A ~ + k3 + 3 + ~ + 7 6 9 10 Log t o (Rn) Figme 4: Wcke ficotion, Wf, es c f motion of fhe Rey olds m mber for the seve~al turtclff~qe models tested ~esclts of the SA, mw md mw v~ models are very similar: At full qcle, f = 2 x 109, the CS, MT md KE also show good cg e ment he axicl velocity isolines et fhe propeller pkme, x = 0.989l, et fhe same f ee Rey olds m mbers, 5 x 106, 108 md2 x 109, are presented infigmes 8 to I O he turbulence models i qlcded are cgain the CS, SA, MT, KW, K\AfM md KE he Ul isolines e hlbit Figme S: Limitmg sheamlmes ctf = S x 106 c d c tic i fiue qe of f At model scale, fhe t pical hOOk shape' does cppear for th k—m models, md to some extent, for fhe one-eq wtion mod 1s5 SA md MT Hqwever, et x = 0.989l, the 'hook shape' tends to discpp ar wifh fhe i q~ecse of fhe Rey olds m mber At f = 2 x 109, nqDe of fhe p~edictioms e h~bits c hOOk shape' md d fferff~qes betweff~ the resclts of fhe variocs m odels, i qhding the clgebrcic CS m odel, are rcther smell his effect of f is ~eMted to the st~etch ng of the b ilge qrt :x, genemte d wif hin fhe ship b o mdary Icyer which red ces its thick ess with the i q~ecse of f Figmes I I to 13 illc trcte the cross- trem velocity field et x = 0.989l for fhe same f ee Rey olds m mbers he plots i qlcde fhe CS, MT md K\AfM models Although there me some dffferences betwen the p~edictions of the f ee models ew~n et f = 2 x 109, xsm co e Deo ,mese remoco om eeas~ylmprov wi h s s mple ooneotlm t the grodcotloc temm o the tr m eqc diOC of he twbulect qc mtity
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From page 563... ...
t taker et -i e R y olds m mbers, 5x 109,2 x 107,109, 5x 109 md2x 109, suggestthe following conclusions: e it is possible to simulate m merically ship stern -I wsfrommodeluptof llsccleRey oldsmmbers with the most popular eddy-viscosity tmbulence models, mcludmg algebraic, oneequation md two-equation models e in global temms, The predictions e habit the same Tend in the flow field with The increase of the Rey olds m mber for all the turbulence models
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From page 564... ...
A AA A na A A 0.0 A A U.U ~U 0.01 0 00 0 01 0 02 0 03 0.04 ~.u A Aq A A A A 1 y/L A A U.U ~ UA MT 0.01 0 00 0 01 0 02 0 03 0.04 .u A Aq A A A A 1 A A t 01 * OZ O Os 04 as ~ as s 07 os os y/L 0.0 0.03 0.02 0.01 0.00 0.00 0.01 0.02 0.03 0.04 y/L 0.03 0.02 0.01 0.00 0 00 0.01 0.02 0.03 0.04 y/L 0.01 0 00 0 01 0 02 0 03 0.04 y/L Fit3~re (i: Axi4i vebc~t isolines 4t ~ = 0.989l ob+ ot *
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From page 565... ...
Do N 0.0 0.0 0.0 A n r/ w.u (S 0.0 0 00 0 01 0 02 0 03 0.04 y/L 0.04 ~ 0.0 0.0 N .1 ~ I 0.0 A ~ oo h oo h 1 0.01 A ~ y/L 0.04 ~ ~ w.w KE 0.01 0 00 0 01 0 02 0 03 0.04 y/L Ot OF 04 04 o~ 06 07 09 09 Fit3~re 8: Axi41 velocit isolines 4t ~ = 0.989l ob hined 4t Rn = 2 x 109 ~ ~A ...
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From page 566... ...
o.ol o.oo . 0.01 M O.01 ; V 0.00 0.01 0.02 0.03 0.04 y/L CS 004 0.01 0 00 0 01 0.02 0 03 0.04 y/L MT 004 0.01 0 00 0 01 0.02 0 03 0.04 y/L KWM Figure 10: Transverse velocity field et ~ = 11.989l obtained et Rn = I 09 ~ i \ N \ N U.U 0.0 ~ fit 0.0 0.0 ::~ ~ 0.00 0.01 0.02 0.03 0.04 y/L CS 0.00 0.01 0.02 0.03 0.04 y/L MT AAt i ;~\ o.ol Alto ~~'~Ir~I~;~ ~~ v \ ~~ N ~ ~ \ 0.00 0 01 0.02 0 03 0.04 y/E KWM ~ ~t3~e 11: Trarisverse velocity field et ~ = 0.989l obtained et Rn = 2 x 109
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From page 567... ...
fhe predictions depend on fhe m merical implementation of fhe t bo mdary conditigm et c solid surfae Some implementations cdvised m fhe open literatme are mass ptable The present ~esults ~ei force fhe need for relictle experimenbl dat4 et full secle Rey olds n mber for 41idation p gposes All eddywiscosity turbulff~se models, used here, w re essentially developed for bo mdary-kyers et modemte R y olds n mbers References [1] Wctson S JP, Bull PW - TheScolingofHigh R y 71du Numben Vuco u Fbow Pnedictio u Using CFD Technigu~ - Thi d Osakc Colloqui m, Osakc, Jcp m [2]
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From page 568... ...
120, September 1998, 457-462 [19] Hoekst~a M, Ega L - PARNASSOS: An Ef i cl mt Me6hod fon Ship Stern Flow Cole dotion - Thi d Osakc Colloqui m on Ad anced CFD Applications to Ship Flow md Hull Fcrm Desig, Osakc, hp m, l 998 [20]
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