Twenty-Third Symposium on Naval Hydrodynamics (2001) / Chapter Skim
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Combining Accuracy and Effciency with Robustness in Ship Stern Flow Computation
Combining Accuracy and Effciency with Robustness in Ship Stern Flow Computation
Pages 882-896
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From page 882... ...
Abstract Usmg c matme RANS-solver for ship tem flows as c starting pomt, we consider some possibilities to erJvmce eflflciency md robustness wifhout sarff cing acuray Among fhem me alterm~tive domcm decompositions md vector izab le preconditi onmg techniques It is show fnat cgoodbahmcebetw enfl:x~bility, robust ess md efflciency c m be obtained md fnat it is ju tffed to negdect dfff sion in mcin sheam di ection md some of fhe cor~ctive terms 1 Introducidon For the use of iscous flow computations in fhe desigm process of ships, robust ess of the m merical solution pmcedure is c pre~equisite Praticclly this me ms that mdemecsoruible dema ds on g id qua lity, convergence of the iterctive sohtion is warrmted in fl w simoktions mder t pical ship opercti g conditions How wv, robust ess is m sewval methods obtained by sacrffcing a uray (for example by using fl st -or der di scret isati on) A d mger ous route , be mse fhe ~esults of c robust code have no pratical value ff not c certam level of acuray c m be ahieved it is one recson why considerale mphcsis is p t nowcdays on veriflcati on md val idat ion Mm e over, in view of usual time conshamts in ship desigm, it is desirale fnat c good quality sohtion ca be obtamed et low cost m c small am o mt of time (eflflciency)
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From page 883... ...
even for full-sccle Rey olds n mbers he i herently ve y high c pect ratio of the cells near The hull puts high d m mds on the solver for The linear systems, which is one of The reasons to mcmtam the cm phngt~w en The equations m The iterative solution For turbulence modelling in PARNASSOS, The concept of the isotropic eddy viscosity is used it is possible to choose from c wide ridge of tmbulence models, varymg fr m algebraic models to seve~al two-equation turbulence models Ego & Hoeksh c 2000) A detailed description of the math maticcl model, The computaticrurl g id md the PDEs in curvilinear coordinates is given m R f Hoekshc 1999 Here w reshict ourselves to c sho t description of The discretization ~ PARNASSOS The foll wing Smite difference approximations are used: e For The m mericcl evaluation of th g id metric terms w use second-order, central difference schemes e in the continuity equation w use c second-order, th Copout backward scheme for the mcinsh eam derivative, md c thi d-order four-pomt scheme with c -i ed bits in nommcl a d gir6h-wise di~ection e For The derivatives of The velocities that occur in The convective terms w use c second-ord r upwind scheme m sheamwise dinection, ad c thi d-order upwind scheme for The nommcl md gir6h-wise direction e For The g adients of the pressure m the moment m conservation equations w use c third-order scheme For stability reasons, the bias ht.
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From page 884... ...
are till solv d using c pi meby-plane stmtegy 2 Imearize tb tANScquations, using {v~) ,jt computed ctov, md c mpute u,jt, v,jt, w,/t, md P,jt by solvmg the remltmg yst m of Imear equrtions to c prescribed acu ay Hen e w c umot expect c quadrctic speed of cor rgence, not even when cll cor ctiv terms bav been linearized second-order acu ctely 3.2 The linear system solver As bas be n mention d ctov, fLe coeflflcient meh i is g times larger tb m in the phneby-phne v rsion of PA tNASSOS, md it contains coeflflcients which acoumt for the couplmg between the separcte 6=const mt phnes bis matrix will be denoted by A Let fLe entries of A be g ouped in blocks so fnat all elements multiplyi g fLe varictles m c ~ =constat pkme form c block Let su h c block, which itself is c square mch ix of size 4 u NY u NZ, be represented byon e tryA~:,inwhichi md/arerow mdcolumn mdices, with i for convenien e taken equal to fLe g idmdex of the relevat ~ =con t mt pkme wibbin the cu r nt mb-domcin As c result of the chosen discretiza tion, fLe block A,l ca onlybe non-7mo if 1/—il ~ 2 Hen e A bas the following pentc-diagorul sh u tme: dl cl fi o o o .
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From page 885... ...
Usuclly, the most expensive operctions in such medhods are th computation of i merpro ducts , vector up date s, md 6he matrixwect or multiplicetion in our particohr situation, 6he mch i vector multiplication is much more expensive th m m i merpod ct or vector update Hen e w should minimi:m the m mber of matrixwector multiplications, md 6herefore we choose G~PS~ es the linear system solver (where ~ indicates 6~t G~PS is restarted eve y Mit step) 3.3 Preconditioning The sped of cor rgence of G~PS trongdy dep nds on 6he eiger~lue spech m of A Ideclly, 6he matrix should have cll eigemvlues clu ter d; 6hen A would be close to the id ntity metri md G~PS will rapidly cor rge in order to brmg c given spech m closer to thet idecl situation, on c m use c pr conditionmg techmique Instecd of solving Az b, on solvesM~IAz M-lb,inwhichMissomeapproximation of A The mch ix M should be ec y to constn t md for c given vector z the comp htion of M-i z should be cheap There me several possibilities: on c m hy to cppro imete the mverse directly, for exemple, by using c poly omicl of 6he origirul matrix A How ver, the totcl n mber of matrixwector multiplicetiom will then not be less th m with full G~PS Therefore, w have chos n c dfffere t kind of preconditioning techmique: we exploit 6he fat that 6he equations me clmost parabolic in 6he str emwise di ection, byn glectmgthetemms e~ adf m (1)
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From page 886... ...
N (6) order to in ecse 6be Adfiop-mte on on processor of the Cmy C90, memory bak co fiicts should be avoided Tber for, ff the lecding dimension of krge arrcys is c pow r of two, on the Cmy C90 it is mcrecsed by on Tbis gives m cdditional mcrecse of 6be memory usage by cpproximately 15% A very robu t shategy is to avoid the use of c marching scheme by choosmg g NX How ver, fi om m mericcl experiments it bas cppeared tbat the number of global itemtion steps does not mcrecse markedly ff g equals NX,, 2 mstecd of NX, while the memory requi ements c m ckmo t be balved 3.4 Including 'negligiblet terms Tbe fiow aroumd mor or less slender bodies I kc ships md cir rafts et high R y olds m mber is charaterised by c domi mt fi w dir ction D ffusion in 6~t dir ction is pratically n glig~ble, md we bave discarded 6be cssocicted temms m the moment m conservation equations in seri g up PARNASSOS Also 6be cor tion temms for 6be domi mt fiow dir ction bave clways be n tr cted somewbat differ ntly fiom 6be cor ction terms m the Ictercl dir ctions Although PARNASSOS bas been decling wibb fiows with b oumd uy Icyer separct ion in f Le sense tbat there iscrgionwher thefi wisagainstthedominmtfi w di ection, w bave cssumed fnat fLe reversed-fiow r gion will be of relatively small extent md fLerefore with low cor ctive h m port cgamst fLe mcin fiow As c consequ n e we bave neglected the corre pondmg cor ction term Wbile in c planeby-pkme marchmg scheme the 'n gligible' terms bave to be in luded explicitly, i e m the right-bmd side w~ctor, m c sub-domcm of g ecter extension fLe neglected terms c m recdily be mcluded implicitly Accordingly, w bave ir tigated fLe role of these terms to confirm fnat fLey are negligible mdeed md to verffy their effect on fLe convergen bebaviou 4 Numerictil results Tbe present comparisons are pe fommed for fLe cclculation of fLe fiow aroumd the HSVA-t mker et model sccle R y olds number, R 5 x I o6, md aroumd the Mystery Tmker both et model sccle R y olds m mber, R S x I o6, md et full sccle R y olds m mber, Rn 2 x 109 Except for fLe HSVA-taker, wher C beci-Smith's algebmic tmbulen e model is used (Cebeci & Smith 1984)
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From page 887... ...
is measmed he CPU-time on one processor of z C zy C90 for the computation cor~espondmg with Fig 2 is zpproximately I O minutes 100 i ID . 10 10 ' 10' i,\ T~_,, `` ; 0~2 ~ ~~ 0 ' ~ , ~\ dp - - - dw dev 10-8 ic 20 i~ lo so Numbar of 9 oba t rat ons Figme 1: Convergence behavior for HSVA-taker, C beci-Smi6h's turbulence model g 4 Rn 5 x Ne t, we st dy the effect of t kmg into a o mt zll convectiv terms on 6he convergence behavior of the global itemtion Fig 3 sh ws 6he convergence behaviour for 6he computation of 6he fl w aro md the Mystery tar ker on m odel-scale Rey olds n mber 6~t m main tream di ection is d opped, whe~eas Fig 4 shows the cor~rgence behavior obtained when zil convectiv temms me t ken mto aco mt It zppears 6~t t king all convectiv terms mto aco mt has z tzbilising i fluence on the global convergence behavim his is c msed by 6he second-order upwind scheme for th g zdients m 6he t~eamwise convection, which giv s z positiv conh ibution to the main diagorurl of the coeffcient matri For comparison, Fig 5 shows the convergence behavior m case g I, also for model-scale Rey olds n mber It zppears 6~t 6he n mber of global itemtion steps slightly mcreases ff g is inmeased from I to 4 How v r, 6he requi cd CPU-time is reduced considerably smce zt eah do
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From page 888... ...
100 _ 10-' 1010 10 ' 10-' 10' 10 ~ _ 10' "~:.'.~\ , _ \~ ~ F~ I I 10 zo 30 do so Numbol otg obal wratons dp dv dw dw Figme 2: Convergence behmior for HSVA-tarDcer, C beci-Smif 's tmbulence model g NX, 2 Rn sxlo6 ~n ~ M nter's tmbulence model No terms oglected g 4 Rn s,~lo6 Figme 3: Co orgence behwior Mystery tarD~r, M nter's turbulence model St oomwise co oction neglectedinr or od-flowzo o g 4 Rn s,~lo6 10' , 1 n 10 10' ~ Figme 5: Co orgence behmior Mystery tar~cer, M nter's tmbulence model No terms oglected g I Rn s~lo6
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From page 889... ...
, are neglected e PNS: Onlyneglects sheamwise dfff sion e PNS : No temms in the RANS |
From page 890... ...
The integrals in the definitions above are calculated with Gaussian quadrature rules assuming a bi-linear variation of the unknowns between the grid nodes. The area Q for the calculation of the wake fraction is the propeller disc which has been defined by: · x = 0.989L · Axis of the propeller at z = 0.0166L · Hub radius equal to zero.
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From page 891... ...
nn4 nn~ 0.02 \ 0.0 1 ] .QO~ 0.01~ 0.020.oo 0.01 0.02 0.03 0.04 y/L + ozo 030 040 K O 50 090 070 9 0 RO Fig;ure 9: Isolines Ul for ~ 0.989l, solid 1ine: PNS, dotted: PNS , d4shed dotted: PNS Rn 5 >~ I O9 O.E 0.02 \ 0.01 .
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From page 892... ...
~ 6 ~25 485 ~ \1~/ / 485 \ 0.01 |j~,// / 000 :~ 00 ) \\\ 002 ~ ~ \ + 0.00 0.01 O.OZ 0.03 0.04 y/L Figme 16: Logl O of th di fe~ences between PNS md PNS m the mcg itude of the tr msv rse v bcitv vctorforphnex 0.989l Rn 2>~109
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From page 893... ...
Rn 34x106. The bottom huff shows isolines for The wave elevation aro md the cfterbody, The top half sh ws the coneponding isolines for The en or m the dynamic FSB C, expressed es c wave elevati m difference The error is sigma mt only m The tem area md is negligible eve wh re else Therefore, for mo t of the domcm w have Weedy fo md c solution equivalent to c f 11 RANS FS solution, with a effort ju t marginally Urger thm that for c doublebody RANS solution This easy md most efficient procedure will soon ~epkce The doublebody Afro toh for m ost of our p~atical computations R m tins, however, the stern are a where viscous effects on The wave pattern me present, es show by the Urger pressme differences et the wave surface Locally, c full RANS FS method is still needed While most such methods use c time-depffndent solution cpproah, msteady waves may deny the cpproah to teddy state considerably We prefer to solve the steady form of The fiee-surfae problem, by ite~ation rather 6 m by time-stepping Refs Raven & m Br mmelen 1999ad mBrummelff~&Rcven2000 derive c new fomm of The FSBC Citable for iterative solution, md show The superior performance for c 2D test case The iterative process converges ve y la t, ad The total effort is just 2 or 3 times That for correspondmg problem without tree smfae The same aproah is n w bemg implemented m PARNASSOS to be tested in 3D, ad will hop huh suppl merit the composite procedure jn t described 6 Conclusions The re mlts for The My te y t mker have show That the effect of the temms that have been negdected so far m PARNASSOS is indeed very small B pecially at full scale R y olds m mber, it is justified to omit diffusion in main tream di ection altogether md to discard part of the convection terms m regions with -I w separation H wever, especially The Utter temms m have a positive i fiuence on the omen 3mx:e behsviour 11 ~relore, it is better to take all convective terms mto taco mt It The cunent production version of PA BASSOS, we also take The sheamwise dfffih non into a o mt Although we have d monshated That The effect of this temm on both the convergence behavior md the computed results is ve y small for The Mystery t mker, taking all temms mto taco mt removes my do ht in this re pect The new feature in PARNASSOS, a spa marching solution with sub-domains of h get size 6 m a single g id plume has improved The efficiency ad The fi:xibility: it is n w possible to choose the sin: of The via domains m the rmge from one-g idphme to even the complete domain The phmebyphme solution sh ateg -has The advmtage that it is ve y efficient with respect to memo y usage A fill scale compohtion using I million g id pomts c m be done onaPCwi6h only 64Mhyte of mainmemo y
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From page 894... ...
, expressed as wave elevation differe pe Contour interval O OOOSl. The multi-pkme tmtegy, on the odher h md, hcs 6he cdvmtage that it is faster, mcmly bec mse the qua siN wton Imearization process pm be shffted to 6he global iterction in praptice, the best tmtegy cm be chosen depff~dent on 6he mcchine a~ihtle for 6he computations Treating 4 phnes simultmeously seems c good choice in mmy pmpticcl situations in 6~t case, 6he memo y requir ments mcrease only by cpproximately c fcotor of 4 compared to the phneby-phne solution shategy Even c flow compubtion at f 11-socle Rey olds m mber c m 6hen till be done on c workstatipm or c PC 6~t contsins et lecst 256 M ytes of m mo y ff enough computer memory is a~ihtle md ff m algeb~aic tmbule pe model I kc the Cebeci-Smith model is used, mcrecsing 6he size of 6he sub-domcms is c good option smce it red pes the m mber of global itemtion teps ffMenter's turbule pe model is used, 6he n mber of global itemtion steps does not decrecse sigmffcmtlybymcrecsingg h 6henearflure,w hope to remedy 6his phenomffmoby hectingthe nonlinear sompe temms diffe~ently More 6 m 80 % of the CPU-time is pent on 6he solution of the systems of linear equations, usmg p~econditioned GMPES The Mflop-mte that is obtamed on c ptor computer like 6he Cmy C90 is quite high: on one processor of 6his mcchme, cpproximately 300 Mflop/sec is cthined This is cchieved by choosing c suitdole ordering m 6he computation which cl c mvents recmre pes in 6he consh ption md cpplication of the p~econditioner We are co fldent that 6he erJumcements i po pomted in PA tNASSOS es descobed m this pep r have brought us closer to the optim m combim~tion of fl:x~bility, cocurapy, effciff~py md robustness in ship tem flow compubtions Acknowledgements The mthorswishto6 mkH Rcvenformmystimubting discussions md m merous mggestions for improving 6he presentation of 6he pcper Th y clso th mk 6he miversity of Utr pht for the fort m routine 6~t implements GMPES~ References [1]
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From page 895... ...
FR Menter, Eddy Viscosity Trasport Equa tions md Thel Rehtion to 6he k—~ Model Joum nol of Fluid Engineering, 119:876 884, L cember 1997 [8] H.C.Raven, ASoluLonMethodfw6heNonlin eor Ship Wove Resut nce Pnoblem Ph 6hesis, MARIN, Wzgenmgen, 1996 [9]
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From page 896... ...
It appears that the computed data hardly changes when the position of the outlet plane relative to the aft-perpendicular is moved from 0.25L to 0.365L. Furthermore, the plots of the isolines of the pressure and velocity distributions obtained with the outlet plane at x/L=1.365 are virtually identical to the plots shown in Section 3.4, where the position of the outlet plane is atx/L=1.25.
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