Turbulent orifice flow in hydropower applications, a numerical and experimental study

Transcription

1 Turbulnt orfc flo n hydropor applcatons, a numrcal and prmntal study Zj Zhang Akadmsk avhandlng som md tllstånd av Kunglga Tknska Högskolan framläggs tll offntlg gransknng för avläggand av tknsk doktorsamn frdagn 4: sptmbr, kl 0:00 sal D, Lndstdtsvägn 7 tr Stockholm. Avhandlngn försvaras på nglska. Fakulttsopponnt: TknD atrk Andrasson Eamnator: rof. Klas Cdrall TRTA-AM HD 047 SSN SRN KTH/AM/HD 047-SE

2 Abstract Ths thss rports th mthods to smulat flos th compl boundary such as orfc flo. Th mthod s for gnral purposs so that t has bn tstd on dffrnt flos ncludng orfc flo. Also t contans a chaptr about th prmnt of orfc flo. Hghr-ordr prcson ntrpolaton schms ar usd numrcal smulaton to mprov prdcton at accptabl grd rfnmnt. Bcaus hghr-ordr schms caus nstablty n convcton-dffuson problms or nvolv a larg computatonal krnl, thy ar mplmntd th dfrrd corrcton mthod. A lor-ordr schm such as upnd numrcal schm s usd to mak prlmnary guss. A dfrrd dfct corrcton trm s addd to mantan prcson. Ths avods th conflct btn prcson ordr and mplmntaton dffculty. Th author proposs a shftng btn upnd schm and cntral dffrnc schm for th prlmnary guss. Ths has bn provn to mprov convrgnc hl hghr ordr schms hav dr rang of stablty. Non-orthogonal grd s a ncssty for compl flo. Usually on can map coordnat of such a grd to a transformd doman hr th grd s rgular. Th cost s that dffrntal quatons gt much mor compl form. f calculatd drctly n non-orthogonal grd, th quatons kp smpl forms. Hovr, t s dffcult to mak ntrpolaton n a nonorthogonal grd. Thr mthods can b usd: local corrcton, shap functon and curvlnar ntrpolaton. Th local corrcton mthod cannot nsur scond-ordr prcson. Th shap functon mthod uss a larg computatonal molcul. Th curvlnar ntrpolaton ths author proposs mports th advantag of coordnat transformaton mthod: asy to do ntrpolaton. A coordnat systm staggrd half control volum usd n th coordnat transformaton mthod s usd as accssory to drv ntrpolaton schms. Th calculaton n physcal doman th non-orthogonal grd bcoms as asy as that n a Cartsan orthogonal grd. Th author appls ths mthod to calculat turbulnt orfc flo. Th usual undr-prdcton of ddy lngth s mprovd th th ULTRA- QUCK schm to rflct th hgh gradnts n orfc flo. n th last chaptr, th author quantfs hydraulc abruptnss to dscrb orfc gomtry. Th abruptnss can hlp ngnrs to ntrpolat stng data to a n orfc, hch savs dtald prmnts.

3 Contnts Acknoldgmnts Chaptr ntroducton Chaptr clt numbr-dpndnt dfrrd corrcton to mplmnt hgh ordr dffrnc schms to mprov convrgnc. ntroducton. Mathmatcal formulatons 3 3. Dfrrd corrcton tchnqu on dffrnc schms 5 4. Tst cass 7 5. Conclusons 0 Chaptr 3 Toards computaton n physcal doman th non-orthogonal grd. ntroducton. Mathmatcal formulaton 3 3. Tst cass 8 4. Conclusons 8 Chaptr 4 Computaton of orfc flo. ntroducton. Mathmatcal formulaton 3. Frst ampl 4 Th Ultra-QUCK schm prforms bttr 3 5 Conclusons 6 Chaptr 5 Hydraulc dscrpton of orfc gomtry. ntroducton 8. Abruptnss of transton n trm of hydraulc quantty 8 3. Conclusons 3 Fnal Conclusons 33 Appnd Appndd paprs and manuscrpts and contrbuton to jont paprs 34

4 Acknoldgmnt Bfor rt ths pag, rfr to thos bautful ords n th countrpart of othr dfndd thss. Authors alays hav many to thank. So do. As usual, apprcat rof. Klas Cdrall for hs support for and suprvson on ths projct. Enthusasm s th most oftn-usd ord for hm n prvous acknoldgmnts. agr. Whn startd ths projct, Doctor Bjan Dargah told m to do somthng orgnal. Ths has bn a stmulus to m. Also thanks Dr. Jams Yang for hs commnts on manuscrpts. Thank Brtt Cho for kpng vrythng n ordr and organzng all th mystrous routns. Also thanks to othr popl n th Dvson of Hydraulc Engnrng, ttr, tr, Frdrck, Ccla, Ola, Krstoffr and Hans. spcally thank th KTH-Lbrary hr gt hlp to track many rfrncs. Th fnancal support from BFR Sdsh Buldng Rsarch,.bfr.s and Elforsk AB Svnska Elförtagns Forsknngns- och Utvcklngs-, AB s gratly apprcatd. Thank rof. Zhuang Zhang at Tsnghua Unvrsty, Bjng for hr shong m th ay of numrcal smulaton th atanka S. V. s book. also ould lk to apprcat rof. Shoutan Zhang, rof. Junm Ca and Eng. Jngmng Fng at Tsnghua Unvrsty. n th nd, must say lov my daughtr Ann.

5 Chaptr ntroducton Ths ntroducton s about th rlaton of th follong chaptrs. Th author ntndd to smulat turbulnt orfc flo th a smpl cod. Durng th dvlopmnt of th cod, som mprovmnts ar mad. Bascally, th thss follos th ay of mprovmnts to th dstnaton turbulnt orfc flo. n th author s opnon, th man mprovmnt s on th ay nstad of at th dstnaton. Th numrcal smulaton of flos nds mthods that rflct th physcs of th flo. Othrs, th smulaton procss s not smooth slo convrgnc or vn dvrgnc. Chaptr ntroducs an mprovmnt to mplmnt hgh ordr schms. Wth ths proposal, hgh ordr schms, f appld to th convcton trm n a convcton-dffuson problm, convrg fast and shar a d rang of stablty. Th author calls t a hybrd platform. Numrcal smulaton mthod should b smpl. Chaptr 3 ntroducs a novl schm to mplmnt ntrpolaton n non-orthogonal grd. Ths nabls computaton n th physcal doman th as. t combns advantags both th convnnc of ntrpolaton n curvlnar coordnat systm and th smpl form of dffrntal quatons n physcal doman. n Chaptr 4, ths mthod s appld to th flo mntond n th ttl of ths thss. On can fnd that advancd schms that r dsgnd for Cartsan orthogonal grd can b usd to non-orthogonal grd, n physcal doman thout coordnat transformaton. Th last chaptr, Chaptr 5, s about prmnt, hch s th bnchmark for th smulatons. t s not th focus pont of th thss, hovr, th hydraulc abruptnss dfnd thr nabls ngnrng socty to us acadmc matral th as.

6 Chaptr clt numbr-dpndnt dfrrd corrcton to mplmnt hgh ordr dffrnc schms to mprov convrgnc Ths chaptr mprovs th dfrrd corrcton tchnqu.. ntroducton n th smulaton of flud flos, convcton-dffuson problm s oftn nvolvd. Th soluton procss of th quaton,.g., th Navr-Stoks quatons, bgns at dscrtzaton. n th dscrtzaton, varabls on dscrt grd ponts ar usd to ntrpolat valus on grd surfacs among thm. Ths ntrpolaton nabls nodal ponts to b numrcally lnkd togthr, n th nd, to th boundary and th ntal condtons. A propr ntrpolaton schm contrbuts to a larg dgr to th accuracy of th soluton and th stablty of th solvng procss. Th radrs of th prsnt papr ar assumd to hav rad books such as by atankar, S. V., n hch numrcal schms for th convcton trm ar ntroducd. Th upnd UDS, th hybrd HDS and th por-la dffrnc schms DS hav frst-ordr prcson. Th cntral dffrnc schm CDS has scond-ordr prcson. Thrd ordr schms ar sn n applcatons to compl flos. Espcally, th QUCK quadratc upnd ntrpolaton for convctv knmatcs schm, hch s th most famous among hghr ordr schms, s vn avalabl n computatonal flud dynamcs CFD commrcal programs. Byond th thrd-ordr schms, fourth ordr schms can b usd, largly found n journal paprs. To knds of fourth ordr schms ar avalabl. Th mor oftn usd s basd on th cubc spln;, 3, 4, 5 th othr typ s that basd on 3rd-por Lagrang polynomal. 6 Th formr s calld as compact fourth ordr schm bcaus t has a smallr computatonal krnl, at th cost of addtonal computatonal ffort. Th lattr somtms s calld th fourth-ordr CDS. Th hghr ordr prcson a schm has, th mor dffcult to mplmnt. Ths s manly attrbutd to thr rasons. Hghr ordr schms ar oftn condtonally stabl. Th cntral dffrnc schm has a ll-knon stablty condton, n convcton-dffuson problm, as clt numbr lss than unform grd. Thrd and fourth ordr schms hav lss stablty. Thrd and fourth ordr schms hav a larg computatonal krnl,.. th ntrpolaton nvolvs mor nodal valus. Th coffcnts n th algbrac quatons form a pnta-dagonal matr. Although most of th popular solvrs for tr-dagonal matr solvrs can b tndd, th ngatv coffcnts for th outr nodal valus mnmz th possblty of convrgnc. 3 Thrd and fourth ordr schms ar mor dffcult to mplmnt n non-orthogonal grd. By dong spac transformaton, ths dffculty can b solvd. Actually, th stablty and convrgnc ar th man challng. For a commrcal CFD cod, thy ar vtal. To mprov th stablty, th hghr-ordr schms ar oftn drvn on a lorordr schm. Ths s th dfrrd corrcton tchnqu. n an tratv calculaton procss, th n CV-fac valu s appromatd as an mplct prsson a lo ordr schm plus a corrcton trm th dffrnc btn th hghr-ordr schm and th mplct part. Th corrcton trm s dfrrd from th prvous old valu

7 and lumpd nto th sourc trm. Th formulaton of th mplct prsson affcts th coffcnts n th algbrac quatons, and n turn, th stablty and convrgnc. Th rsult stll has th hgh ordr prcson of th orgnal schm. By ths ay, th conflct btn prcson and stablty s dssolvd. Th dfrrd corrcton s usually attrbutd to Khosla and Rubn. 0 Latr authors gav ths nam to Khosla and Rubn s schm. t s mor dly knon as dfct corrcton out of th socty of flo smulaton. t as frst appld to th soluton of ordnary dffrntal quaton. Latr t as dvlopd to solv partal dffrntal quatons. Th man da s, to solv an oprator quaton n a smplfd vrson of th problm th frst guss, th ths nformaton to calulat an approprat corrcton quantty and to apply th corrcton to th frst guss to obtan a n and oftn bttr appromaton. [] Th dfrrd corrcton tchnqu should b optmzd, hch s th objctv of ths chaptr. rvously, th frst-ordr upnd dffrnc schm s oftn usd as th mplct prsson. 5,8 A larg corrcton trm s addd to th sourc trm, hch drags th convrgnc spd. An mprovmnt s proposd upgradng th mplct prsson to CDS onc th clt numbr s lss than to. Ths maks a prlmnary guss that s closr to th orgnal schm. Th prsnt proposal s usd as a platform to tst thr schms, rspctvly th scond, thrd, and fourth ordr prcson. Ths s compard th th smpl mplmntaton hch alays uss th UDS to mak th frst guss. ts advantag s dmonstratd n tst flos.. Mathmatcal formulatons Th Navr-Stoks quatons n -D rad: u u u u p ρ u ρv = µ µ, v v v v p ρ u ρv = µ µ, hr u and v ar mass flo spds; µ s flud dynamc vscosty and p s prssur. Th mass flo spds satsfy th contnuty quaton: u v = 0. 3 Th ntrpolaton schm s usd to prss control volum CV surfac vlocty th nodal valus. Abov dffrntal quatons ar dscrtzd nto algbrac quatons,.g. for u-quaton: a u = anghbors unghbors Sourc. 4 Wth th notatons n Fgur, a numrcal schm appromats th ast CV surfac valu as: u =... β, u β, u β 0, u β, u β, u n ma Usually, β. 6 nmn n, = n hch n ma nmn gvs th computatonal krnl n on drcton. Th CDS has coffcnts β as n, 3

8 4 mn n = 0, ma n =,, 0 f = β, 7, f = β ; hr f s th ntrpolaton factor: f = = δ. 8 E EE W WW - - j j j n s E EE W WW - - δ NN N S SS Fgur Spac dscrtzaton and notatons. Th control volum for nod s dran th dash lns. Th arros at CV surfacs ndcat on of possblts of CV-surfac vlocty drctons. dos not ncssarly l at th CV cntr. Th QUCK schm s dpndnt on th drcton of u. For u >0, mn n = -, ma n =,, = β, 9, = β,,,, 0 = β β β. For u < 0, th coffcnts tak smlar forms but for dffrnt nodal ponts. Th fourth ordr CDS has coffcnts as mn n = -, ma n =,, = β,, = β, 0, = β,,,,, 0 = β β β β. Ho to mplmnt ths schms affcts th coffcnts a and nghbors a n quaton 4. Drctly applyng ths schms to th convcton trm s not stabl. Wth hgh ordr schm n 5 for convcton trms, quaton s dscrtzd as

9 ρu [ β, u β, u β0, u β, u β, u ] ρu [ β u β u β u β u β u ],, 0,,,, u u u u = µ µ sourc trms n hch subscrpt and rfr to th ast and st surfacs, rspctvly. Arrang as a u = aww u W u aeu aeeu sourc trm, n hch a WW = β, ρu β, ρu, µ a ρu, W = β0, ρu β, µ µ a p = β0, ρu β0, ρu µ a E = β, ρu β, ρu, EE = β, ρu β, ρu. a Thn an algbrac quaton solvr s usd to gt th soluton. Most quaton solvrs rqur postv coffcnts n. Som vn rqur th domnanc of a ovr nghbors. Hovr, coffcnts n can b ngatv, spcally th to standng for th outr-lyng nodal valus. For th CDS n unform grd, stablty condton s: u = < =. 3 µ / ρ f For non-unform grd, th condton s a lttl complcatd. For th thrd or fourth ordr schms, th stabl condton s so strct that th schms can not b drctly mplmntd for a compl flo. 3. Dfrrd corrcton tchnqu on dffrnc schms Wth th dfrrd corrcton tchnqu, a dffrnc schm s rrttn as lorordr hghrordr lorordr u [ ] old = u u u, 5 lorordr n hch u stands for th prlmnary guss th lor-ordr schm and hghrordr u th appromaton th advancd schms. Th trm n th squar brackts s valuatd th valus from th prvous traton, as th suprscrpt old lorordr mpls. Th upnd schm s oftn usd for u : u [ β, u β, u β0, u β, u β, u ], f u 0 u = u [ β, u β, u β0, u β, u β, u ], f u < 0 mplct plct 6 Th algbrac quaton has th follong coffcnts: a = 0, WW 5

10 µ aw = ma[ ρu, 0], µ µ a p = ma 0, ρu ma 0, ρu µ a E = ma[ 0, ρu ], [ ] [ ], 7 a EE = 0, and th sourc trm ll nclud th corrcton trm. Bcaus ths coffcnts ar all non-ngatv, th vlocty fld can b solvd th stablty. Ths form of dfrrd corrcton tchnqu has bn appld to hgh ordr schms. n a compl flo, th orgnal hgh-ordr schms and th UDS gv much dffrnt stmatons for th CV-surfac valu,.. th UDS maks a bad frst guss ths s hy hgh ordr schm s usd. n th follong a nodal pont th C = ρu < 0, C = ρu > 0 Fgur s usd as ampl to plan th consqunc of th upnd schm as mplct part. For th u-momntum quaton, th fact that C < 0 and C > 0 mans that u E s ngatv Fgur. Th UDS gvs an appromaton that could b on-ordr largr than that gvn by th orgnal schm. A corrcton hghrordr UDS old old C u u C u s lumpd nto th sourc trm. Th n valu old n of u gts an rror that s of th ordr of α u u n hch α s th undrrlaaton factor. n CVs n hch C and C gt th sam sgns, such tra rror stll sts but could b smallr. Ths tra rror should b mnmzd. stramln -grd ln n W j-grd ln C s C E u W u Hgh-odr schm E W CDS UDS u E u Fgur A complcatd cas n -D flo, C = ρu < 0, C = ρu > 0, to sho that th CDS maks bttr appromaton to hgh-ordr schm than UDS. Th stratgy s to mprov th frst guss to upgrad th mplct prsson from th UDS to th CDS. n a compl flo, ths upgradng s possbl. For ampl, th nodal ponts th C >, C < 0 oftn hav lo vlocty. n such rgons, th grd s also rfnd. Ths nabl a small local clt numbr. Th CDS can b usd as th mplct prsson. Upgradng th mplct prsson s trggrd th <, 6

11 ffctvly n th compl rgons. Thus th hybrd dfrrd corrcton can actly compnsat for th draback of smply usng UDS. An attractv pont s that, f th orgnal schm s th CDS and th mplct prsson s upgradd to CDS, th corrcton trm s zro. Ths agrs th th fact that th CDS s condtonally stabl. 4. Tst cass Thr numrcal schms ar mplmntd on th prsnt platform th hybrd dfrrd corrcton tchnqu: th CDS, th QUCK and th fourth ordr CDS. Thr schm coffcnts hav bn ntroducd n 8, 9, and 0, rspctvly. n th tst, th prsnt proposal s compard th th popular on UDS s alays usd as mplct prsson. To solv a flo problm, many othr dtals ar nvolvd: th control volum mthod, th SMLEc algorthm 6 and forardly staggrd A-grd, undr-rlaaton and TDMA tr-dagonal matr algorthm, hch ar not dtald n ths papr. Convrgnc spd quantfd th th outr-traton numbr and stablty quantfd th th undr-rlaaton factor α ar compard. Th SMLEc algorthm s usd nstad of th SMLE, 7 hch avods sttng undr-rlaaton factor for th prssur quaton. n th comparson, th soluton gvn by an accssory schm s prsntd to plan th advantag of th prsnt proposal. Ths s th hybrd dffrnc schm thout droppng th dffuson trm dnotd as HDSc n th follong. t has th sam shftng mchansm as n th prsnt proposal, du to hch th prsnt author borros th nam from th ll-knon hybrd schm. t s mplmntd just by sttng th corrcton trm as zro. ts soluton ll b usd to rprsnt th mplct prsson. As for th tratmnt of boundary condtons, corrspondng prcson s mantand for all th thr schms. Th tratmnt of th QUCK schm on sold boundary has bn dtald n rfrnc [5]. For CV facs nar th boundary, th UDS s alays usd thout schm upgradng. 4. Ld-drvn cavty flo Ths s a popular tst flo. Th tst vrfs th mprovmnt of th prsnt proposal on convrgnc spd. Fgur 3 shos th rlaton btn th outrtraton numbr and th undr-rlaaton factor for R = 00. A unform grd th 0 0 CVs s usd. Th prsnt proposal taks lss outr-traton to gt convrgnc for all th thr schms. On th othr hand, stablty s not lost. Th prsnt proposal mprovs th convrgnc bcaus th mplct prsson dos a bttr frst guss. Fgur 4 shos th stramln pattrns calculatd th 0 0 unform CV, th dffrnt schms. Thy ar compard to th rsults th a rfnd grd Fgur 46. Th lattr flo pattrn agrs th bnchmark. 9 On can smply us th cntral stram functon as th nd to compar. Hghr ordr schms gv bttr soluton than lor-ordr schms. HDSc s closr to hgh ordr schms, hch savs th traton. Usng an mplct part that s closr to th orgnal schms to mak dfrrd corrcton s th ky pont of th prsnt proposal. Th prformanc at hghr R of ths flo s rfrrd to th to appndd manuscrpts clt numbr-dpndnt dfrrd corrcton to mplmnt th cntral dffrnc schm to mprov convrgnc and A n form to mplmnt th QUCK schm to mprov convrgnc and stablty. 7

12 outr-traton numbr th UDS platform th prsnt proposal undr-rlaaton factor, α outr-traton tms 0 th prsnt proposal th UDS platform undr-rlaaton factor, α outr-traton numbr th UDS platform th prsnt proposal undr-rlaaton factor, α 3 Fgur 3 Th outr-traton numbr vs. th undr-rlaaton factor α n th ld-drvn cavty flo R = 00 calculaton th to mthods to mplmnt th CDS, th QUCK schm and 3 th fourth ordr CDS. Th curvs nd at th mamum α allod Fgur 4 Stramln pattrns at R = 00, calculatd th dffrnt schms. Unform 0 0 CVs ar usd for, UDS, HDSc, 3 CDS, 4 QUCK, 5 fourth ordr CDS; unform CVs ar for 6 th th fourth ordr CDS. 8

13 4. Turbulnt asymmtrc suddn-panson flo n ths flo, flud ntrs a pp panson th damtr rato as :. Rynolds numbr R s 0 5 basd on th ara-avrag vlocty V av and pp damtr R bfor th panson. Th govrnng quatons and th th k-ε modl ar rfrrd to th manuscrpt clt numbr-dpndnt dfrrd corrcton to mplmnt th cntral dffrnc schm to mprov convrgnc. Only th 4 th ordr CDS s rportd hr. Th HDS s usd for th k- and ε-quatons, du to th non-ngatv rqurmnt durng th traton, and also du to th pak valus of k and ε. To non-unform grds ar usd for s panson pp damtrs: 30 6 CVs and CV, rspctvly. Ecpt n th rcrculaton rgon, th flo s rlatvly smpl -- mprovmnt potntal s not larg. Hovr, th mprovmnt can b sn n Fgur 5. Th prsnt proposal outplays th UDS platform at all α, ncludng at hgh α. Th fastr convrgnc n momntum quatons hlps th th k- and εquatons. Ths offsts th slor convrgnc of contnuty quaton at hgh α. Ths s lcom bcaus th practcal applcaton oftn nvolvs mor transport quatons hos convrgnc dpnds on th momntum quatons. Th man advantag s at th lft branch of th curvs, hch s mor sgnfcant for practc. n practc, mult-lvl grds ar usd to acclrat th convrgnc anothr applcaton of dfct corrcton mthod, th bttr prformanc of th prsnt proposal s avalabl on dffrnt grd lvls. Th flo fld s contourd n Fgur 6 for th rfnd grd. Th ddy lngth for th to grds ar 3.86R and 4.R, rspctvly. Th lattr valu s ndpndnt of grd rfnmnt. Th turbulnt kntc nrgy Fgur 6 agrs th prmnt n [7]. Th ffctv knmatc vscosty Fgur 63 can b usd to chck f th schm shftng mchansm can b trggrd at ths hgh Rynolds numbr flo. As for th undr-stmatd ddy lngth, on can mport som curvatur-corrcton 8 to th standard k-ε modl. Ths corrcton prdcts ddy lngth as 4.6R Fgur 64, hch s clos to th prmntal valu 4.75R. Th rsult s shon n Fgur 64. outr-traton numbr th UDS platform th prsnt proposal undr-rlaaton factor, α outr-traton numbr th prsnt proposal th UDS platform undr-rlaaton factor, α Fgur 5 Th outr-traton tms vs. th undr-rlaaton factor α n th turbulnt asymmtrc suddn-panson flo R = 0 5 calculaton th to mthods to mplmnt th fourth-ordr CDS. To grds ar usd, 30 6 CV, convrgnc crtron 0-4 ; CV, convrgnc crtron

14 stramln pattrn Kntc nrgy k / Vav Effctv knmatc vscosty µ ff / ρ Flo pattrn th curvatur-corrcton Fgur 6 Smulaton rsults on asymmtrc suddn-nlargmnt flo at R = 0 5 th fourth ordr CDS and non-unform grd. 4 shos th rsult th curvatur-corrcton. 5. Conclusons An mprovd dfrrd corrcton mthod s proposd to mplmnt hgh ordr dffrnc schms. Th mplct prsson s upgradd from th UDS to th CDS f th clt numbr s lss than. On ths platform, hgh ordr schms shar a d rang of stablty. Thy convrg fastr than on th smpl UDS platform. Thr hgh ordr schms, namly, scond-ordr CDS, th QUCK schm and th fourth-ordr CDS, ar tstd n a gnral form on th ld-drvn cavty flo and turbulnt suddnpanson flo. From th pont of v of th dfrrd corrcton tchnqu, th upgradng mchansm n th currnt proposal nabls th mplct trm to gv a bttr frst appromaton, hch mprovs th convrgnc. Th schm-upgradng mchansm has plotd th local grd rfnmnt n compl flo. Ths amnds th draback of usng UDS n such rgons. 3 Th prsnt proposal mprovs convrgnc for th momntum quatons n compl flos, provdd that th grd s rfnd nough for hgh ordr schm to gt corrct soluton. Rfrncs. atankar, S. V., 980, Numrcal Hat Transfr and Flud Flo, McGra-Hll.. Lonard, B.., 979, A stabl and accurat convctv modlng procdur basd on quadratc upstram ntrpolaton, Computr Mthods n Appld Mchancs and Engnrng, Vol. 9, pp Hnz, Z. 975, Turbulnc, nd dton, McGra-Hll, N York. 4. Zhang, Zj, On th bnchmark to chck flo dvlopmnt n pp, Submttd to J. Hydraulc Rsarch, AHR, 00. 0

15 5. Hayas, T., Humphry, J. A. C. and Grf, R., 99, A consstntly formulatd QUCK schm for fast and stabl convrgnc usng fnt-volum tratv calculaton procdurs, J. Computatonal hyscs, Vol. 98, pp Van Doormaal, J.. and Rathby, G. D., 984, Enhancmnts of th SMLE mthod for prdctng ncomprssbl flud flos, Numrcal Hat Transfr, Vol. 7, pp atankar, S. V. and Spaldng, D. B., 97, A calculaton procdur for Hat, mass, and momntum transfr n thr-dmnsonal parabolc flos, nt. J. Hat, and Mass Transfr, Vol. 5, p Thompson, M.C. and Frzgr, J.H., 989, An adaptv multgrd tchnqu for th ncomprssbl Navr-Stoks Equatons, J. Computatonal hyscs, Vol. 8, pp Gha, U., Gha, K. N., and Shn, C. T., 98, Hgh-R solutons for ncomprssbl flo usng th Navr-Stoks Equatons and a multgrd mthod, J. of Computatonal hyscs, Vol. 48, pp Khosla,. K., and Rubn, S. G., 974, A dagonally domnant scond-ordr accurat mplct schm, Computrs and Fluds, an ntrnatonal Journal, Vol., pp Böhmr, K. and Stttr, H. J., 984, Dfct Corrcton Mthods, thory and applcaton, Sprngr-Vrlag, Wn. Murphy, J. D., and rntr,. M., 985, Hghr ordr mthods for convctondffuson problms, Computrs & Fluds, Vol. 3, pp L, M., Tang, T., and Fornbrg, B., 995, A compact fourth-ordr fnt dffrnc schm for th stady ncomprssbl Navr-Stoks quatons, nt. J. for Numrcal Mthods n Fluds, Vol. 0, pp Gupta, M. M., Manohar R.., and Stphnson, J. W., 984, A sngl cll hgh ordr schm for th convcton-dffuson th varabl coffcnts, nt. J. for Numrcal Mthods n Fluds, Vol. 4, pp Batson, R. K., 986, On th convrgnc of som cubc spln ntrpolaton schm, SAM, J. Numrcal Analyss, Vol. 34, pp Tagaa, T., and Ozo, H., 996, Effct of randtl numbr and computatonal schms on th oscllatory natural convcton n an nclosur, Numrcal hat transfr, art A: Applcaton, Vol. 303, pp Mnh, Ha, and Chassang,., 979, rturbatons of turbulnt pp flo, n: Durst, F. t al. d., Turbulnt Shar Flo, slctd papr from st nt. Symp. on Turbulnt Shar Flos, hold at th nnsylvana Stat Unv., Aprl 979, pp Lschznr, M.A., and Rod, W., 98, Calculaton of annular and tn paralll jts usng varous dscrtzaton schms and turbulnc modl varatons, J. Fluds Engnrng, ASME, Vol. 03, pp

16 Chaptr 3 Toards computaton n physcal doman th nonorthogonal grd Ths chaptr ntroducs a novl mthod to carry out ntrpolaton n non-orthogonal grd.. ntroducton Computatonal flud dynamcs prdcts flos by solvng a st of partal dffrntal quatons. At arly tm, flos to solv r confnd by boundary th smpl gomtry. Cartsan grd as usd. Advancd numrcal mthods, charactrzd by advancd numrcal schms such as th QUCK schm, [] hav bn proposd. Thy ar pctd to dal th compl flos. Hovr, compl flo oftn has compl boundary. n hydraulc or nvronmntal ngnrng, th flo boundary s oftn compl vn f som smplfcatons hav bn don. Cartsan grd s not applcabl. Usually non-orthogonal grd s usd to dscrtz th spac and ft th boundary. Most of th numrcal mthods dvlopd for Cartsan grd cannot b drctly usd n th physcal doman. For a long tm, th computaton s carrd out n an magnary doman th curvlnar coordnat systm. Wth som mappng corrspondnc, non-orthogonal grd n th physcal doman gts to b orthogonal n th transformd systm. Advancd tchnqu can b n th transformd spac. Rh and Cho [] took ths stratgy. So do many commrcal computatonal flud dynamcs programs. Th cost s rlatd to th fact that dffrntal quatons ar mor compl n th computatonal doman. All th frst-ordr drvatvs n th physcal doman turnd to b to drvatvs -D flo plus transformaton paramtrs. Scond-ordr drvatvs gt mor trms. n 3-D cas, quatons ar vn mor compl. Som quatons,.g., Rynolds strss transport quatons, ar complcatd n physcal doman. Thr forms n th transformd doman ar a grat challng to rt. Th control volum mthod can dscrtz dffrntal quatons n physcal doman rgardlss of grd gomtry. Ths advantag nabls th mthod to dal th compl flo boundary th smpl form of quatons. Thortcally, no coordnat transformaton s ncssary bcaus th ara vctor of control volum surfac carrs nformaton of local grd orntaton and dstorton. Thy play th rols of transformaton paramtrs. Th only dffculty s ho to prform ntrpolaton n th non-orthogonal grd. Th prsnt author mports th advantag from th coordnat transformaton mthod to drv th ntrpolaton schms. Th ntrpolaton s carrd out n a staggrd transformd systm. Th dffrntal quatons ar solvd n th physcal doman n smpl form. Ths chaptr ntroducs ho to mplmnt ths proposal.

17 . Mathmatcal formulaton Th gnral form of a consrvatv quaton, oftn nvolvd n th smulaton, rads φ φ φ φ ρu ρv = Γφ sφ, hr u and v ar mass flo spds for convcton, Γ φ dffuson coffcnt and s φ sourc trm. Control volum mthod starts dscrtzaton at ntgral form of : φ φ φ φ ρ u ρv dv = Γφ dv V V sφ dv. y y V n hch ntgral doman V s th control volum CV. At ths stag, t s assumd that th flo fld and th flud proprty ar knon n som ay. Not that Cartsan vlocts ar dfnd at cll cntr. Wth Gauss thorm, r r r ρφu ds = Γφ gradφ ds sφ dv, S S CV n hch th ntgral doman S s th surfac of CV. Th control volum mthod poss no rstrcton on th gomtry of V or S. n th quaton, both th convcton flu and dffuson flu ar projctd to th ara vctor of cll fac. Th ara vctor,.g. of th ast fac Fgur, s r r r s = yn ys n s y, 3 n hch r and r y ar unt vctor for -as and y-as, rspctvly. s r has a unt vctor n r. ts norm s dnotd as s. n n s E n s N n s E y s S η S ξ Fgur Gnralsd curvlnar coordnat rght to mplmnt ntrpolaton for CV ntrfac md-pont on a non-orthogonal grd lft. n a -D Cartsan grd cas, th ast and st facs ar paralll to y as, and north and south facs paralll to as. Th projcton s much smplr. Equaton has form: ρ u y φ ρ u y φ ρ u φ ρ u φ = φ φ φn φs Γφ, y Γφ, y Γφ, n - Γφ, s sφ y, n hch th subscrpt,, n, and s rprsnts th cll surfacs. n Cartsan grd, th ara of th ast and st facs s qual to y and th ara of th north and south facs n n n s s s 3

18 φ s. Th cll fac valu,.g. φ and ts gradnt hav to b stmatd th th nodal valus n CV cntr. Ths stmaton s prformd th som numrcal schms. For ampl, cntral dffrnc schm CDS stmat φ as φ φe φ = n a unform grd, or gnrally E φ = φe φ = f φe f φ, 4 E E n a non-unform grd. Ths schm assums lnar dstrbuton of φ btn th to CVs: and E. Th gradnt on th cll fac can b calculatd as φ φe φ =. δ f th CDS s appld to th cll fac valu, on gts th dscrtzd quaton: [ Fd f Fc Fd f Fc Fd f y Fc n Fd f y Fc s ] φ, = f Fc Fd φe f Fc Fd φw f yfc Fd nφn f yfc Fd sφs sφ y 5 facs fac n hch convcton flu Fc = ρu facs fac and dffuson flu F = Γφ, d n hch δ δ s th dstanc btn th prsnt nodal pont and ts nghbor. Ths nb algbrac quaton srs, th boundary condtons, can b solvd for th fnal soluton. n non-orthogonal grd, th calculaton can tak th sam ay. Th ntgrals n quaton ar calculatd on all th four cll facs. ntgral on any cll fac s appromatd th md-pont rul. Th convcton trm on th ast fac s r r r r ρφ u ds = ρu ds φ ast fac n hch th subscrpt rprsnts th cll fac cntr Fgur. Also th dffuson trm, appromatd th md-pont rul, r r Γφ gradφ ds = Γφ s gradφ. ast fac Up to no, t s on th ay for quaton n non-orthogonal grd to hav th form r r smlar to q. 4. What s ndd ar th convcton flu ρ u ds, th gnral varabl φ and gradnt gradφ. Thy should b ntrpolatd. n Cartsan grd, th CDS s avalabl. n non-orthogonal grd, ths gts dffcult bcaus th currnt cll cntr, ast fac cntr and th ast cll cntr E do not st n a straght ln Fgur. To mplmnt th lnar ntrpolaton, Frzgr and rć [3] proposd a local corrcton mthod. Th corrcton s basd on accssory pont hr th straght ln E crosss th ast fac. Th man da of th local corrcton mthod, n trm of quaton, s φ φ φ = φ y y, 6 nb 4

19 n hch φ s for accssory pont. φ and th to gradnts s lnarly ntrpolatd bcaus, and E ar n a straght ln, rgardlss of th grd non-orthogonalty: E φ = φe φ. E E Ths mthod dos dgrad to quaton 3 for orthogonal grd. t plots Taylor srs panson nar for th ntrpolatd varabl: φ φ φ φ = φ y y y y HOT. On can fnd that quaton 6 s not scond-ordr accurat. Frst, no co-drvatv trm occurs n 6 bcaus t s not avalabl. Ths lors th accuracy of th local φ corrcton mthod to lss than scond ordr. Scond, th corrcton for gradnt : φ φ = nds scond ordr gradnt φ φ φ y y and th accssory pont s usd n stad,.., φ φ, hch ar unavalabl. Th gradnt at φ =. t dos not affct much on th corrcton of φ. Hovr, t dos for th dffuson trm bcaus rplacng φ th φ causs rror that ncrass th grd non-orthogonalty. Ths s th dffculty of ntrpolatng n non-orthogonal grd. f on dos coordnat transformaton to curvlnar coordnat systm, a usual ay to dal th compl gomtry, ths dffculty s rsolvd. Th prsnt author stcks to th stratgy of prformng calculaton n physcal doman but sks som ntrpolaton mthod n transformd systm. f on slcts th curv that passs,, and E tc. as so-ξ coordnat as,,, and E st on a straght ln n th n coordnat systm. Ths coordnat systm s not th sam as n th coordnat transformaton mthod hr th grd lns form th curvlnar coordnat. Th prsnt accssory systm staggrs half CV. Ths dffrnc hlps on do lnar ntrpolaton on ths accssory doman, as follong. Jacoban matr paramtrs lnk th coordnat systm -y and th gnralsd systm ξ-η: = ξ η = ξ ξ η η, 7 ξ η y = ξ η = yξ ξ yη η. 8 ξ η Transformaton paramtrs, tc. dscrb ho th grd n th physcal doman s ξ non-orthogonal. Thy ar calculatd n scond-ordr form: 5

20 yn ys =. η Subscrpt N and S stand for nods north and south to th prsnt nod, rspctvly. Th ncrmnt n th magnary doman ξ η s prssd n trm of ts countrparts n th physcal doman y. yη η y ξ =. 9 ξ yη yξ η Consdr sgmnt : yη η y y ξ ξ =, 0 ξ yη yξ η and sgmnt E: yη E E η E ye y ξ E ξ =. ξ yη yξ η E Bcaus and E ar n th sam straght ln, lnar ntrpolaton can b don n th magnary doman: φ = f φe f φ, n hch th ntrpolaton coffcnt s ξ ξ f =. 3 ξ E ξ ξ ξ Cntral dffrnc schm CDS has th sam form as q. 4 for orthogonal grd, nvolvng only to nodal ponts. Wth th coffcnts, on can ntrpolat varabls, thr gradnts, flud proprts and tc. Smlarly on can drv th QUCK schm [] n non-orthogonal grd. Th drvaton s rfrrd to manuscrpt Toards computaton n physcal doman th non-orthogonal grd: th QUCK schm. Wth ntrpolaton schms, dscrtzaton s largly smlar to that n Cartsan grd. Th convcton trm s appromatd as r r ρφu ds = Fc φ fac, 4 S,, n, s fac n hch mass flu du to convcton r r r Fc = ρ CV surfac u fac v facy s fac. 5 Th valus th subscrpt fac ar valuatd at th md-pont th th CDS. Th dffuson trm r r Γ gradφ ds = Γ s gradφ n S φ fac fac fac fac,,, n, s fac φ r φ r n hch gradφ fac = fac fac y s ntrpolatd th CDS. To trat th dffuson trm mplctly th a dffuson flu so that th program looks lk that for r Cartsan grd, a lor-ordr appromaton to gradφ fac n fac should b usd to do "dfrrd corrcton". Ths "dfrrd corrcton" follos Muzafrja, S. [4] ctd n [3]. Th lor-ordr appromaton s φnb φ. l nb 6

21 n hch l nb s th lngth of th straght sgmnt btn and ts nghbor nb.g. E. Ths lor-ordr appromaton has ntrpolatd valu as gradφ fac nb / l nb,.. gradnt projctd to th drcton of -nb. Th dfrrd corrcton taks th form: r φnb φ r gradφ fac n fac = [ gradφ fac n fac gradφ fac nb / lnb ]. lnb Th trms n th brackt ar lumpd nto th sourc trm. Th frst trm to th rght forms mplct part of dffuson trm: r Γ gradφ ds = F φ φ, 6 th dffuson flu S φ F Γ d,, n, s fac s nb fac fac d =. 7 lnb Ths s th form smlar to that for Cartsan grd. Th coffcnt matr of algbrac quatons s smlar to that n orthogonal grd. Th condtonal stablty of CDS and hghr-ordr schms for th convcton trm can b tratd th clt numbrdpndnt dfrrd corrcton proposd by ths author. Th couplng of prssur and vlocty nvolvd n th latr tst cas s tratd th th SMLE algorthm, [5] hch s also smlar to that n Cartsan grd. Th dscrtzd momntum quatons hav form Apu = Anbunb su, A v p =,, n, s Anbvnb s,, n, s * Wth th to quatons, on gts th vlocty u and v *, as th frst appromaton. Wth CDS, on gts th mass flu at ast r fac as: r * * * r m = ρ u v y s. 8 Along th ts countrparts at othr surfacs, ths mass flu dos not fulfll th contnuty rqurmnt: * m fac = m 0. 9,, n, s To nforc zro mass balanc, mpos a corrcton on vlocts to gt n valus: * * u = u u ', v = v v '. 0 Du to th momntum quaton, th corrcton taks th form as: V p' V p' u' =, v' =, A A n hch p s th prssur corrcton. Ths s th man appromaton of th SMLE algorthm. Th vlocty corrctons ffctvly caus mass flu corrcton at CV surfac: V p m ' = ρ s ', A n v. 7

22 p' p' r p' r r n hch V s th volum of CV and = y n n y s th corrcton prssur gradnt at th drcton normal to th surfac along n r p '. s n dscrtzd as p' p' E p ' = r. n E n Th mass flu corrcton m ' should balanc m n quaton 9: V p' nb p ' ρ fac fac s fac r m = 0. 3,, n, s facs A Nb n hch lnks nodal corrcton prssurs togthr. Ths corrcton prssur quaton has coffcnt matr that s smlar to that n orthogonal grd. Onc th corrcton prssur s solvd, th vlocty s corrctd. Bcaus nodal varabls ar collocatd, an oscllaton-dtctor trm s appld to mass flu m * suggstd by Rh and Cho [] r r * * * r V pe p p m = ρ u v y s r A. 4 E n n p n hch s th ntrpolatd prssur gradnt th nodal valus at CV cntrs. n Smlar dtctor s also nstalld on mass flus through othr CV surfacs. f thr s chssboard-lk prssur, ths dtctor s trggrd and th oscllaton s smoothd out. 3. Tst cass Th tst cas for th CDS s lamnar flo n a U-turn channl. Ths flo s govrnd by th Navr-Stoks quatons. Fgur shos th rsult at R =00. Th dtals can b rfrrd to manuscrpt Toards computaton n physcal doman th non-orthogonal grd: th cntral dffrnc schm. Th tst cas for th QUCK schm s th ld-drvn cavty flo. Th grd s dran as non-orthogonal. Hgh Rynolds numbr s shon. Th dtals can b rfrrd to manuscrpt Toards computaton n physcal doman th nonorthogonal grd: th QUCK schm. 4. Conclusons A n mthod s ntroducd to calculat compl flo n physcal doman. An accssory curvlnar coordnat systm s usd to drv ntrpolaton coffcnts. Ths curvlnar coordnat systm s staggrd half CV from that usd n th coordnat transformaton mthod. Ths n systm nabls on to calculat ntrpolaton coffcnts. Wth ntrpolaton avalabl, dscrtzaton of quatons n physcal doman th non-orthogonal grd s smlar to that n Cartsan grd. Th advancd mthods dvlopd n Cartsan grd can b usd n non-orthogonal grd. On can calculat flos n physcal doman th non-orthogonal grd as asly as th Cartsan grd. 8

23 Fgur Non-orthogonal grd th CV dran on grd ln n vry to n and grd-ndpndnt flo pattrn at R = 00. Outr-traton s trmnatd at rsdual of algbrac quaton lss than Fgur 3 Non-orthogonal grd th CV th on grd ln dran n vry four n, flo pattrn at R = 00 n and flo pattrn at R = 000 n 3. Th QUCK schm s usd. Outr-traton s trmnatd at rsdual of algbrac quaton lss than

24 Rfrncs. Lonard, B.., 979, A stabl and accurat convctv modlng procdur basd on quadratc upstram ntrpolaton, Computr Mthods n Appld Mchancs and Engnrng, Vol. 9, pp Rh, C. M., and Cho, W. L., 983, Numrcal study of th turbulnt flo past an arfol th tralng dg sparaton, AAA Journal, Vol., pp Frzgr J. H., and rć, M., 996, Computatonal Mthods for Flud Dynamcs, Sprngr, Brln. 4. Muzafrja, S., 994, Adaptv fnt volum mthod for flo prdctons usng unstructurd mshs and multgrd approach. hd Thss, Unvrsty of London. 5. atankar, S. V. and Spaldng, D. B., 97, A calculaton procdur for Hat, mass, and momntum transfr n thr-dmnsonal parabolc flos, nt. J. Hat, and Mass Transfr, Vol. 5, p.787. Nomnclatur A th matr coffcnts n algbrac quatons dv, V control volum,, n, s th four CV surfacs f, f lnar ntrpolaton coffcnts n - and y-drcton y F c convcton flu F dffuson flu d r, r y unt vctor for and y- as n r unt vctor of s r nb th nghbors of th prsnt cll, can b W, E, N or S p prssur p ' prssur corrcton n th SMLE algorthm, W, E, N, S th prsnt control volum and ts four nghbors s r ara vctor of th ast surfac u flud vlocty at drcton v flo vlocty at y drcton * * u, v vlocty obtand from momntum quatons u ', v ' vlocty corrcton n SMLE algorthm, y coordnat systm n physcal doman Γ φ dffuson coffcnt φ gnral varabl ρ dnsty µ dynamc vscosty ξ, η curvlnar coordnat n th accssory systm 0

25 Chaptr 4 Computaton of orfc flo Th mthod n th prvous chaptr s appld to turbulnt orfc flo.. ntroducton Orfc flo also has compl boundary. t should b calculatd th body-fttd non-orthogonal grd. rvously, Cartsan grd as usd by som authors. Cartsan grd nabls asy dscrtzaton. Hovr, ths asts much of th grd rfnmnt bcaus th orfc body tslf s also cut nto control volums and calculatd as f t as flo fld. [] Usng non-orthogonal grd s a propr choc. Coordnat transformaton s oftn don, as n []. Th physcal doman s mappd nto rgular doman. Wth th mthod n th prvous chaptr, th calculaton can b don n physcal doman. Th mthod ll hav both th advantags of th to mthods smpl form of quatons and smpl ntrpolaton schm. Th calculaton of non-orthogonal grd s no longr a dffculty. Ths chaptr focuss on slctng corrct schm to rflct th complty of th flo. Orfc flo has local hgh gradnt n th shar layr. Thr s pak on th dstrbuton of turbulnt kntc nrgy and dsspaton rat. A schm th lmtr, Ultra-QUCK, [3] s appld to th flo. Ths schm has long bn rcommndd by Lonard, B.. Hovr, du to th complty of ths schm, t s not dly appld n turbulnt flo smulaton. Th applcaton of ths schm n physcal doman th non-orthogonal grd has not bn documntd so far.. Mathmatcal formulaton Th flo s govrnd by momntum quatons: u u p u u ρu ρv = µ ff rµ ff r r r r, u v µ ff rµ ff r r r v v p v v ρu ρv = µ ff rµ ff r r r r r, u v v µ ff rµ ff µ ff r r r r r and th contnuty quaton: u v = 0. 3 r Th ffctv vscosty s th sum of flud dynamc vscosty µ and ddy vscosty µ : t µ ff = µ µ t, n hch µ t s gvn by an ddy-vscosty modl, such as k-ε modl:

26 k µ t = ρc µ. 4 ε As for th k-ε modl, to quatons ar nvolvd. On s for th turbulnt-moton kntc nrgy k: k k µ ff k µ ff k ρu ρv = r { k ρε}. 5 r σ k r r σ k r Th othr s for th dsspaton rat of th turbulnt nrgy, ε: ε r ε µ ff ε µ ff ε ε ε ρu ρv = r C k c ρ. 6 r r σ ε r r σ ε r k k Th trm k s th rat of gnraton of k: u v v u v k = µ t. 7 r r r Coffcnts C, C, σ k, σ ε and C µ hav standard valus as: C =.44, C =.9, σ k =.0, σ ε =.3 and C µ =0.09. n th abov transport quatons,, 5 and 6, th trms n th squar brackts ar tratd as dffuson trms and thos n th bracs ar sourc trms. Th boundary condtons ar: u r r /8 nlt: =,.. por-la vlocty profl; u r = 0 R v = 0; r 3 r k = Tc u r = R R n hch T c s th turbulnc ntnsty basd on local man vlocty at th pp cntr. [6] T c s st as 4%; 3 r r.07 u* r ε =.6 for 0. 9 and thn kps constant up to th R R R R all; [9] u * taks valu as u r = 0 at th prsnt Rynolds numbr; [7] k : th masurd data s usd; [9] Outlt: ϕ / = 0 for u, k and ε, and v ; / Sold boundars: u = 0, v = 0, and th all la for k = τ ρcµ and ε τ 3/ = / k ; ρ p cntr: φ / r = 0 for u, k and ε, and v = Frst ampl n ths ampl, to orfcs ar nstalld n th pp th 3.0 pp damtrs ntrval. Thr damtr ratos ar both Rynolds numbr R s 0 5 basd on th ara-avrag vlocty n pp and pp damtr D = R. Th flo fld has bn publshd lshr. [0] Thr s on ddy aftr ach orfc. Th upstram on s longr than th donstram on. Th smulaton doman s nn pp damtrs long. t s dscrtzd nto CV hch nabls grd-ndpndnt soluton. Th CDS s usd for th convcton and

27 dffuson trms of momntum quatons. Th upnd dffrnc schm and cntral dffrnc schm ar usd for th convcton trm and dffuson trm, rspctvly for k- and ε-quatons. Fgur shos th grd and rsults. On th flo pattrn, th to dds ar.57d and 0.94D long, rspctvly. Th prmnt gvs.8d and 0.98D, rspctvly. [0] Th mamum vlocty n th flo fld, du to flo convrgnc, s u v ma =.47V, n hch V s th ara-avrag vlocty n th orfc bor. Th prmnt gav rsult as.6. Th mamum turbulnt kntc nrgy gvs valu as k / ma uma = Ths valu s clos to th prmntal valu 3 3 k ma / uma = 0.9. Hovr, du to th lor ma u, ths mamum turbulnt nrgy s also undr-prdctd. n th follong, th prdcton s mprovd by usng hghr-ordr schm. a Grd, on ln s dran n vry to b Flo pattrn c k- contour d ε contour Fgur Grd, flo pattrn and k- and ε contours for flo th doubl orfcs. Som donstram part of th computatonal doman s not shon. Th lngth scals n th to drctons ar th sam. 4 Th Ultra-QUCK schm prforms bttr Th scond tst flo s confnd n pp th an orfc th a : approachng slop. Th flo fld and prssur dstrbutons ar publshd lshr. [4,5] Wth 430 CVs, grd-ndpndnt soluton s avalabl. Ths covrs a computatonal doman as long as.5d,.5d bfor th tralng dg and 0D aftr. Ths tra long donstram doman hlps th flo to rcovr to nlt vlocty profl. Th Ultra-QUCK schm [3], hghr ordr than n th abov ampl, s usd for th convcton trms. Th cntral dffrnc schm CDS s stll usd for th 3

28 dffuson trms. Th Ultra-QUCK schm uss th ordnary QUCK schm [8] f th gnral varabl φ s n monotonous varaton rgon. A lmtr s actv f th varabl s at local mamum or mnmum valu. Ths lmtr forcs th schm to choos th upnd schm, cntral dffrnc schm or upnd cntral dffrnc schm nstad of th QUCK schm so that ovr-shootng dos not occur f a hgh gradnt sts nar th cll fac. Ths nabls th schm to prdct hgh gradnt of varabls. For ampl, th vlocty has a hgh gradnt n th shar layr. Wth th Ultra- QUCK schm, th vlocty profl has no ovr-shootng or undr-shootng othrs causd by CDS or th QUCK schm. Th vlocty profls at 0.5D donstram th panson ar compard n Fgur. Vlocty profl s normalzd th th ara-avrag vlocty V n th orfc bor. Th CDS causs oscllaton on th profl bfor th shar layr. Also, th CDS undr-rports th mamum vlocty aftr th panson n Fgur. Ths causs undr-stmaton of ddy lngth. Ths also plans hy th ddy aftr th frst orfc n th abov ampl s undr-prdctd. Th pntraton of th oscllaton causd by th CDS s proportonal to cll clt numbr. n th frst ampl, thr s hgh ffctv vscosty bfor th scond orfc. Ths lors th local clt numbr. Ths plans hy th CDS can gv corrct prdcton for th ddy aftr th scond ddy. To prdct both th ddy lngth and vlocty profl, th Ultra-QUCK schm s th corrct choc. ur / V oscllaton Ultra-QUCK CDS Eprmnt r/r Fgur Comparson of th CDS and th Ultra-QUCK schm for th vlocty profl nar th shar layr 0.5 D donstram from th donstram fac of th suddn-panson, obtand at half grd rfnmnt. Th Ultra-QUCK also gvs prdcton on th turbulnt kntc nrgy th hgh prcson. Fgur 3 shos th rsults obtand th th Ultra-QUCK schm. Not that CDS cannot b usd for k-quaton at all. Ths s th rason hy no CDS rsult s dran n Fgur 3. Mor nformaton of usng Ultra-QUCK s n th manuscrpt Control volum mthod to calculat n physcal doman th non-orthogonal grd. 4

29 k 0.3 / V r/r Fgur 3 Comparson of turbulnt vlocty profl at D aftr th panson hr mamum kntc nrgy s locatd. Crcl markrs prmnt; sold ln Ultra-QUCK th th lmtr stchd off nar th pak. Th grd s shon n Fgur 4, along th th smulaton rsults. f th cntral dffrnc schm CDS s usd, th ddy lngth s.d Fgur 4b. Wth th Ultra-QUCK schm, th ddy lngth s.35d Fgur 4b, compar th th masurd lngth.4d. As comparson shos, th smulaton corrctly prdcts th mamum vlocty and mamum turbulnt kntc nrgy. agrd, on ln s dran n vry four 0.5D b Flo pattrn th CDS for th convcton trm 0.5D c Flo pattrn th th Ultra-QUCK schm for th convcton trm d k- contour th th Ultra-QUCK schm ε contour th Ultra-QUCK schm Fgur 4 Grd, flo pattrn and k- and ε contours. Som donstram part of th computatonal doman s not shon. Th lngth scals n th to drctons ar dffrnt. Th horzontal scal s shon n b on 0.5D. 5

30 Fgur 5 shos th all prssur dstrbuton. rssur s rlatv to th upstram valu and normalzd usng th vlocty V. Th smulaton prdcts th rcovry of th prssur. p ρ / V r/r Fgur 5 rssur dstrbutons at th pp all, crcl prmnt, sold ln smulaton. 5 Conclusons Th mthod, calculaton n physcal doman and ntrpolaton n transformd doman, s tstd on turbulnt orfc flo. To prdct th hgh varabl gradnts n th flo and to mantan a hgh ovrall prcson, th Ultra-QUCK schm s usd for th convcton trms n all dffrntal quatons. Th Ultra-QUCK prforms bttr than th cntral dffrnc schm. Th flo fld clos to th panson s corrctly prdctd, spcally th ddy lngth. Vlocty, kntc nrgy and prssur agr th prmnt. Wth th mthod proposd n prvous chaptr, th calculaton of non-orthogonal grd gts asy. Th advancd schms can b appld to nonorthogonal grd. Rfrncs. Morrson, G. L., anal, D. L. and DOtt, R. E., Jr., 993, Numrcal study of th ffcts of upstram flo condton upon orfc flo mtr prformanc, Journal of Offshor Mchancs and Arctc Engnrng, ASME, Vol. 5, pp Zhang, Zhuang, and Ca, Junm., 996, Body-fttd numrcal modl for multorfc flo and ts applcaton, Journal of Hydrolctrc Engnrng, Vol. 55, pp Lonard B.. and Mokhtar, S., 990, Byond frst-ordr upndng: th ULTRA- SHAR altrnatv for non-oscllatory stady-stat smulaton of convcton, nt. J. Numrcal Mthods n Fluds, Vol. 30, pp Zhang, Z., Ca, J. and Shn, X., 99, A study of Flo fld donstram assymmtrc suddn-nlargmnt by usng LDV systm, roc. nt. Sym. on Hydraulc Rsarch n Natur and Laboratory, Nov., 99, Wuhan, pp

31 5. Zhang, Z. and Ca, J., 999, Comproms orfc gomtry to mnmz prssur drop, J. Hydraulc Engnrng, ASCE, Vol. 5, pp Zhang, Z., On th bnchmark to chck flo dvlopmnt n pp, Submttd to J. Hydraulc Rsarch, AHR, Hnz, Z. 975, Turbulnc, nd dton, McGra-Hll, N York. 8. Lonard, B.., 979, A stabl and accurat convctv modlng procdur basd on quadratc upstram ntrpolaton, Computr Mthods n Appld Mchancs and Engnrng, Vol. 9, pp Laufr, J., 954, Th structur of turbulnc n fully dvlopd pp flo, NACA TN Shn, X., Yan Y., Gao, J. and Dng, Z., 988, Turbulnt vlocty masurmnts n orfc pp flo th an mprovd on-as LDV systm, prsntaton to 4th ntrnatonal Symposum on Applcaton of Lasr Anmomtry to Flud Mchancs, nsttuto Supror Técnco, Lsbon. Nomnclatur C, C, σ k, σ ε and C µ : modl paramtrs for th k-ε modl D pp damtr p prssur r asymmtrc coordnat R pp radus u flud vlocty at drcton u ma th mamum vlocty; v flo vlocty at y drcton u * all frcton vlocty V ara-avrag vlocty n th orfc bor coordnat systm n physcal doman k turbulnt kntc nrgy ε nrgy dsspaton ρ dnsty µ ffctv vscosty ff t µ turbulnc vscosty µ dynamc vscosty 7

32 Fnal Conclusons n ths thss,. a platform th hybrd dfrrd corrcton mthod s proposd to mplmnt hghordr dffrnc schm for th convcton-dffuson problm. Th mplct part of th dfrrd corrcton mthod should upgrad from th upnd dffrnc schm UDS to th cntral dffrnc schm CDS f th local grd Rynolds numbr allos th lattr. Ths amnds th draback of purly usng th UDS. Th convrgnc gts fastr;. a staggrd curvlnar coordnat systm s usd to drv ntrpolaton schms to carry out flo smulaton n physcal doman th non-orthogonal grd. Th schms for th non-orthogonal grd thus hav th form smlar to that for Cartsan orthogonal grd. Th computaton can b don n physcal doman hr quatons hav smpl forms, th non-orthogonal grd as asly as th Cartsan grd. 3. th abov mthod s appld to turbulnt orfc flo. Advanc schm can b usd n physcal doman th non-orthogonal grd. Th Ultra-QUCK schm can mprov th prdcton of orfc flo; 4. a hydraulc abruptnss s proposd to dscrb orfc gomtry. Wth ths quantfcaton, stng orfc flo masurmnts can b asly usd to gt nformaton for a n orfc, savng dtald masurmnts. 33

33 Appnd Appndd paprs and manuscrpts and contrbuton to jont paprs. Zhang, Z. and Ca, J., 999, Comproms orfc gomtry to mnmz prssur drop, Journal of Hydraulc Engnrng, ASCE, Vol Frst author carryng out prmntal ork and rtng th papr.. Zhang, Z., 00, A n form to mplmnt th QUCK schm to mprov convrgnc and stablty, n rv by ntrnatonal Journal of Numrcal Mthods n Fluds 3. Zhang, Z., 00, clt numbr-dpndnt dfrrd corrcton to mplmnt th cntral dffrnc to mprov convrgnc, n rv by Journal of Computatonal hyscs, 4. Zhang, Z., 00, Control volum mthod to calculat n physcal doman th nonorthogonal grd, n rv by Journal of Hydraulc Rsarch, AHR 5. Ca, J.M., Fng, J. M., and Zhang, Z., 000, Flood tunnl th orfcs to control flo rat, rocdng of 4 th nt. Confrnc on Hydro-Scnc and Engnrng, CHE 000, Soul, Spt, Thrd author, rtng th papr. 6. Zhang, Z., 00, Toards computaton n physcal doman th non-orthogonal grd:cntral dffrnc schm, Submttd to Numrcal Hat Transfr. 7. Zhang, Z., 00, Toards computaton n physcal doman th non-orthogonal grd: th QUCK schm, Submttd to ntrnatonal Journal of Numrcal Mthods n Fluds. 8. Zhang, Z. 00, Scond-ordr ntrpolaton schm basd on shap functon for control volum mthod, Submttd to Th ntrnatonal Journal of Numrcal Mthods n for Hat & Flud Flo. 34