Grid Transformations for CFD Calculations

Transcription

1 Coll of Ennn an Comput Scnc Mchancal Ennn Dpatmnt ME 69 Computatonal lu Dnamcs Spn Tct: 5754 Instuct: La Catto Intoucton G Tansfmatons f CD Calculatons W want to ca out ou CD analss n altnatv conat sstms. Most stunts hav alt wth pola an sphcal conat sstms. In ths nots w want to tn ths noton of ffnt conat sstms to cons abta conat sstms. Ths ppas th wa f th consaton of bo-ftt conats n CD poblms. Bo ftt conat stat b fttn a conat ln to th phscal bo un consaton. Ths bo ma b an abta shap. On consqunc of th us of bo-ftt conats s that th sultn conat sstm ma not b thoonal. In an thoonal conat sstm th unt vcts n an conat cton wll b ppncula to th unt vcts n th oth ctons. In a nal thoonal conat sstm w can pss th unt vcts n th th conat ctons an as th follown column vcts. [] W usuall us th notaton to psnt on componnt of a vct. H w us th notaton nsta of f th unt vcts to mphas that ths a n fact vcts wth th componnts; w a not fn to th nvual componnts th s an s of ths vcts h. Wth ths fnton of th unt vcts f th thoonal conat sstm th vct ot pouct of an two unt vcts ma b psnt as follows. T f [] f In quaton [] th fst notaton s th convntonal ot-pouct notaton f vcts; th scon notaton s th mat multplcaton notaton f column vcts. Both opatons pouc th Konc lta fn as th last tm n quaton []. ou Catsan conats th unt vcts a usuall not as an. W wll call ths unt vcts an n ths nots. Conat tansfmatons qu two ffnt tms. Th fst s a tansfmaton of conats. Whn w hav a vatv of som functon wth spct to n a Catsan sstm how o w pss that vatv n a nw conat sstm? Th scon qustons lats to componnts of vcts. If th vloct vct has ctan componnts alon th as n a Catsan sstm what a th componnts of th vloct vct n th tansfm sstm? In aton to vcts w hav to cons th sha stss σ. Ths two mnsonal quantt s nown as a tns an ts nn componnts must hav appopat tansfmatons to t nto th nw conat sstm. Whn w pat fom an thoonal sstm w hav an atonal complcaton. Th a two concpts that a th sam n thoonal sstms. Th fst s th tannt to a conat ln. Ennn Buln Room Mal Co Phon: E-mal: lcatto@csun.u 848 a:

2 Conat tansfmatons ME 69 L. S. Catto Spn Pa Th scon s th nmal to a sufac wh a conat s constant. Ths s not alwas th cas n nonthoonal sstms. ampl a conat ln n th cton of anula splacmnt θ n a clncal conat sstm scbs a ccl. A tannt ln to a ln alon whch onl th θ conat vas s a tannt to a ccl. In ths conat sstm a sufac on whch θ s a constant s a plan. A vct nmal to ths plan of constant θ wll pont n th sam cton as th tannt to th θ conat ln. In nonthoonal conat sstms ths s not th cas. Dfnton of Gnal Conat Sstms W a concn about tansfmatons btwn two ffnt conat sstms. Ultmatl th analss of two nal conat sstms wll allow us to obtan latons wth th m famla Catsan conat sstm. W us th notaton an to fn on conat sstm an th notaton an to fn th scon conat sstm. ampl w coul hav th Catsan conat sstm = = an = as on sstm. * Th oth sstm mht b th clncal conat sstm: = = θ an =. In nal an on of th nw conats can pn on all th of th ol conats. Thus w can wt th follown nal latonshp. [] W can wt all th latonshps as shown blow. [4] ampl w can wt th usual quatons f th convson fom a Catsan conat sstm to a clncal conat sstm n tms of th usual θ vaabls an th notaton ntouc h n quaton [5]. tan tan [5] W want to nvt th nal functonal latonshp n [] an [4] so that w o bac an fth btwn th two conat sstms. That s w want to hav latonshps l th follown. [6] As bf th functonal latonshps shown n quaton [] a quvalnt to th follown th latonshps. * Lat n ths nots w wll us th notaton to scmnat btwn a Catsan sstm an an oth conat sstm. That s w wll wt = = an = f a Catsan conat sstm.

3 Conat tansfmatons ME 69 L. S. Catto Spn Pa [7] Th quatons shown n [5] f tansfmn a Catsan sstm nto a clncal conat sstm hav th nvs latonshps shown n quaton [8]. As n quaton [5] w wt ths quatons n tms of th usual θ vaabls an th nal conat sstm ntouc h. sn sn cos cos [8] Th nvson of ths functonal latonshp f th conat tansfmaton qus th acoban tmnant f th tansfmaton b nono. Ths tmnant s pss b th follown smbolc notaton. [9] Th actual fnton of ths tmnant s shown blow. [] Th functonal latonshps n quaton [] allow us to pss a ffntal chan n th nw Catsan conats n tms of th ffntal chans n th nw conat sstm b th usual latonshp f total ffntals n tms of patal vatvs. [] W can wt th nvs latonshp f th total ffntal b a smla latonshp. [] Usn th summaton convnton that pat ncs n a snl tm a summ ov all th conat valus w coul wt quatons [] an [] as follows.

4 Conat tansfmatons ME 69 L. S. Catto Spn Pa 4 [] In both cass th pat n s summ ov ts th possbl valus an. To stat ou nal conat sstm w cons a poston vct that s fn n a Catsan conat spac as follows wth unt vcts an. Th poston vct s fn as follows n a Catsan sstm. W us th as ou Catsan conats.. [4] Th vatv of th poston vct wth spct to a patcula nw conat s vn b quaton [5] whch s also us to fn th bas vct n th nw conat sstm. [5] In th last psson w us th summaton convnton ov th pat n. Th th bas vcts fn n quaton [5] a th quvalnt of th usual unt vct that w hav n ou Catsan conat sstm. W can us ths bas vcts to comput th ffntal vct lnth alon an path n ou nw conat sstm. [6] In th two ml tms of quaton [6] w hav us th summaton convnton th summaton ov pat ncs. W can wt an lmnta lnth n Catsan spac s as th mantu of th vct. Ths s th squa of th mantu of th vct whch s th absolut valu of th ot pouct of th vct wth tslf. s [7] Not that all th tms nvolvn ncs an hav both ncs pat. Thus w sum ov both ncs an w hav nn tms n th sum f s. Th ot pouct of two bas vcts an s fn on of th nn componnts of a quantt nown as th mtc tns. om th fnton of n quaton [5] w can obtan a scala quaton f as follows. In th follown w us quaton [] that fns th thoonalt conton f th Catsan unt vcts. m m m m m m [8] ampl n a clncal conat sstm = cos = sn an =. W hav th follown patal vatvs.

5 Conat tansfmatons ME 69 L. S. Catto Spn Pa 5 cos sn sn cos [9] Usn quaton [8] w can comput som of th componnts f th clncal conat sstm. sn sn cos sn cos sn sn cos [] Th mann unqu off-aonal tms an can both b shown to b o. Th mann off aonal tms an a sn to b smmtc b th basc fm of quaton [9]. Ths tms wll also b o. Whn th mtc tns has o f all ts off-aonal tms th sultn conat sstm s thoonal. In an thoonal sstm ach bas vct s ppncula to th oth two bas vcts at all ponts n th conat sstm. Th ffntal path lnth vn b quaton [7] whch w us to fn a nw tm f thoonal sstms onl h =. h h h h s [] In th quaton [] ampl of clncal conats w ha = h = = h = an = h = =. Thus th th tms n quaton [] a θ an. W s that h = multpls th ffntal conat θ an sults n a lnth. Ths s a nal sult f an h coffcnt; ths coffcnt s a fact that tas a ffntal n a conat cton an convts t nto a phscal lnth. Ths fact also appas n opatons on vct componnts f thoonal sstms. Ths facts a usuall wttn n tms of Catsan conats an b th follown quatons whch a a combnaton of quatons [] an [8]. h h h []

6 Conat tansfmatons ME 69 L. S. Catto Spn Pa 6 Now that w hav an psson f th ffntal lnth n ou nw conat sstm w can v quatons f ffntal aas an volums. om quaton [7] w s that th lnth of a path alon whch onl on conat sa chans s vn b th quaton no summaton ntn; th vct psntaton of ths path s. To t an ffntal aa fom two ffntal path lnths w ta th vct coss pouct of ths two ffntal lnths. Th vct coss pouct vs th pouct of two ppncula componnts of th ffntal path lnths to calculat an ffntal aa S. S no summaton cclc Th vct that sults fom th coss pouct s n th plus mnus conat cton pnn on whch cton th sufac s facn. Th noton that an a cclc mans that w us onl th follown th combnatons = = = = = = = = =. In to comput th mantu of th sufac aa w n to comput th mantu of th vct coss pouct =. To obtan a usful sult fom ths fnton w n to us th follown vct ntt. A B C D A C B D A D B C [] [4] Usn A = C = an B = D = vs th follown sult f th coss pouct of bas vcts. Wth ths psson w can wt th mantu of th ffntal sufac aa n cton as follows. [5] S no summaton cclc [6] Nt w obtan an quaton f th ffntal volum lmnt n ou nal conat spac. Ths s on b tan th vct ot pouct of ffntal aa lmnt n quaton [] an th ffntal lnth lmnt nmal to th aa. Ths vs th ffntal volum lmnt b th follown quaton. V no summaton cclc [7] ust as w f th ffntal aa lmnt w also s th mantu of th vct tm n th volum lmnt quaton. Ths qus that w fn th tm =. To stat ths w n th follown vct ntt. A B C A A B C B C A B C [8] W can us th ntt n quaton [4] to substtut f th tm B C B C. W can also us th follown ntt to substtut f th A B C tm. A B C B A C C A B [8]

7 Conat tansfmatons ME 69 L. S. Catto Spn Pa 7 Snc w hav onl th bass vcts w wll us th follown bas vcts fom quaton [7] n quaton [8]: A = B = an C =. Man ths substtutons an conn that th ot pouct = th mtc coffcnt vs th follown sult. In aann quaton [9] w hav ma us of th smmt latonshp f th mtc tns componnts = n obtann th th ln. W s that ths fnal ln s ust th quaton f th tmnant of a aa. If w wt ths tmnant as w hav th follown sult f th volum lmnt. [9] V [] Dt Th valu of s th sam as th valu of th acoban tmnant n quaton []. You can show ths b wtn th acoban tmnant as a mat an squa that mat. Th componnts n th sultn pouct mat wll b th componnts of th mtc tns. Bcaus th tmnant of a mat pouct s th sam as th pouct of th tmnants of th two matcs w hav th sult that =. If w tun to ou pvous ampl of clncal conat sstms f whch = = = = an = = = = = = th valu of s smpl th pouct of th aonal tms whch s qual to n th convntonal notaton. ths sstm quaton [] f V vs th usual sult f th ffntal volum n a clncal conat sstm V = θ. Ecs: th sphcal pola sstm th th conats a th stanc fom th n to a pont on a sph th countclocws anl on th - plan fom th as to th pocton of th conat on th - plan an = th anl fom th vtcal as to th ln fom th n to th pont. Ths conats a m convntonall call θ an φ. Th tansfmaton quatons fom Catsan conats an to sphcal pola conats a vn b th follown quatons: = + + = tan - / an = tan - [ + / ]. Th nvs tansfmaton to obtan Catsan conats fom sphcal pola conats s: = cos sn = sn sn an = cos. n all componnts of th mtc tns f ths tansfmaton. Vf that ths s an thoonal conat sstm. What a th th possbl ffntal aas f ths sstm? What s th volum lmnt f ths sstm? Vct componnts n nal conat sstms Th smplst vct to cons n a nal conat sstm s th vloct vct v whos componnts a th vatvs of th conats wth spct to tm. W can fn th vloct componnt n a patcula cton b th smbol v. Th fnton of v n th abta conat sstm an ts latonshp to th Catsan conat sstm s shown blow wh w hav us quaton [] [] f th conat tansfmaton substtutn th notaton of f th Catsan conats. v t t t t t []

8 Conat tansfmatons ME 69 L. S. Catto Spn Pa 8 W s that th tms /t on th ht-han s of quaton [] a ust th vloct componnts n th Catsan conat sstm. In aton th s no patcula ason to assum that th nal sstm s Catsan w coul quall wll us th notaton f th altnatv conat sstm an th notaton v f th vloct componnts n that sstm. Ths vs th follown quaton f th tansfmaton of vloct componnts fom on conat sstm to anoth. v v v v t t t t v [] Ths tansfmaton quaton f componnts of th vloct vct can b contast wth th tansfmaton quaton f th componnts of th ant vct. Equaton [] s th quaton f th ant of a scala A; wttn as A n Catsan conats. A A A A [] If w not on componnt of ths vct as a w can wt ths componnt an ts conat tansfmaton nto a nw sstm a A a A A A A A a [4] If w compa quaton [4] f th tansfmaton of th componnts of a ant vct wth quaton [] f th tansfmaton of th componnts of a vloct vct w s that th s a subtl ffnc n th quatons. tansfmn th ant vct fom th ol ā componnts to th nw a componnts th patal vatvs of th conats hav n th numat. th tansfmaton of th vloct componnts fom th ol conat sstm v nto th v componnts of th nw sstm th ol conats appa n th nomnat. It thus appas that w hav two ffnt quatons f th tansfmaton of a vct. What w hav n fact s two ffnt ns of vcts fn b th tansfmaton quatons. A vct that s tansfm fom on conat to sstm to anoth usn quaton [] s call a contavaant vct. On that tansfms accn to quaton [4] s call a covaant vct. You can mmb ths nams b f ou mmb that covaant vcts hav tansfmaton latons f vct componnts n whch th ol conats a collocat wth th ol vct componnts n th numat of th tansfmaton. Th tansfmaton latons f contavaant vcts hav th ol conats an th ol vct componnts locat n th oppost locatons ol vct componnts n th numat an ol conats n th nomnat. In accanc wth ths nams w call th vloct a contavaant vct an th ant a covaant vct. Althouh th a natuall two tps of vcts accn to th tansfmaton latonshps ths ffncs sappa f an thoonal conat sstm. In aton on can pss a covaant vct b ts contavaant componnts an vc vsa. Th covaant vct componnts psnt th componnts alon th conat lns. Th contavaant componnts psnt th componnts alon nmal to a plan n whch th conat valu s constant. A vct such as vloct alwas has th sam mantu an cton at a vn locaton n a flow. Th onl thn that vas n ffnt conat sstms s th sa n whch w choos to psnt th vct. In an thoonal sstm onl ou choc of conat sstm chans th psntaton of th vct. In a nonthoonal sstm w choos not onl th

9 Conat tansfmatons ME 69 L. S. Catto Spn Pa 9 conat sstm but also whth w want to psnt th vct b ts covaant contavaant componnts. Althouh much of th nal w on bouna ftt conat sstms us ffnt psntatons of vloct componnts most cunt a appoachs us a m fmulaton. Th conat sstm s nonthoonal but w us Catsan vct componnts. Ths s l usn a -θ- conat sstm but lavn th vloct componnts as v v an v. Ths s not a ws choc but t s possbl. Whn w a aln wth compl bouna-ftt conat sstms th us of Catsan vct componnts os pouc smpl sults f th CD calculatons. Tansfmn th basc quatons Th analss of bouna-ftt conats usuall stats wth th quatons n a Catsan bass an spas of a tansfmaton of a mnsonlss conat sstm ξ η ζ. Ths s somtms psnt as a tansfmaton fom a Catsan sstm to th mnsonlss sstm ξ ξ ξ. Th latt fm of th tansfmaton allows th us of th summaton convnton. Th tas of tmnn th nw conat sstm s th tas of fnn th appopat tansfmatons ξ = ξ η = η an ζ = ζ. Ths tansfmatons can b compactl wttn as th vct tansfmaton ξ = ξ. Th tansfmatons convt ou nown omt n spac nto a computatonal spac whos conats a ξ η ζ. Th tas of fnn a tansfmaton whch s scuss blow s on of fnn Catsan conat valus f ach pont fn n ξ-η-ζ spac such that th -- bouna conats of omt bn mol bcoms conat bounas f th computatonal. Th sultn qus us to tansfm th nal quatons nto th tansfm conats ξ η ζ ξ ξ ξ to nvo th convnnc n vatons allow b th summaton convnton. In to complt th tansfmaton of ou balanc quaton w must fst obtan som latons amon th vatvs of conat sstms. Th latons amon th vatvs a foun b wtn th nal quatons [] an [] n tms of th patcula conat schm scb h. Thos quatons pss th fact that a ffntal chan n an of th conats n th nal conat sstm can caus a ffntal chan n on of th ξ conats. Th nal quaton f ξ s vn blow. [5] In th scon fm of quaton [5] th s an mpl summaton ov th pat n. Equaton [5] appls f = an. If w loo at th nvs poblm of tmnn th ffntal chans n ou nal conat sstm fom ffntal chans n th ξ ξ ξ conat sstm w woul hav th follown analo of quaton [5]. [6]

10 Conat tansfmatons ME 69 L. S. Catto Spn Pa W can wt both quatons [5] an [6] as mat quatons to show that th patal vatvs an a lat to ach oth as componnts of an nvs mat. In mat fm quaton [5] bcoms. [7] Convtn quaton [6] to mat fm vs th follown sult. [8] Equatons [7] an [8] can onl b cct f th two th-b-th matcs that appa n ths quatons a nvss of ach oth. That s th patal vatvs a lat b th follown mat nvson. [9] If a mat B s th nvs of a mat A th componnts of b a vn b quaton [4]. In that quaton M nots th mn tmnant whch s fn as follows. If A s an n-b-n mat t has n mn tmnants M whch a th tmnants of th n- b n- matcs fm f ow an column a lt fom th nal mat. Th mn tmnant s us to fn th cofact A = - + M. Th componnts of th nvs mat a fn n tms of ths cofact an th tmnant of th nal mat A. A Dt M A Dt A b A B [4] Th mat on th ht han s of quaton [9] has th sam fm as th acoban tmnant fn n quaton []. Thus th tmnant of ths mat s th acoban. Wth ths psson f th cofact w can wt th follown nn quatons f th nvs componnts

11 Conat tansfmatons ME 69 L. S. Catto Spn Pa fom quaton [9]. Ths vatvs a call th mtc coffcnts f th tansfmaton. In th quatons blow w wt ths coffcnts n both th nal fm wth numcal subscpts an usn th an ξ η ζ notaton. Th fnal tm n ach quaton s an altnatv notaton f patal vatvs. [4a] [4b] [4c] [4] [4] [4f] [4] [4h] [4]] Th latonshps f two-mnsonal flows can b foun fom ths quatons b conn that n such flows th s no vaaton n th th mnson. Ths mans that th s no vaaton of wth ζ. Thus all vatv of an wth spct to ζ a o. W st th vatv ζ = f appln quaton [4] to two-mnsonal flows. Ths s quvalnt to assumn a conat tansfmaton of = ζ f such flow omts. Th sults of convtn quatons [4a] [4b] [4] an [4] to two-mnsonal fms a shown blow [4a]

12 Conat tansfmatons ME 69 L. S. Catto Spn Pa [4b] [4c] [4] Not that quaton [4] s cct. Th lft han s ζ s qual to on. Th tms n bacs on th lft-han s a ust th fnton of th acoban f th two-mnsonal cas. Thus both ss of quaton [4] a qual to on n th two-mnsonal cas. W a now a to tansfm th balanc quatons nto ou computatonal spac. To ca out ths tansfmaton w con that a chan n an of th tansfm conats can b flct as a chan n an of th nal conats. Thus w wt th follown quaton to convt fst vatvs n ou Catsan conat sstm wth spct to an Catsan conat to fst vatvs n tansfm spac. [4] Th scon fm of ths quaton has an mpl summaton ov th pat n. W fst appl ths to th convcton tms wh w wt = ρu φ. In tansfm spac th convcton tms wth th summaton convnton a wttn as follows. [44] Ths quaton can b convt nto an altnatv fm b fst multpln b th acoban of th tansfmaton an wtn th sultn A tm as A A. Ths vs th follown sult. [45] Th summaton ov th two pat ncs n th fnal tm vs nn spaat tms f th tansfm convcton quaton. W can show that th fnal tm multpl b s o f ach valu of b usn th mtc coffcnt latonshps n quaton [4]. W fst t th follown sult f = usn quatons [4a] [4] an [4]. [46]

13 Conat tansfmatons ME 69 L. S. Catto Spn Pa Can out th ncat ffntatons vs th combnaton of m scon- patal vatvs shown blow. Each of ths vatvs occus two tms onc wth a plus sn an onc wth a mnus sn. Th of ffntaton s also ffnt but m scon vatvs a th sam alss of th of ffntaton. A ltt blow th tm wth a plus mnus sn ncats th matchn tms that cancl. ampl th tm labl +A has a plus sn n th quaton that cancls th tm labl -A. C E D B A E D C B A [47] Ths shows that th tm n quaton [45] s o whn =. Th poof that ths tm s o f = an = follows th sam appoach us abov an s lft as an cs f th ntst a. Wth all ths tms o quaton [45] vs th follown sult f th tansfm convcton tms. [45] Rcall that = ρu φ; w can fn th vloct U K as follows. u u u u U [46] Ths quaton has th sam fm as quaton [] that fn th nal tansfmaton f vloct componnts. At that pont w not that th vloct was a contavaant vct whch mant that th cton of th componnt U s was ppncula to a sufac wh th th conat s constant. Ths contavaant vloct componnt appas natuall n th tansfm quatons. Howv w can stll plac th nal tanspt vaabl φ b ou choc of vloct componnts whn w wt th momntum quatons. Wth ths fnton of U th convcton tms n ou momntum quaton bcom. U u [47] Th tansfm convcton tms cpt f th nw fnton of U n plac of th usual Catsan vloct componnts hav th sam fm as thos n th nal quatons.

14 Conat tansfmatons ME 69 L. S. Catto Spn Pa 4 Th nt stp s th tansfmaton of th scon-vatv ffusv tms. W can us th sult obtan abov n quaton [45] f w chan th fnton of to appl to th scon vatv tms. so that Dfn [48] Th sult n quaton [45] not pn on th fnton of ; t onl pn on th fact that th was an mpl summaton ov all th componnts of n th tm. Thus w can appl quaton [45] to th fnton of n quaton [48] to obtan th follown sult. [49] To complt th tansfmaton of th scon vatv tms w hav to plac th mann fst vatv of φ n b vatvs wth spct to th tansfm conats ξ. W us quaton [4] f th tansfm of ths fst vatv to obtan th follown sult. so that [5] W us th n f th scon mpl summaton n to t th sult that th a th pat ncs n th fnal vson of quaton [5]. W can smplf th notaton of ths poblm b fnn th coffcnt B as follows. B [5] Wth ths fnton w can wt quaton [5] as follows. B [5] Wth th tansfm convcton an ffuson tms n quatons [47] an [5] th nal balanc quaton n tansfm conats bcoms S B U t [5] Th scon vatv tm n ths quaton aft applcaton of th summaton convnton has nn spaat tms. Th of thm a pu scon vatvs whch w a us to sn n ou nal tanspt quaton. Th oth s a m scon- vatvs. In mplct fnt-ffnc appoachs ths m tms a usuall tat plctl. Th coffcnts B whch a lat to but not th sam as th mtc tns coffcnts n quaton [8] a o f thoonal s f. Ths B coffcnts wth that multpl th m scon-

15 Conat tansfmatons ME 69 L. S. Catto Spn Pa 5 vatv tms can caus convnc poblms s th anls a fa fom thoonal f th a la aspct atos f th clls. Th spacn on th computatonal s usuall tan as ;.. ξ = η = ζ =. At ach pont on th computatonal w wll hav a nown valu of an. A naton pam tmns th latonshp btwn ths actual conats an th mnsonlss computatonal conats. Th an valus at th mnsonlss conats a us to comput th patal vatvs that appa n th fnton of U fn n quaton [46] an B fn n quaton [5]. Ths quatons qu th valuaton of. Bcaus th computatonal has ξ η an ζ as th npnnt vaabls w cannot valuat ths vatvs ctl. Insta quaton [4] s us to lat to vatvs of th fm whch a valuat usn convntonal scon- fnt-ffnc pssons such as th ons shown blow. [5] Tanspt quatons l quaton [5] can b on b fnt ffnc mthos usn convntonal fnt-ffnc fms. An altnatv appoach to nonthoonal s uss a fnt volum appoach wth a conat sstm that hanls tansfmatons locall. Ths qus an analss of th spcfc omt of a tpcal cll. Ths nots hav not cons th poblms of naton whch s an ntl spaat topc.