Numerical Simulations of Unsteady Navier-Stokes Equations for Incompressible Newtonian Fluids using FreeFem++ based on Finite Element Method

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1 Aals of Pre ad Aled Maemacs Vol. 6 o ISS: 79-87X P ole Pblsed o 7 Ma 4 Aals of mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod M. M. Rama ad K. M. Helal Dearme of EEE Soeas Uvers Daa aglades Emal: mafss59@aoo.com Dearme of Maemacs Comlla Uvers Comlla aglades Emal: maloelal@aoo.com Receved 3 Marc 4; acceed 3 Marc 4 Absrac. Te goal of s aer s cocered o mercal aroac of e sead aver-soes eqaos for comressble ewoa flds based o fe eleme meod ad we rese ere e mercal smlaos mlemeed w FreeFem. We frs gve e cosve formlao of ese eqaos. Te ows are e veloc ad e ressre. Te cosve eqaos lead o a o-lear ellc ssem of aral dffereal eqaos for. We fd e varaoal formlao of e sead aver-soes eqaos ad oba e resls of mercal smlaos rog a rogrammg code develoed FreeFem. Te aromao of e veloc ad ressre are P coos ad P coos fe eleme resecvel. Kewords: aver-soes Eqaos Fe Eleme Meod FreeFem AMS Maemacs Sbec Classfcao : 76D 76D99 76E9. Irodco I s aer we sd e mercal solos of e sead aver-soes eqaos for comressble ewoa flds based o fe eleme meod FEM ad we se FreeFem see Hec [] o oba e mercal smlaos. We dedce e cosve eqaos of sead aver-soes roblem. Tese cosve eqaos coss of gl o-lear ssem of aral dffereal eqaos of ellc e. Te veloc ad e ressre are e ows. We assme a e solo s reglar eog. Te aromae veloc ad ressre are resecvel P coos ad P coos fe eleme. We frs rodce e coservao of laws see Sagess Kaz ad Scaffer 5 [4] Cor ad Marsde [6] Qarero ad Vall 994 [] ad formlae e cosve eqaos of sead aver- Soes for comressble ewoa flds. Te e varaoal formlao of ese 7

2 mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod cosve eqaos s derved ad fe eleme aromao of s roblem s rodced see [5 6]. All meses ad smlaos are doe FreeFem. Usg e varaoal formlao we develo a rogrammg code FreeFem o fd from e aver-soes eqaos. We cosder a well ow becmar flow roblem amel e Km-Mo model roblem wose eac solo s ow o valdae e code. Fall some coclsos ad ersecve of fre wors are dscssed.. Te coservao laws Coservao laws saes e scal rcles goverg e fld moo. Tag o acco e Lavoser law: are og s creaed og s los everg s rasformed we ca dedce e basc rcles of coservao. Accordg o e coservao laws a arclar measrable roer of a solaed scal ssem does o cage as e ssem evolves. We cosder flows of a comressble ewoa omogeos fld a boded doma w bodar. Te maemacal formlaos of ese coservao laws are as follows: Coservao law of mass: Coservao of mass s a fdameal rcle of classcal mecacs. Ts meas a mass s eer creaed or desroed. Ts wa drg moo e mass of e bod rema caged. I a fed rego e oal me rae of cage of mass s decall zero. Te dffereal eqao eressg coservao of mass s ρ ρ were ρ s e des of e fld s e veloc vecor. Ts eqao s also called e co eqao. If e des s a cosa e e flow of e fld s comressble ad e coservao of mass s eressed as Coservao law of momem: Te coseqeces of bod moo cao be descrbed ol b a veloc e also deed o e mass. So we se e momem of mass mass veloc o relae em. Te coservao law of momem s e eeso of e famos ewo s secod law of moo force mass accelerao. For a movg flow feld s law descrbe a e oal me rae of cage of lear momem or accelerao of a fld eleme s eqal o e sm of eerall aled forces o a fed rego. Te eqao of coservao of momem s gve b ρ T ρf 3 were T s e smmerc esor feld called Cac sress esor ad f s a eeral force. 3. Te cosve law ad roblem formlao 7

3 M. M. Rama ad K. M. Helal Te cosve law relaes e Cac sress esor w e emacs of dffere qaes arclar e veloc feld. Tese relaos allow s o caracerze e mecac beavor of fld. I s wor we are cocered w flds obeg a ewoa beavor. Te ewoa flds are a sbclass of soroc dreco deede vscos flds o wc e sress esor T s e sm of e eso cased b e ermodamc ressre e fld e eso a cases deformao fld ad e eso de o volmerc easo. Tese flds are called Soesas Flds. ewoa flds are soroc vscos flds o wc e sress esor T s gve b T I η µ D were η e volmc vscos mlles e eso de o volmerc easo ad µ e drodamc vscos mlles e eso wc corbes o e moo of e fld. Tese vscoses verf e relaos 3η µ ad µ. I a ewoa comressble fld e Cac sress esor s a lear fco of e sra esor. Te Cac sress esor ca be wre e form T I µ D PI µ [ ] 4 were e erm µ D s ofe referred as vscos sress comoe of e sress esor. As eamle of comressble ewoa flds we refer e followg gases: oge droge ar meae ad ammoa. As eamle of comressble ewoa flds we refer e followg lqds: waer gasole olve ol. Cosderg a µ s cosa ad T as 4 for e ewoa comressble fld coservao law of momem 3 ca be wre as ρ T ρf ρ [ I µ D ] ρf Afer smlfg we ge ρ µ D ρf µ Cosderg ρ as a cosa we defe e emac vscos ν m /s ad e ρ scaled ressre ρ m /s sll deoed b ad we oba ν D f 5 Te aver-soes eqaos for comressble flds s e ssem of eqaos formed b e aral dffereal eqaos of e law of coservao of mass ad e momem eqaos 5 7

4 mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod ν f 6 If from e D [ ]. So e coservao of momem ca be wre as ν f 7 Usg 7 we ca rewre e aver-soes eqaos for e comressble flds as follows: ν f 8. Here s a boded doma of 3 w Lscz coos bodar.to close maemacal formlao ad oba a well-osed roblem e above eqaos eed o be slemeed b some bodar codos. For smlc we cosder e case wc e ssem of dffereal eqaos 8 s eqed w e Drcle bodar codos g o aderece codos. Te codo g s called e omogeeos Drcle bodar codos or o-sl bodar codos.e. o wc descrbes a fld cofed o a doma w fed bodar e bodar s a res. So w e omogeeos Drcle bodar codos defed over we ca wre e sead aver-soes roblem as follows: ν f 9 o. 4. Varaoal formlao We se dffere fco saces w dffere oaos deals of wc ca be fod Adams ad Forer 3[] rezs [4]. Wo loss of geeral we cosder a comressble fld cofed o a doma w fed bodar. Maemacall for eac [ T ] o smlf we ae from ow we wre e sead aver-soes eqaos as Gve f fd sc a 73

5 M. M. Rama ad K. M. Helal ν f [ T ] [ T ] o [ T ]. were f s a gve eeral force feld er mass s e veloc feld s e ow al veloc feld s e rae bewee e ressre ad e des ad ν s e cosa emac vscos. Te varaoal or wea formlao of aver-soes eqao cosss of e egral eqaos over obaed b egrao afer mllg e momem eqao ad co eqao b arorae es fcos. Le s sose a C [ T ] ad C [ T ] are e classcal or srog solo of. Cosder wo Hlber saces V H ad Q L ad ae v V ad q Q be wo arbrar es fcos. Alg e Gree s formla for e egrao b ars ad ag o acco a v vases o e bodar ad afer smlfg we ge e varaoal formlao of e aver-soes roblem as: [ T ] gve f L T; H ad H w fd L T; V L T; Q sc a v. v. v ν D : D f. v q T for all v q H L. Here D s e deformao esor. Tag o e defos of e followg blear ad rlear forms: ad a ν D D ν D : D b v v c w w w v we ca reformlae e varaoal formlao of e aver-soes roblem as follows: 74

6 mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod [ T ] gve f L T; H ad H w fd L T; V L T; Q sc a a c b v f b q for all v q H L. I ca be roved [9] a e roblem s well-osed ad eqvale o. Te esece ad qeess of eorem for e solos of aver-soes ssem ca be fod Gald 994 [7] Gral ad Ravar 986 [8] Temam 984 [5] Gral ad Ravar 979 [9]. 5. Fe eleme aromao We cosder fe eleme meod FEM o aromae e mercal solos of aver-soes roblem. Te FEM s a meod wc aroaces e solo of aral dffereal eqaos PDEs ad s a geeral ecqe for cosrcg aromae solos o bodar vale roblems dmeso d d 3. All resls wll be reseed ere for e wo-dmesoal case were we wll do e alcao of ese coces ad reseao of mercal smlaos. Alog ere are several es of fe elemes e followg we deal ol w e dscrezao of e aver-soes roblem sg a Lagrage Fe Eleme of e P P. Te solo of e roblem lves a sace of fe dmeso. I s crcmsace s geerall mossble o calclae e eac solo. Te we deerme a aromao of ad resecvel ad eac oe defed fe dmesoal arorae saces V sc a dm V I lm I ad deede o a arameer >. Tese saces are formed b olomals ad for all fco v V arclar ad for e arorae saces we ave v I α α IR I were } s a bass of V. { I Ts s e rcle of e Fe Eleme Meod. Te FEM ca be sded deals reer ad Sco 994 [3] [] ecer e al. 98 []Gral ad Ravar979 [9]. We se classcal Galer meod o fd e solo. We cosder Galer s meod for cosrcg aromae solos o e varaoal bodar-vale roblem or s absrac formlao. Galer s meod cosss of seeg a aromae solo a fe-dmesoal sbsace V of e sace of 75

7 M. M. Rama ad K. M. Helal admssble fcos were e solo les s sbsace raer a e wole sace. Te aral Galer aromao for roblem s a med meod wc s based o Lagrage mller formlaos of cosraed roblems. We refer o med aromao meods as ose assocaed o e aromao of saddle o roblems wc ere are wo blear forms ad wo aromao saces sasfg a comabl codo see []. We dscreze roblem. Le{ Τ } > be a faml of raglaos ad deoes a dscrezao arameer ad le V ad Q be wo fe dmesoal saces sc a V H ad L Q. We le V : V H ad M : Q L. I ese saces e dscree fe eleme aromao roblem of ca be wre as follows: for eac [ T ] V fd V M sc a d a c b v f v V d b q q M 3 wc ca be wre as for eac [ T ] V fd V M sc a v. v. v ν D : D f. v v V 4 q q M As moo s o-saoar we eed o dcreze e aver-soes eqaos over me. Tere are several meods of me dscrezao. I s aer we se Caracersc Galer Meod wc assocaes bacward Eler sceme of frs order defed b... Te Caracersc Galer Meod evalaes me dervaves of vecor feld o Lagraga frame aealg o caracersc les or raecores descrbed b a maeral arcle we as bee drve b e feld a e veloc of e feld. We descrbe e moo of maeral arcle of ewoa fld drg e me erval wc was oso a sa b ; : ; ad defe s caracerscs le or raecor w e same flow dreco b e ol solo of Cac roblem ; ; ; ; 76

8 mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod 77 ow ag a form mes of [T] defed b beg e me se ad alg e bacward Eler sceme we ca wre e sceme for e roblem deog ν.. 5 Te dscree varaoal formlao of 5 s as follows: for eac ] [ T gve fd M V sc a. v g v D D v v ν 6 were f g. Le } { ad m } { ψ be e Lagrage bases of e saces V ad M resecvel. Gve we eress e corresodg aromae solos ad e bass of V ad M m ψ ad w e es fcos V ad M ψ we oba e followg lear algebrac ssem marcal form as: b b A A A A 7 were [ ] ad [ ] m are e vecors of ow degree of freedom ad [ ] ν A A [ ] A A [ ] m m ψ [ ] m m ψ g b We ca rewre e above marcal eqao a more smle wa

9 M. M. Rama ad K. M. Helal A b A A w A A A b ad b b 6. mercal smlaos Te solo ca be evalaed sg a drec meod or erave meod aled o smmerc marces as CG cogae grade meod. Deals ca be fod Saad 3 [3]. All meses ad smlaos were doe FreeFem wc s a free sofware w s ow g level rogrammg lagage based o e fe eleme meod FEM o solve aral dffereal eqaos. A aomac mes geeraor s sed FreeFem based o Delaa-Voroo algorm were e mber of eral os s rooroal o e mber of os o e bodares. Te drec mercal smlao from e varaoal formlao for e me dscrezao ca be sragforwardl mlemeed o e geeral fe eleme solver FreeFem wc we se o mae mercal eermes. Te gracs were geeraed FreeFem ad Maemaca. We develo e rogrammg code FreeFem from e varaoal roblem ad se Cro meod ad Cogae Grade meod as solvers o solve e ssem. Towards e valdao of e code coosg solver ad error aalss we cosder e Km-Mo model roblem w ow eac solo gve b cos s ν s cos ν 8 cos 4 cos 4 ν 9 Te veloc ad ressre feld rema sace ad decrease moolcall w me. Te Km-Mo model roblem s solved o e sqare [.5.5] [.5.5] ad rescrbes e eac veloc accordg o 8 ad 9 alog e bodar of e fld doma. Te calclaos ave bee erformed w a emac vscos of ν. wc resls e ll eeral force. Te roblem as bee dscrezed sace w wo meses w 48 ad 89 Hood-Talor elemes. For e fer mes we ave 45 odes P for e veloc ad 89 odes P for e ressre ad for e coarse mes we ave 664 odes P for e veloc ad 45 odes P for e ressre. Te me erval [ ] was dscrezed o sbervals of eqal amlde w of e [ ]. We se a lear drec ad a erave solver avalable FreeFem: Cro meod s a drec meod. I s a vara of e facorzao meod LU were U s a er raglar mar w ar dagoal ad L mar s a lower raglar w s coeffces defed b 78

10 mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod eg A a osglar mar as well as e mar L ad U s wa e dagoal elemes are o ll. I s case we ca sl e ssem A b o raglar ssems of smler resolo as follows A b or LU b or Cogae Grade meod CG s a erave meod a ales o lear ssems wc e mar s smmerc osve defe. Ts meod s ormall sed large sarse marces. Tag e dffere meses ad sg e wo solvers Cro ad CG FreeFem we oba e followg mercal resls: Frs es case: I s es case we ae e mes w 48 elemes ad w ad solve e roblem w Cro meod. Te fgre. sows e errors of e fld veloc feld ad e ressre for eac sa of me evalaed e L -orm.e. ev 48 L / 48 e L. / ad Fgre.: Error o e fld veloc feld o e lef ad e ressre o e rg L -orm for eac sa of me for dffere sg a mes w 48 elemes. Secod es case: I s es case we ae e mes w 89 elemes ad w ad solve e roblem w Cro meod. Te fgre. sows e ev ad e e errors of e fld veloc feld ad e ressre resecvel for eac sa of me evalaed e L -orm. 79

11 M. M. Rama ad K. M. Helal Fgre.: Error o e fld veloc feld o e lef ad e ressre o e rg L -orm for eac sa of me for dffere sg a mes w 89 elemes. Comarso of bo ess: Comarg e solos comed b e wo meses for dffere me ses we oce a ere s o remarable dfferece bewee bo solos alog e error L s decreasg wc ca be observed from e fgre. ad fgre. wc we ca easl coclde from e fgre.3. Te fgre.3 sows e error ev ad ev of e fld veloc feld o e lef ad e error e ad e of e ressre o e rg for e sa of me 48 evalaed e L -orm. Fgre.3: Comarso of errors of e fld veloc feld o e lef ad e ressre o e rg L -orm for eac sa of me for dffere sg e bo meses. Gve e beavor of e error we ca seclae a we we ave e48 e89. Comarg e CPU me for eac es clearl e fe mes demads for a large CPU me for dffere me ses fgre.4. From e o of vew of CPU me s beer o emlo a mes lle refed ad a smaller o aceve e same accrac level. 8

12 mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod Fgre.4: Comarso of CPU of mes sed as a fco of solvg e roblem b e meod of Cro for e wo meses. Trd es case: I s es case we ae e mes w 48 elemes ad w ad solve e roblem w CG meod. We comare e resls w e frs es. Te fgre.5 sows e comarso of errors of e fld veloc feld ad e ressre resecvel for eac sa of me evalaed e L -orm for e bo meods Cro ad CG. I s evde a e rese e same beavor ad e same recso. Te dfferece of error vales bewee e Cro meod ad CG 7 meod are of e order. Fgre.5: Comarso of errors of e fld veloc feld o e lef ad e ressre o e rg L -orm for eac sa of me for dffere sg a mes w 48 elemes for e bo meods: Cro ad CG. Fgre.6: Comarso of errors of e fld veloc feld o e lef ad e ressre o e rg L -orm for eac sa of me for dffere sg e bo meods. As was eeced b a-ror esmaes for e error me we oba a lear covergece order o me fgre.6. 8

13 M. M. Rama ad K. M. Helal For es case: I s es case we ae e mes w 89 elemes ad w ad solve e roblem w CG meod. We comare e resls w e rd es. We also oce for s meod a ere s o remarable dfferece bewee bo solos alog e error L decrease. Te fgre.7 sows e comarso of errors of e fld veloc feld ad e ressre resecvel for eac sa of me evalaed e L -orm for e bo meods Cro ad CG. I s evde a e rese e same beavor ad e same recso. Te dfferece of 7 error vales bewee e Cro meod ad CG meod are of e order. Fgre.7: Comarso of errors of e fld veloc feld o e lef ad e ressre o e rg L -orm for eac sa of me for dffere sg a mes w 89 elemes for e bo meods: Cro ad CG. Sce e bo meods ave e same accrac we ave o decde wc meod o se based o CPU me. As we ca see fgre.8 e meod CG eeds mc CPU me a e meod Cro e case of mes o be refed. I e case of less fe mes e me se b e wo meods are aromael eqal alog s case e Cro meod s slgl faser fgre.9. Fgre.8: Comarso of CPU of mes sed as a fco of roblem w e wo meses. 8 solvg e

14 mercal Smlaos of Usead aver-soes Eqaos for Icomressble ewoa Flds sg FreeFem based o Fe Eleme Meod Fgre.9: Comarso of CPU of mes sed as a fco of solvg e roblem b e meod of Cro for e wo meses. I large sarse mar e Cogae Grade meod sold be more effce a e meod of Cro b we do ow f e mlemeao of s solver s omzed. Gve e revos sd we coose o se e meod of Cro. Te followg fgres llsrae e eac solo ad corresodg aromao obaed for a mes w 48 elemes ad / 8a me. Fgre.: Eac solo a. Frs comoe of veloc o e lef secod comoee of veloc o e cere ad ressre o e rg. Fgre.: Aroac solo a. Frs comoe of veloc o e lef secod comoe of veloc o e cere ad ressre o e rg. 7. Dscsso ad coclsos We ave reseed s aer e Gler fe eleme meod o smlae e moo of fld arcles wc sasfes e sead aver-soes eqaos. Tme dscrezao s obaed sg Caracersc Galer meod. We se FreeFem o mleme e smlaos sragforward from e varaoal formlao. Te veloc ad ressre.e. e solo of aver-soes eqao s obaed. Comarg dffere es cases sg dffere meses ad solvers we coclde a e Cro solver FreeFem s more effecve. From fgre. ad fgre. we observe a e eac solo ad aroac solo s aromael same. Te aromao of e veloc 83

15 M. M. Rama ad K. M. Helal ad ressre are P coos ad P coos fe eleme resecvel. All e smlaos were doe wo dmesoal case. We ca eed o e reedmesoal case sg 3-D aver-soes solver FreeFem for frer wor. Frermore dscoos Galer fe eleme meod ca be sed case of srcred mes ad smlaos ca be mlemeed w FreeFem. REFERECES. R.A.Adams ad J.F.Forer Sobolev Saces ed Academc Press Y 3.. E.ecer G.Care ad J.Ode Fe Elemes. A Irodco Vol. Prece Hall Ic. Eglewood Clffs ew Jerse S.reer ad L.R.Sco Te Maemacal Teor of Fe Eleme Meods Srger-Verlag ew Yor H.rezs Fcoal Aalss Sobolev Saces ad PDE Srger. 5. G.F.Care ad J.T.Ode Fe elemes Vol.VI. Fld mecacs Te Teas Fe Eleme Seres VI. Prece Hall Ic. Eglewood Clffs ew Jerse A. J. Cor ad J. E. Marsde A Maemacal Irodco o Fld Mecacs 3 Ed. Srger. 7. G.P.Gald A Irodco o e Maemacal Teor of e aver-soes Eqaos: olear Sead Problems Srger Tracs aral PlosoVol. 38 Srger ew Yor V.Gral ad P.A.Ravar Fe Eleme Aromao of e aver-soes EqaosComaoal Maemacs Srger-Verlag erl V.Gral ad P.A.Ravar Fe Eleme Aromao of e aver-soes Eqaos Lecre oes Ma. 749 Srger-Verlag erl F. Hec FreeFem 3 rd Edo Verso 3.7 :// P.Lesa ad P.A.Ravar O a fe eleme meod for solvg e ero rasor eqao C. de oor edor Maemacal Asecs of Fe Elemes Paral Dffereal Eqaos 89-3 ew Yor Academc Press A.Qarero ad A.Vall mercal Aromao of Paral Dffereal Eqaos Srger-Verlag Y.Saad Ierave Meods for Sarse Lear Ssems ed Soce for Idsral ad Aled Maemacs Pladela E.J.Sagess I.M.Kaz ad J.P.Scaffer Irodco o Fld Mecacs Oford Uvers Press R.Temam aver-soes Eqaos Teor ad mercal Aalss 3ed or- Hollad Amserdam M.Pal mercal Aalss for Scess ad Egeers: Teor ad C Program arosa ew Del ad Ala Scece Oford Ued Kgdom M.A.Rama M.A.Alm ad Md. Jarl Islam ermooress effec o MHD forced coveco o a fld over a coos lear srecg see resece of ea geerao ad ower-law wall emerare Aals of Pre ad Al. Ma Aa ad Md.M.Hoqe Termal boac force effecs o develoed flow cosderg all ad o-sl crre Aals Pre ad Al. Ma