Analytical approximate solutions for two-dimensional incompressible Navier-Stokes equations

Transcription

1 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol Aalical aroimae solios for wo-dimesioal icomressible Naier-Sokes eqaios A. S. J. Al-Saif Dearme of Mahemaics Collge of Edcaio for Pre Sciece Basrah Uiersi Basrah Iraq Absrac: Aalical aroimae solios of he wo-dimesioal icomressible Naier-Sokes eqaios b meas of Adomia decomosiio mehod are reseed. The ower of his maageable mehod is cofirmed b alig i for wo seleced flow roblems: The firs is he Talor decaig orices ad he secod is he flow behid a grid comariso wih High-order wid comac fiiedifferece mehod is made. The merical resls ha are obaied for wo icomressible flow roblems showed ha he roosed mehod is less ime cosmig qie accrae ad easil imlemeed. I addiio we roe he coergece of his mehod whe i is alied o he flow roblems which are describig hem b sead wo-dimesioal icomressible Naier-Sokes eqaios. Kewords: Naier-Sokes eqaios Adomia decomosiio wid comac differece Accrac Coergece aalsistalor's deca orices flow behid a grid. Mahemaics Sbjec Classificaios [MSC]: 76S5 65N99 35Q35 - Irodcio The roblem of real flid flow is of grea comlei de o he ma hsical effecs ad a cosiderable se of o-liear arial differeial eqaios ioled. For eamle Naier-Sokes eqaios NSEs are oe good eamle ad hae he wides of alicaio as he goer he moio of eer flid be i a gas or liqid or a lasicized solid maerial aced o b forces casig o chage shae. So secific adaced echiqes ms be alied o obai he solios of his roblem. I or cosidered oliear roblem we eed good mahemaical rocedres o simlif or liearize roblem ad sole i sch as fiie differece mehod ad Adomia decomosiio mehodadm. A he rese ime he eed o se ADM i solig arial differeial eqaios became more obios b sig i i solig arios roblems of differe fields sch as hsics egieerig ad alied mahemaics [656] eseciall i he las decade. There is a challege i sig ad alig his mehod o sole he comlicaed roblems ha iclde o-liear differeial eqaios like flids flow roblems rereseed b a ssem of o-liear arial differeial eqaios called NSEs. 69

2 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol I his aer we al he merical mehod which is kow as Adomia decomosiio mehod o obai aalical aroimae solios for Naier-Sokes eqaios. These eqaios are elliic ad o-liear icrease o-lieari wih icreasig olds mber. The eqaios of he moio of a icomressible flid are µ ρ ρ µ > Ω a i Ω where Ω is a smooh boded domai wih bodar Ω ad are eloci comoes i direcio ad direcio reseciel. is he ressre is he ime ad are b he sace coordiaes µ is he kiemaic iscosi ρ is he flid desi i j is he gradie oeraor ad is he alacia oeraor. A mber of merical mehods for solig seeral es of mli-dimesioal imedeede icomressible Naier-Sokes eqaios were gie i[ ]. A lo of sdies hae idicaed o he imora role of Adomia decomosiio mehod i is alicaio for solig arios roblems o arios scieific models[ ]. The alicaio of he ADM was eeded o secific mli-dimesioal flow roblems sbjec o a secific daa heoreicall b Seg e al. [7]. cel Sadighi e al.[6] alied he ADM o sole NSEs models hese sorces are differe from he model of or roblem i case of aalsis of coergece heoreicall ad he mahemaical formaio of a roblem. For he simle case of orici roorioal o he sream fcio Talor obaied a aalical solio for sead flow ha rereseed a doble ifiie arra of orices decaig eoeiall wih ime. Koasza eeded Talor's idea b errbed he sream fcio b a iform sream ad he was able o liearize he NSEs ad obai a eac solio for sead flow which resembles ha dowsream of a wo-dimesioal grid. From he lierare reiew ad b deedig o or hmble kowledge we obsered ha he ADM o e sed o sd hese wo roblems his maer was moie for s o se i here o fid aalical aroimae solio i case of sead flows. 7

3 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol The aim of his aer is o eed he alicaio of he ADM roosed b Adomia [] o sole wo-dimesioal icomressible NSEs ad comare is reliabili ad efficiec wih a high-order wid comac differece mehoducdm[93] The resls ha we obai from sig he mehods will be saed ad comared o roe he efficiec of each mehod i accrac seed of coergece ad ime. - Adomia Decomosiio Mehod To show he basic ideas of ADM[] we will sd he algorihm alicaio of his mehod i aroimae oe-dimesioal o-liear iiial ale roblem. This roblem is wrie b sig he differeial oeraors as follows: R N g R a b where R is real mbers. The liear erms decomosed io R while he oliear erms are rereseed b N where is a easil ierible liear oeraor R is he remaiig liear ar ad is iiial codiio. B akig he ier of liear differeial oeraor which is deoed b for he wo hads of Eqaio a we obai g R N 3 Here for iiial ales roblem for he oeraor is defied as;.. dτ 4 From 4 we hae; 5 Hece Eqaio 3 became g R N 6 The mehod cosiss of decomosig he solio io sm of a ifiie mber of comoes defied b he decomosiio series [ 9] as; where he 's are calclaed recrrel. 7 The oliear oeraor N Ψ is decomosed as: 7

4 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol N A 8 where A are Adomia's olomials for he secific olieari [687]. These Adomia's olomials deeded o comoes of ad he fas coerge formla for he series [98]. The A are gie as; A d i [ Ψ λ ] i λ! dλ i... 9 There are differe algorihms o come Adomia olomials which hae bee discoered b he coios imroeme of his mehod i fidig he aalic solios or similar formaios of good acceleraio b ma researchers [ 7]. Sbsiig Eqaios 7 ad 8 io Eqaio 6 we obai R A Coseqel i ca be wrie as: φ g R R R A A A where φ is he iiial codiio. Hece all he erms of are calclaed ad he geeral solio obaied accordig o he ADM as. The coerge of his series has bee roed i [ ]. Howeer for some roblems his series ca' be deermied so we se a aroimaio of he solio from rcaed series: M U M wih M U M lim The acceleraio for his coerge meas he eed o few erms of Eqaio for obaiig he formla which earb o he eac solio[ 6]. 7

5 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol Algorihm Aalsis of ADM for NSEs Afer clearig he simle ad basic ideas of sig ad alig Adomia decomosiio mehod algorihm o sole he differeial eqaios we will eed his alicaio for a ssem of o-liear eqaios; ha describes he algorihm o NSEsa. I order o faciliae he aalsis he followig dimesioless ariables are cosidered ; U U U P where U is a referece eloci ad is a referece legh. We he dro he rimes. The eqaios become ρ P U 3a R e 3b where U is he olds mber. The dimesioless eqaio of coii b gies µ 3c This eqaio eables s o defie a sream fcio ψ sch ha ψ ad ψ 3d Now we sar alig he ADM algorihm for Eqaios3ab sbjec o he iiial codiios ad. Followig we defie he liear oeraors ad. Therefore we rewrie Eqaios 3ab wih oeraor form as 4a 4b B defiig he ierse oeraor which are gie i 4 we ca wrie he Eqaios 4a ad 4b as; 5a 5b B sig Eqaio 7 he comoes solios ca be wrie as; 73

6 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol ad he oliear oeraors are N Ψ N Ψ The associaed decomosiio mehod is gie b 6a Ψ 6b Ψ 6c We decomosed Ψ ad Ψ accordig o he series A ad B reseciel where A ad B are calclaed b he Adomia's olomials which are defied i Eqaio 9 he we obai A A 7a A similarl B B 7b B ad So o. B sig Eqaio we hae A B 8 A B ad so o. From Eqaios 3 ad coii Eqaiob we obai; P 9a

7 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol We smbolize he righ side b z. Therefore we rewrie Eqaios 9a wih oeraor form as; z 9b Now from Eqaio 9b we calclae he ressre. Solig for oeraor wih.. d d we hae ad ierig he z where he oliear ar z is calclaed from he Adomia's olomial of Eqaio 9. Wriig ad ideifig where he 's are calclaed recrrel. Hece z for From Eqaio we rodced; where C C C are Adomia's olomial ad calclaed b Eqaio 9 as; C C ad we ca wrie 3 h erm aroimaio for b i i or. Similar eqaios ca be wrie for ϕ which coerges o where.. d d we hae z for 4 Almos of reios relaios which are made i [7]b wiho he merical comaios ad wiho heoreical roe of coergece. Now we ca smmarize he algorihm comig as follows: where ad are obaied hrogh iiial codiios he from Eqaios 8 are comed deedig o ad. Also from Eqaios 8 he comoes ad are comed b deedece o he 75

8 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol ales of ad where is obaied b Eqaio so o. Moreoer also b sig his algorihm ad he resls ha obaied we ca comed he sream fcio3d ad he orici ω. 4- Aalsis of Coergece I his secio we will sd he aalsis of coergece i he same maer as [33 4] of he decomosiio mehod o he oliear Naier-Sokes Eqaios 3 a or b ad Eqaio 9a. e s cosider he Hilber sace H which ma be defied as H Ω [ T ] he se of alicaios; : Ω [ T ] R wih Ω [ T ] dω < Ad scalar rodc ad idced orm: where R is real mbers. Ω [ T ] dω ad We cosider he oliear Naier-Sokes eqaios he he oeraor of a oliear Naier- Sokes Eqaios 3ab ad Eqaio 9a are; 5a 5b f 6 where f is he oeraor of Eqaio 9a. Followig we defie he differece oeraor z z zˆ for a qai sch as z.the Adomia decomosiio mehod is coerge if he followig codiios are saisfied ; Ι : k ad k k k H. f > ˆ ˆ ΙΙ : Whaeer ma be M > here eis a cosa C M > sch ha for ˆ ˆ H wih M M M ad M ˆ M we hae: w C M w ad f w I w for eer w H ad I <. Ι : k3 ad k k3 k H. f > ˆ ˆ ΙΙ : Whaeer ma be M > here eis a cosa C M > sch ha for ˆ ˆ H wih M M ˆ M ad M ˆ M we hae: 76

9 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol w C M w ad f w I w for eer w H ad I <. Now we will se he followig heorems o saisf he aboe codiios as [33] i addiio we will erif he ressre hohesis which iclded hese wo hohesis. Theorem : If Ι ad ΙΙ are saisfied he ADM of Eqaios 5a ad 6 is coerge. Proof: Firsl we will erif he coergece of codiio Ι for he oeraors ad f : ˆ ˆ [ ˆ ˆ ] [ ˆ ] [ ] [ ] [ ] [ ] [ ] ˆ [ ] 7 ˆ f 8 Therefore f 3 Sice δ δ 5 δ 6 ad are differeial oeraors i H he here eis cosas δ > sch ha δ δ δ δ 3 ad accordig o he Schwarz ieqali we ge δ η δ M δ 3 where < η < ˆ M ad δ 3 > is he iscizia cosa ad herefore M δ M δ 33 3 also δ M 4 ˆ 4 3 δ 34 where M ad δ 4 > is he iscizia cosa ad herefore δ M δ Sbsiig Eqaios 3-35 io Eqaios 9 ad 3 ields 77

10 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol δ 3 δ4 M δ δ k 36 5 δ 6 f δ k 37 where k δ 3 δ 4 M δ δ ad k δ 5 δ 6. The codiio Ι holds. Secodl we will erif he coergece of codiio ΙΙ for he oeraors ad f. For ha we hae: w w w w M w M w w w C M w 38 w w I w f 39 where C M M I ad he codiio ΙΙ is saisfied. Hece he roof is comlee. Theorem : If Ι ad ΙΙ are saisfied he ADM of Eqaios 5b ad 6 is coerge. Proof: I he same maer of Theorem we ca roe he coergece of Eqaio3bb erifig he coergece of codiio Ι ad ΙΙ for he oeraors ad f. 5- Nmerical Tes ad Discssio The heoreical aalsis of ADM doe i he reios secios will be alied i his secio o fid he aalical aroimae solios for wo sead sae flow roblems: he firs is he Talor decaig orices ad he secod is he NSEs wih a eriodici i oe direcioflow behid a grid i order o es he alidi of he rese mehod. 5. The Talor's deca orices The roblem of Talor's deca orices is sed so mch o reesig he efficiec of he merical mehods for hadlig he flow roblems[49333]. To describe he flow 78

11 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol of Talor's deca orices he NSEs5 ab sig wih iiial codiios cos si si cos ad [ cos4 si4 ] for π π. 4 The obaied ieraie solios of decomosiio series of Eqaios 5 ab are sig he relaios 8where hese relaios rerese he ieraie solios for D NSEs. The efficiec ad a high accrac i fidig he eac ad aroimae solios for he iiial ad bodar ales roblems are cosidered osiie ois for ADM. The merical comaios of es roblem which rerese he flid flow codc iside sqare cai are alied wih some ad ales b sig ADM algorihm. Figre : a Shows Profiles of π ad π elociies for differe imes 37 b elaied he ideificaio bewee he eac ad merical solios ad ha idicaio has roed he efficiec of ADM i sole NSEs wih good coergece ad c Shows Coor drawig for orici ω a ad for ADM ad UCDM. I addiio he measremes of maimm error for he eloci ad orici fcios which are showed i Table esre he abili of sggesed mehod ad is accrac i fidig he solios. From he measres of maimm error he able shows he reqired eidece o elai a high accrac for mehod where as he ADM accrac icrease wih icreasig olds mber a. From or comaios b sig ADM we oice ha he coergece of hese comaios correlae wih he ariables ad i iersel relaio which goer he solio. For eamle a ad < 5 or > 5ad he coergece of ADM becomes weakl i he solio. All obaied resls b he secod ieraio of ADM rerese aroimae solio is eqiale ad ideical o eac solios for he roblem. The oher osiie ois of ADM are sorage of imecpu abo.7 ad effor ha is elaied from he ablar resls i his aer. Figre c rereses he ssem of eddies arraged i he sqare aer each roaig i he oosie direcio o ha of is for eighbors ad his fac is cofirmed b ma ahors[ 36493]. 79

12 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol a 4 6 Eac ADM b.5 Eac ADM 4 6 ADM c UCDM ω ω Figre. avelociies ad for 37 b Eac ad merical solios of ad for ad. c Vorici ω for ad. 8

13 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol Table : Errors eloci ad orici ω of he rese sd for Talor's orices roblem Ma U Ma ω For he comariso addiio o eac soliofigre b we selec he bes aroimaio for solig NSEs ha ca be sed is high-order wid comac differece mehod. Eqs.3 ab ad 9 a ca be sall chaged io discree differece eqaios for deails see refs.[ 7993] ad he be soled hrogh ieraiel mehod. For solig resl discree fiie differece eqaio corresods o Eqaios3 ab ad 9 a we sed Gass Seidel ad sccessie oer-relaaio ieraie mehods reseciel. We irodced he comariso of he obaied resls bewee ADM ad UCDM. The comariso is rereseed b he sd of errors Table : Errors comariso of he rese sd ad UCFDM a. for Talor's orices roblem. Grids Velociies Mehod UCDM Prese Mehod Vorici ω UCDM Prese Mehod for orici ω ad elociies ad also b ieraios mber ad CPU ime. We oice ha he merical solios of orici fcio ad he elociies b sig ADM ad UCDM are corresode. The sggesed wo mehods cofirm is efficiec i solig he wo dimesios NSEs. The accrac of hese wo mehods icreases wih icreasig olds mber a. Moreoer From Table we see ha accrac of ADM is higher ad beer ha UCDM for differe olds mber ales 5. Besides ha CPU ime.7 ad ieraios mber of ADM is beer ha CPU ime < CPU< 6.7 ad ieraios mber8 < No. of Ieraios< of UCDM. We ca sa ha ADM is faser coergece ad more accrae ha UCDM. 8

14 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol Usead flow behid a grid we cosider he lamiar flow of iscos flid Eqaio behid a wo-dimesioal grid wih -ais ormal o he grid ad he eloci field is assmed o be sch ha : U ad : where U is he mea elocireferece of eloci i he -direcio. Ths; he wodimesioal Naier-Sokes eqaios wih a eriodici i oe direcio which ma rerese he wake of a wo dimesioal grid as he same as Eqaios ab wih relacig he coefficies of coecie erms i -direcio b U ;ha is: U ad i he o-dimesioal NSEs3ab hese erms become U ad ad.the lamiar flow behid a eriodic arra of medim[9] is sed o eamie he erificaio of accrac of ADM. To come he merical resls for sead sae of his roblem b sig he algorihm secio3 he iiial ales ha are adoed is he sead sae wo-dimesioal eac solio of his roblem [8343];which is gie as P e P Ρ e 6 π 6π e 4π cos π 6 π 6 π si π where Ρ is a referece ressre a arbirar cosa. We comed he aalical aroimae solio b sig AD algorihm for sead of his roblem sig recrrece relaios8& ad he relaios are relaed o is sch as sream fcio ad orici. The calclaios are r b Mahcad 4 sofware. The comed sreamlies ad orici coors for 54 are show i Figre. The airs of bod eddies geeraed behid he sigle elemes of he grids ad a large disace dowsream howeer he sreamlies become arallel ad eqidisa as show b he shor lies o he righ side of he figre for all ales of as i he case of a iscos flid. From he figre of he sreamlies ad oriciies we oe ha whe he old mber icreases he whole flow aer is eeded iforml i he direcio of mai flow. we obsere ha he rae of chage of he flow is er grea ad he legh of orices icrease wih 8

15 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol a ψ ω b ψ ω c ψ ω Figre. Sreamlies ad orici coor los for. ad a5b c4 83

16 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol he icrease i he old mber owards he dowsream flow. Tables 3 show grid refieme es resls for wid comac differece ad Adomia decomosiio Mehods. The comariso is rereseed b he sd of errors for sream fcio ad orici ad also b ieraios mber ad CPU ime. I is oed ha he magides of he orici gradies ad sreamlies i he rese sd are similar o hose obaied b Shah e al. [9] whe solig he sead flow ad sead flow b[343]. Moreoer The accrac of hese wo mehods icrease wih icreasig olds mber a.. From Table3we see ha he accrac of ADM is higher ad beer ha UCDM for differe olds mber ales 4 ad mber of grid ois. Besides ha CPU ime.3 ad ieraios mber of ADM is less ha CPU ime < CPU<.6 ad ieraios mberno. of Ieraios < of UCDM. We ca sa ha ADM is faser coergece ad more accrae ha UCDM. Table 3. Errors Comariso of he rese sd ad UCDM for flow behid a grid roblem. Grids Mehod 4 4 UCDM Prese Mehod 6 6 UCDM Prese Mehod Coclsios The Adomia decomosiio mehod is esed for wo-dimesioal ime-deede icomressible Naier-Sokes eqaios ha describe he Talor's orices wih low o moderae olds mbers ad flow behid grid wih comared he resls of boh of hese roblems wih he UCDM. The alicaio of ADM gies a simle owerfl ool o obai he solios wiho a eed for large size of comaios like UCDM. The resls show ha ADM has high accrac ad efficiec i fidig he eac ad aroimae solios wih less comaio workload. Also we coclde ha ADM is efficie ad beer ha UCDM i ieraios mber ad CPU ime a leas i he crre cases. There are ideificaio bewee he aroimae solios of UCDM ad ADM for solig NSEs a ad. Beside ha he accrac of solios b sig hese wo mehods icrease wih icreasig olds mber wih fied ime ad he rae of coergece is a er high. Adaages of ADM oer he classical echiqes. For eamle i aoids discreizaio ad roides a efficie aalical aroimae solio wih high accrac ad low comaioal load. 84

17 Adaces i Phsics Theories ad Alicaios ISSN 4-79X Paer ISSN Olie Vol fereces [] Adomia G. " Solig Froier Problems of Phsics: The Decomosiio Mehod Klwer Academic" 994. [] Ali A. H. ad Al-Saif A. S. J."Adomia decomosiio mehod for solig some models of o-liear arial differeial eqaios" Basrah J. Sci. A 68-. [3] Alkalla I.. Abd-Elmoem R. A. ad Gomaa A. M. "Coergece of discree Adomia mehod for solig a class of oliear Fredholm iegral eqaios" Al. Mah [4] Al-Saif A. S. J. " A efficie scheme of differeial qadrare based o wid differece for solig wo-dimesioal hea rasfer roblems Al.Com.Mah [5] Al-Saif A. S. J. ad Zh Z..Y. "Uwid local differeial qadrare mehod for solig coled iscos flid flow ad hea eqaio"al. Mah.ad Mechs [6] Brasos A. Ehrhard M. ad Famelis I.Th." A discree Adomia decomosiio mehod for he discree oliear Schrödiger eqaio"al. Mah. Com [7] Chrisie I. Uwid Comac Fiie Differece Schemes J. Com. Phs [8] El-Kala I.." Error aalsis of a Adomia series solio o a class of o- liear differeial eqaios". Alied Mahemaics E-Noes [9] El-Saed S.M. ad Kaa D."O he merical solio of he ssem of wo-dimesioal Brger eqaios b he decomosiio mehod " Al. Mah. com [] Errk E." Comariso of wide ad comac forh order formlaio of he Naier-Sokes eqaios" I. J. Nmer. Mehs. Flids [] Errk E. ad Gokcol C."Forh-order comac formlaio of Naier-Sokes eqaios ad drie cai flow a high olds mbers " I. J. Nmer. Mehs. Flids [] Gibo F.Mi C.ad Ceiceros H.D."No-Graded adaie Grid aroaches o he icomressible Naier-Sokes eqaios" FDMP [3] Ic M. " O merical solios of oe-dimesioal oliear Brger's eqaio ad coergece of he decomosiio mehod " Al. Mah. Com [4] Jae B."Solios o he o-homogeeos arabolic roblems b he eeded HADM" Al. Mah. Com [5] Jae B." Eac solios o oe dimesioal o-homogeeos arabolic roblems b he homogeeos Adomia decomosiio mehod"al. Mah. com [6] Jebari R. Ghami I. ad Bokricha A. Adomia decomosiio mehod for solig oliear hea eqaio wih eoeial oliearii. Joral of Mah. Aalsis [7] Kha W. Yosafzai F. Choha M. I. Zeb A. Zama G. ad Jg I. H. "Eac solio of Naier-Sokes eqaios i oros media"i. J. Pre ad Al. Mahs [8] Koasza.I.G. " amiar flow behid a wo-dimesioal grid " Proc. Comb. Pbli. Soc [9] i M. Tag T. ad Forberg B. A Comac forh-order fiie differece scheme for he sead icomressible Naier-Sokes Eqaios I. J. Nmer. Mehod i Flids

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