Approximated Solutions to Tow-dimensional and Axisymetric Jet Impinging Flows, Using Adomian Decomposition Method

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1 Proceedigs o the 9th WSES Iteratioal Coerece o pplied Mathematics Istabul Turkey May (pp9-44) pproximated Solutios to Tow-dimesioal ad xisymetric Jet Impigig Flows Usig domia Decompositio Method M NJFI M TEIBI-RHNI KH JVDI ND SF HOSSEINZDEH Mathematics Departmet Ket State Uiversity Ohio State US erospace Egieerig Departmet Shari Uiversity o Techology Tehra IRN Khayyam Research Istitute Tehra IRN bstract: pproximated solutios to two-dimesioal ad axisymmetric jet impigig lows have bee preseted here ssumptios have bee made to reduce the related ull Navier-Stokes equatios to a o-liear ordiary dieretial equatio The dmomia decompositio method (DM) has bee employed to obtai a approximated solutio to this dieretial equatio trial ad error strategy has bee used to obtai the costat coeiciet i the approximated solutio The results were compared with accurate umerical results which show that the DM is a high perormace ad accurate method to solve such low equatios Key-Words: - Navier-Stokes Jet Impigemet domia Decompositio Method alytic solutio Itroductio chievemet o exact solutio or o-liear partial dieretial equatios such as Navier-Stokes (N-S) equatios is a ambitious ad perect goal or egieers ad mathematicias However computatioal iite discrete approaches such as Fiite Volume Method Fiite Dierece Method ad Fiite Elemet Methods have bee widely used to solve the N-S equatios i the past decades Computatioal methodologies have bee the oly way to solve the N-S equatio or predictig low behavior i most lows Some o the problems with computatioal methodologies are: - they are low depedet - To simulate real lows they are very time cosumig - stability ad covergece are ot resolved easily i may cases 4- we eed to use modelig ad approximatio o some ukow i order to achieve a closed system o equatio i turbulet lows Which could be a source o error i the computatioal methods y type o approximated solutio would be very valuable or mathematicias ad egieers To this ed DM is oe the techiques which was itroduced to solve o-liear ordiary ad partial dieretial equatio [] advatage o this method is that it ca provide approximated or approximated solutio to a rather wide class o o-liear (ad stochastic) equatios without liearizatio perturbatio closure approximatio or discretizatio methods Ulike the commo methods ie weak o-liearity ad small perturbatio which are chage the physics o the problem due to simpliicatio DM gives the approximated or approximated solutio o the problem without ay simpliicatio Thus its results are more realistic [] Durig the past ew years several researchers have tried to modiy the dmoia decompositio method Zhag [] preseted a modiied DM to solve a class o o-liear sigular boudary value problem which arise as o-liear ormal modal equatios i o-liear coservative vibratory systems He veriied the eectiveess o his method by solvig three examples Wazwaz [] developed a ast ad accurate algorithm or solutio o sixth-order boudary value problems Luo eal [4] revised DM or cases ivolvig ihomogeeous boudary coditios usig a suitable trasormatio They solved ihomogeeous heat ad wave equatios Jaari ad Varsha [5] modiied DM to solve a system o o-liear equatios which yielded a series solutio with aster accelerated covergece tha the series obtaied by the stadard DM Zhu [6] etal preseted a ew algorithm or calculatig domia polyomials or o-liear operators by parametrizatio The algorithm requires less ormula tha the previous method developed by domia bbasbady [7] preseted some eiciet umerical algorithms to solve a system o two o-liear equatios (with two variables) based o Newto s method Their modiied domia decompositio method was applied to costruct the umerical algorithms Some umerical illustratios were give to show the eiciecy o algorithms Luo [8] proposed a eiciet modiicatio to DM amely two-step domia decompositio method (TSDM) that acilitated the calculatios He coducted a comparative study betwee the TSDM ad previous methods with the help o several illustrative examples Their results idicated that the TSDM was eective ad promisig Recetly several researchers have used DM to solve a wide rage o physical pheomea i various egieerig ileds such as heat ad mass traser [9] [] [] vibratio ad wave equatio

2 Proceedigs o the 9th WSES Iteratioal Coerece o pplied Mathematics Istabul Turkey May (pp9-44) [] [] luid low predictio [4 [5] [6] [7] porous media simulatio [8] ad other o-liear systems [9] [] [] I this work we are goig to obtai a approximated solutio to two-dimesioal ad axisymmetric jet impigemet lows For this purpose we use sel similar assumptio to reduce the Navir- Stokes equatios to a o-liear ordiary dieretial equatio The we use DM to obtai a approximated solutio or this problem trial ad error strategy was implemeted to obtai the costat coeiciet i the approximated solutio Problem Descriptio Whe a jet impiges oto a surace very thi hydrodyamic ad thermal boudary layers orm i the impigemet regio (Figure ) Cosequetly extremely high heat traser coeiciets are obtaied withi the stagatio zoe Jet impigemet arrays are geerally used as a eective source o coolig o the leadig edge ad mid-spa regios o gas turbie blades ad vaes to ehace the covective heat traser The large rates o heat geeratio i itegrated circuit (IC) chips pose severe thermal maagemet challeges or the semicoductor idustry Recetly micro-jet heat siks have bee iterested i by researcher These micro-jets ca be ully ecapsulated which removes the problems with larger spray coolig systems Figure the Schematic o a jet impigemet o a lat plate Problem Formulatio Cosider equatio F(u(t))=g(t) where F represets a geeral o-liear ordiary or partial dieretial operator icludig both liear ad o-liear terms The liear terms is decomposed ito L+R where L is easily ivertible (usually the highest order derivative) ad R is the remaied o the liear operator Thus the equatio ca be writte as: Lu + Ru + Nu = g () where Nu idicates the o-liear terms By solvig this equatio or Lu sice L is ivertible we ca write: L Lu = L g L Ru L Nu () Solvig equatio (-) or u we get: u = + Bt + L g L Ru L Nu () where ad B are costats o itegratio ad ca be oud rom the boudary or iitial coditios domia method assumes the solutio u ca be expaded ito iiite series as: u = u (4) = lso the o-liear term Nu will be writte as: = Nu = (5) where are the special domia polyomials Fially the solutio ca be writte as: (6) u = u L R u L = = = where u is idetiied as: +Bt+L - g [] I Eq (6) the domia polyomials ca be geerated by several meas Hear we used the ollowig recursive ormulatio: d = N λiu i (7)! dλ i= λ = = Sice the method does ot resort to liearizatio or assumptio o weak o-liearity the solutio geerated is i geeral more realistic tha those achieved by simpliyig the model o the physical problem pplicatio to Impigemet Jet Flows For plae icompressible two-dimesioal low the goverig equatios are two-dimesioal Navier- Stokes equatios ivolvig cotiuity ad mometum Usig similarity solutio strategy the goverig equatio or this type o lows i two-dimesioal ad axisymmetric orm ca be reduced to: + β ( ) + = (8) where u Bx( ) v B ( ) υ ad B is the iverse o the low characteristic time scale lso β= or two-dimesioal ad β= or axisymmetric lows [] The boudary coditios are u=v= at the wall (=) ad u=bx at large distaces rom the wall This meas: ( ) = () = & ( ) = as (9) Equatio (8) cotais two o-liearities ad o aalytic solutio has ever bee oud or it [] To apply the decompositio method we write equatio (8) i a operator orm as: L = β ( ) + () = y where d L = ssume the iverse o the operator as d L = () d d d B ()

3 Proceedigs o the 9th WSES Iteratioal Coerece o pplied Mathematics Istabul Turkey May (pp9-44) pplyig the iverse operator to Eq () yields: L L = L L ( ) + L β () Two-dimesioal Impigig Jet Flow I we cosider β= i Eq () the twodimesioal jet impigig low will be obtaied as: = L + () ( ) () () () / ( ) pplyig the boudary coditio (9) at = we get: ( ) = () / /6 + L ( + ) (4) The DM cosider the ollowig expressio or ( ) : ( ) = ( ) (5) = lso the method assumes the o-liear uctio F( ( ) )as a iiite power series o polyomials as: F ( ( )) = (6) = I Eq s (5) ad (6) s are calculated rom (7) ad s ca be obtaied by substitutig (5) ad (6) ito (4) to get: ( ) = () / + L ( = = ) (7) I Eq (7) to calculate the compoets o ad we eed () as ollows: ( ) = () / (8) To id () we will utilize the boudary coditio at iiity (see Eq 9) To this ed let () =α Thus Eq (8) becomes: ( ) = α / (9) Now the remaiig compoets o ( ) i Eq (7) ca be determied recurretly as: ( ) ( ) + = L () Recall that k s ca be geerated rom Eq (7) as: = + = + = + + = + + Similarly ca be determied rom equatios (9) ad () as: = α / = α + α = α + α α Thereore = () Here α is yet to be determied (see results ad discussio i sectio 4) xisymmetric Impigemet Jet Flow I β= i Eq (8) the goverig equatio or axisymmetric jet impigig low are obtaied as []: ( ) = () + () + () / /6+ L ( + ) () Usig DM ad applyig the same approach as discussed or two-dimesioal case the domia polyomial ad closed orm solutio i the axisymetric case ca be obtaied as: = + = + () = + + Similarly s ca be determied rom Eq (9) ad () as: = α / 7 6 = + α = + α α = + α α + α + (4) Fially = (5) Now to id the complete solutio to Eq s () ad (-) we eed to id α ccurate predictio o α is cosiderably importat sice it aects the accuracy o the ial solutio Recall is related to the shear stress ad thus () is related to the shear stress at the wall Thereore its accurate predictio is very importat rom ski rictio view poit Wag [] metioed that Pade approximatio ca ot be applied or this type o equatio He trasormed the Blasius equatio ito a ew space ad calculated α which was rough Later Hashim [4] showed that DM Pade approach ca be used ad gives a more accurate α or Blasius equatio tha Wag s However i this work either we used Pade approximatio or Wag s trasormatio We employed a trial ad error approach To achieve this goal we relied o the physics o the problem ad we looked at the physical meaigs o ad Recall that is the dimesioless velocity iside

4 Proceedigs o the 9th WSES Iteratioal Coerece o pplied Mathematics Istabul Turkey May (pp9-44) the boudary layer ad is related to the shear stress s expected ad rom the boudary coditios (Eq s 9) approaches uity as becomes very large Here oe must ask how large is iiity To aswer this questio we should look at which approaches zero outside the boudary layer Thus the aswer is: whe becomes very small Now i we assume as a ew variable ad cosider ( ) = as a ew equatio the we would have a system o two o-liear algebraic equatios with ukows α ad (see below) which ca be solved by the trial ad error strategy metioed above: ( ) = as (6) ( ) = as 4 Results ad Discussio DM was used to achieve a approximated solutio to two-dimesioal ad axisymmetric jet impigemet lows s metioed earlier α ad were obtaied by a trial ad error approach rom Eq (-5) as: Two-dimesioal Jet Impigemet: ( ) = α = 97 ad = 485 xisymmetric Jet Impigemet: ( ) = α = 95 ad = Table () compares our approximated DM solutio with the reliable umerical data o reerece [] s show there is a excellet agreemet betwee the two dieret results Table Compariso o ( ) with the reliable umerical data o reerece [] Numerical Data DM Error -D ( ) =59 ( ) xisymmetric ( ) =94 5 () =95 Recall aother boudary layer eect is the displacemet thickess as the distace the outer iviscid low is pushed away rom the wall by the retarded viscous layer The ormal deiitio o the boudary layer displacemet thickess is []]: * ( ) d = lim( ) = ( (7) = ) s listed i Table () the predicted boudary layer displacemet thickess i our solutio is cosiderably close to the umerical predictios o Re [] Table Compariso o with the accurate umerical data o reerece [] Numerical Data DM Error -D =6479 =648 4 xisymmetric =5689 = Tables () ad (4) compare our solutio ad the umerical data o reerece [] or twodimesioal ad axisymmetric jet impigemet lows respectively s is see rom these tables there is a excellet agreemet betwee the two results 4 Sesitivity o the Solutio to the Number o Polyomial Terms Used Obviously as the umber o terms i the expasio polyomial series o DM method icreases the closed orm solutio gets closer to the exact solutio lso usig iiite umber o ters the DM closed orm solutio coicides with the exact solutio However whe DM is used oe must irst aswer the questio how may terms are eeded It is ot so easy to id a geeral aswer to the questio sice it is problem depedet We have show that as icreases ad as the sesitivity o the low variables become higher we eed to keep more terms i the polyomial expasio series Figure shows the behavior o the solutio with respect to the umber o terms i the expasio polyomial series Note rom this igure that: -s icreases the dierece betwee our solutio ad the exact solutio icreases ad thus it is ecessary to keep more terms i our closed orm solutio -s the sesitivity o the variables icreases (eg ) the sesitivity to the umber o terms i the closed orm solutio icreases s show i Tables ad 4 we selected (which has the highest sesitivity with respect to ad ) or compariso with other umerical results Thus whe usig DM to solve such problems havig a good physical kowledge o the problems is very importat Sice it is practically impossible to cosider iite umber o terms i the polyomial series it eeds to be determied how ar rom the wall (i directio) our DM results are physically acceptable This matter is more importat or problems havig at least oe boudary at iiity However obviously the iiity boudaries are dieret i mathematics ad i physics So that a closed orm solutio obtaied or a physical problem which is very accurate iside the iiity boudaries may ot be true outside o it I

5 Proceedigs o the 9th WSES Iteratioal Coerece o pplied Mathematics Istabul Turkey May (pp9-44) this work we obtaied =485 ad = or two-dimesioal ad axisymmetric lows respectively ad as Tables to 4 show the results are very accurate or However or > the results are ot physical I we are iterested i achievig physical ad accurate results or > (48 or i this work) we must cosider more terms i DM This meas that by addig more terms i the DM the rage o iiity boudary ( ) goes arther away rom the wall I other words addig more terms the results are acceptable or a wider rage 5 Coclusio dmomia decompositio method has bee employed to obtai a approximated solutio to two-dimesioal ad axisymmetric jet impigemet lows trial ad error strategy has bee used to obtai the costat coeiciet (α) i the closed orm approximated solutio To veriy our solutio the results were compared with reliable umerical data I this regard the sesitive variable ad also the displacemet thickess ( * ) were compared with the umerical results These comparisos showed excellet agreemets lso the results showed that as the sesitivity o the low variables rises the eed or keepig more terms i the closed orm polyomial solutio is icreased For the problems which ivolve at least oe boudary coditio at iiity the applicable rage o the solutio i directio (iiity boudary) must idicated For greater tha the boudary layer thickess more terms eeds to be cosidered i our DM series solutio Table Compariso o our results with those o Re [] or two-dimesioal low Num DM Error E Table 4 Compariso o our results with those o Re [] or axisymmetric low Num DM Error E E E E E E E E E E E (a) (c) Terms 5Terms 6Terms 7Terms 8Terms 9Terms Num [4] (b) Fig o solutio to the umber o DM terms used (a) (b) ad (c)

6 Proceedigs o the 9th WSES Iteratioal Coerece o pplied Mathematics Istabul Turkey May (pp9-44) Reereces: [] domia G review o the decompositio method i applied mathematics J Math al ppl Vol 5 (988) [] Zhag X modiicatio o the domia decompositio method or a class o oliear sigular boudary value problems J Comp ad ppl Math Vol 8 (5) [] Wazwaz M The umerical solutio o sixth-order boudary value problems by the modiied decompositio method ppl Math ad Comput Vol 8 () 5 [4] Luo XG Revisit o partial solutios i the domia decompositio method ppl Math ad Comput Vol 7 (5) [5] Jaari H ad Datardar-Gejji V Revised domia decompositio method or solvig a system o o-liear equatios ccepted or Pub i ppl Math ad Comput [6] Zhu Y Chag Q ad Wu S ew algorithm or calculatig domia polyomials ppl Math ad Comput Vol 69 (5) 4 46 [7] bbasbady S Exteded Newto s method or a system o o-liear equatios by modiied domia decompositio method ppl Math ad Comput Vol 7 (5) [8] Luo XG two-step domia decompositio method ppl Math ad Comput Vol 7 (5) [9] Pamuk S a pplicatio or liear ad o liear equatio by domia s decompositio method ppl Math ad Comput Vol 6 (5) [] l-khaled K Numerical solutios o the Laplace equatio ppl Math ad Comput Vol 7 (5) 7 8 [] Saha Ray S ad Bera RK alytical solutio o a ractioal diusio equatio by domia decompositio method ccepted or Pub i ppl Math ad Comput [] Helal M Mehaa MS compariso betwee two dieret methods or solvig KdV Burgers equatio Chaos Solitos ad Fractals Vol 8 (6) 6 [] Momai S No-perturbative aalytical solutios o the space- ad time-ractioal Burgers equatios Chaos Solitos ad Fractals Vol 8 (6) 9 97 [4] Wag L ew algorithm or solvig classical Blasius equatio ppl Math Comput Vol 57 (4) 9 [5] Bulut H Ergut M sil V ad Bokor RH Numerical solutio o a viscous icompressible low problem through a oriice by domia decompositio method ppl Math ad Comput Vol5 (4) 7 74 [6] Fathi M Kamel K approximatio o the aalytic solutio o the shock wave equatio ccepted or Pub i Comput ad ppl Math [7] Soh CW No-perturbative semi-aalytical source-type solutios o thi-ilm equatio ccepted or Pub i ppl Math ad Comput [8] Pamuk S Solutio o the porous media equatio by domia s decompositio method Physics Letters 44 (5) [9] Hashim I Noorai MSM hmad R Bakar S Ismail ES ad Zakaria M ccuracy o the domia decompositio method applied to the Lorez system ccepted or Pub i Chaos Solitos ad Fractals [] J Biazar J Solutio o systems o itegral dieretial equatios by domia decompositio method ppl Math ad Comput Vol 68 (5) 8 [] Jaari H ad Datardar-Gejji V Solvig a system o o-liear ractioal dieretial equatios usig domia decompositio ccepted or Pub i ppl Math ad Comput [] White MF Viscous luid low d Ed McGraw-Hill Ic 99 [] Wag L ew algorithm or solvig classical Blasius equatio ppl Math Comput Vol 7 (4)-9 [4] Hashim I Commets o ew algorithm or solvig classical Blasius equatio by L Wag ccepted or Pub i ppl Math ad Comput