Verification of a Chemical Nonequilibrium Flows Solver Using the Method of Manufactured Solutions

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1 Avalable onlne at SeneDret Proeda Engneerng 00 (014) APISAT014, 014 Asa-Paf Internatonal Symposum on Aerospae Tenology, APISAT014 Verfaton of a Cemal Nonequlbrum Flows Solver Usng te Metod of Manufatured Solutons L Wang*, We-jang Zou,Cu-qun J Cna Aademy of Aerospae Aerodynams, No.17 Yungang West Rd. Fengta Dstrt Bejng,100074,Cna Abstrat Ts paper presents ode verfaton of a emally nonequlbrum flows solver usng te metod of manufatured solutons.te Metod of Manufatured Solutons(MMS) s a general approa for reatng exat solutons to te governng equatons and an be used n te ode verfaton proess. In te MMS, te analytal solutons for te flow varables are frst onstruted, ten te governng equatons are modfed to satsfy tese solutons by addng approprate soure terms w are generated by applyng te governng equatons to tese solutons. After tat, ode verfaton proess wll start. Te order of auray of te alulatons wll be omputed and ompared wt teoretal order of auray to determne f te ode passes te verfaton test. We reated manufatured solutons for two dfferent sets of Euler equatons. One set nludes te total densty equaton plus ns-1 spees equatons and te oter ontans only spees ontnuty equatons. Te results sow tat te form of ontnuty equatons as lttle nfluene on te beavour of global onservatve varable errors as te mes s refned. Our study also ndates tat a omplete emal reaton model s preferred to ensure te onvergene of observed order of auray s smoot n te order of auray test. 014 Te Autors. Publsed by Elsever Ltd. Peer-revew under responsblty of Cnese Soety of Aeronauts and Astronauts (CSAA). Keywords: metod of manufatured solutons,ode verfaton,emal nonequlbrum flows * Correspondng autor. Tel.: E-mal address: txwangl@yea.net Te Autors. Publsed by Elsever Ltd. Peer-revew under responsblty of Cnese Soety of Aeronauts and Astronauts (CSAA).

2 L Wang, We-jang Zou, Cu-qun J / Proeda Engneerng 00 (014) Introduton As Computatonal Flud Dynams(CFD) plays more and more mportant role n te feld of aerospae, te evaluaton of orretness of CFD predtons as reeved nreasng attentons. Te prmary tools to buld onfdene n CFD results are verfaton and valdaton[1]. Te verfaton and valdaton deal wt dfferent pases n te modelng and smulaton atvtes. Te verfaton proess assesses te orretness of te numeral solutons to te matematal equatons. On te oter and, te valdaton proess determnes te degree to w a CFD model represents te real flud pyss from te perspetve of te ntended uses of te model. Te verfaton of CFD smulatons must be performed before valdaton. Te verfaton of CFD smulatons an be furter lassfed as ode verfaton and soluton verfaton[]. Soluton verfaton s te proess of gaterng evdene to demonstrate te alulaton of partular ase s orret. Soluton verfaton usually nvolves te estmaton of numeral soluton error. Code verfaton s te proess of elmnatng odng mstakes n a omputer ode by omparng numeral solutons wt pror known solutons to a gven set of equatons. Te most rtal rteron for ode verfaton s order of auray test w eks f te order of alulatons s approang te formal order of auray as te mes s refned. In order of auray test, te dsretzaton error, defned as te dfferene between dsrete solutons and te exat solutons, s requred to ompute te observed order of auray. However, te exat solutons to a spef partal dfferental equaton rarely exst. Furtermore, most of te exat solutons are not approprate for ode verfaton beause te smple form of tese solutons makes tem mpossble to exerse all te terms n te equatons beng solved. In order to overome ts dffulty, Roae and Stenberg[3] proposed a new, general approa to generate exat solutons to a system of governng equatons. Ts metod s alled te metod of manufatured solutons(mms). Instead of searng an exat soluton to te governng equatons wt gven ntal and boundary ondtons, te MMS modfes te governng equatons to satsfy a pror osen analytal solutons for flow varables. Ts metod s rooted n te fundamental prnple of ode verfaton w empaszes ode verfaton only dealng wt matematal ssues of a gven problem. Tus, for ode verfaton purpose, weter te manufatured soluton be related to a realst problem or not s nessental. Te MMS as been appled to ode verfaton of several CFD solvers [4-6]. Tese works use te MMS to verfy te solutons to Euler equatons, N-S equatons, or RANS equatons. Relatve lttle work as been done to perform verfaton for governng equatons of nonequlbrum flows wt te MMS. In ts work, we present a ode verfaton exerse of an nvsd, nonequlbrum reatng flows solver usng te metod of manufatured solutons. Te emal soure term onstruton and alternatve governng equatons set are nvestgated to study ter nfluene on te order of auray test.. Numeral Metod In ts seton, we wll gve a detaled desrpton of te governng equatons and present a bref desrpton of te flow solver..1. Governng Equatons Te D Euler equatons for a flud flow n emal nonequlbrum an be wrtten n onservatve form as follows Q E F Ω (1) t x y were te dependent varable vetor Q, x dreton nvsd flux vetor E, y dreton nvsd flux vetor F, and vetor of emal soure terms Ω are defned as:

3 L Wang, We-jang Zou, Cu-qun J/ Proeda Engneerng 00 (014) u v 0 u u p uv 0 v uv v p 0 Q E E uh F vh Ω 0 u v u v ns 1 ns 1 ns 1 ns 1 In te above,,,, u, v, and p are total densty, t spees densty, t spees mass fraton, mass produton rate of spees, x veloty omponent,y veloty omponent and pressure respetvely. ns s te number of spees. Te gas mxture s assumed to be perfet gas. Tus, te total pressure s gven as te sum of te partal pressures of ea spees aordng to Dalton s Law: ns ns Ru p p T M 1 1 Here R u s unversal gas onstant and E e u v / represents total energy per unt mass and H u v / denotes total entalpy. e and are alulated from termodynam propertes of ndvdual spees troug te followng relatons. e s nternal energy and s entalpy of spees. ns ns Ru, e e, e T M M s moleular wegt of spees. 1 1 s determned from polynomal urve ft relatons[8]: Ru a a a a T a T T T T a T W Flow Solver Te flow solver for nvsd, emally reatng flows s based on fnte volume framework. Te AUSMPW seme s used to onstrut te nvsd flux and MUSCL type nterpolaton norporated wt Vanalbada lmter s employed to rea seond order auray n spae. For tme ntegraton, LU-SGS metod s used. 3. Te Metod of Manufatured Solutons Te metod of manufatured solutons s a general approa to generate exat solutons to gven equatons. In ts metod, te solutons to slgtly modfed governng equatons are frst onstruted n analytal form. Ten dfferental operator for te governng equatons s appled to te analytal solutons to generate soure terms. Ts work an be done wt symbol manpulaton software, su as Matlab or Matemata. Fnally we wll obtan a system of new governng equatons wt addtonal analytal soure terms. Te analytal solutons tat we reated prevously exatly satsfy te new governng equatons. Tus, tese manufatured solutons an be used n te ode verfaton proedure. Oberkampf and Roy ave proposed 4 dfferent aeptane rtera for ode verfatons wt varyng rgor[7]. Among tese rtera, order of auray test s te most rgorous one. Order of auray test examnes f te dsretzaton error wt respet to te manufatured solutons tends to zero at expeted rate as te mes s refned. Ts expeted rate s alled formal order of auray and t s usually found by performng a trunaton error analyss of te numeral seme. Te atual rate at w te dsretzaton error s redued s alled observed order of auray. Te onssteny of te observed order of auray and formal order of auray ndates tat te ode verfaton s passed. One te order of auray msmates ea oter, furter nvestgaton s needed to detet odng mstakes.

4 4 L Wang, We-jang Zou, Cu-qun J / Proeda Engneerng 00 (014) Te observed order of auray s evaluated from te dsretzaton errors on dfferent mes refnement levels. Te dsretzaton error s defned as te dfferene between te exat soluton to te governng equatons and te dsrete soluton obtaned on gven mes level. In MMS, te exat solutons to te governng equatons are known and only two suessve refned meses are requred to ompute te observed order of auray. Consder a seres expanson of te dsretzaton error ò n terms of, were s mes spang measurement[7]. p p 1 ò O( ) () Now we an wrte te dsretzaton errors on two dfferent mes levels as p p1 ò O( ) ò r p p ( r) O r Here r / f represents te rato of oarse to fne grd mes spang. Note tat s ndependent of and an be smply understood by reognzng tat t s just te partal dervatve of dsrete soluton u wt respet to n te lmt as tends 0. After some algebra, we an fnd te expresson for te observed order of auray[7]: ò r ln pˆ ò ln r In ts study, te L norm of onservatve varables error s used to ompute te observed order of auray, w s defned as, N Qnum, Qmms, 1 L (5) N Mes refnement study s essental n te order of auray test. Te mes sze levels must be arefully osen n order to rea te asymptot range so tat te dsretzaton errors wll be redued at te teoretal rate assoated wt te dsretzaton seme. In our experene, te mes levels requred to rea te asymptot range are gly dependent on te manufatured solutons. We use 6 mes levels n ts study, w are 9 9, 17 17, 33 33, 65 65, All te grds are unformly spaed n ea oordnate dreton(fg.1). p1 1/ (3) (4) 4. Results Fg. 1 Computatonal mes(65 65) Te verfaton of a nonequlbrum reatng flows solver usng te metod of manufatured solutons s tedous. Varous fators an affet te fnal verfaton results. Te unexpeted results an be aounted for eter wrongly applaton of te MMS metod or odng mstakes. We sould not ontrbute te MMS applaton mstakes to programmng problems. Consderng tese onerns, te verfaton work s onduted n exerses wt nreasng omplexty. In te frst exerse, we present verfaton results for Euler equatons nvolvng 4 spees

5 L Wang, We-jang Zou, Cu-qun J/ Proeda Engneerng 00 (014) wtout emal soure terms. After tat, te emal soure terms omputatons are atvated and te same manufatured solutons are used to arry out te verfaton. In ts exerse, we enountered a serous problem. Te observed order of auray sows relatve large osllatons as te grd s refned altoug ts value s very lose to te teoretal value on te fnest grd. We are not sure weter ts beavour s normal, so two attempts are made to address ts problem Euler Equatons Wtout Cemal Soure Terms As dsussed n last seton, analytal form solutons must be reated to te governng equatons. Te manufatured solutons need not ave pysal mplatons, beause te fundamental plosopy of ode verfaton s to examne te orretness of te solutons to te governng equatons. In ts ase, smootness, dfferentable, and general enoug to exerse all te terms n te governng equatons are more mportant. We oose to use trgonometr funtons to onstrut te expressons for all te varables n te governng equatons. In urrent work, 4 spees, O, N, O, and N are assumed to present n te flow. Analytal solutons for p,, u, v, O, N, and O,are expressed n te form sown below[4]: 3x 4 y 0 1 f 1 f L L Were s a dummy varable representng p,, u, v, O, N, and O, s onstant and f represents osne or sne funtons. Te form of tese expressons wll be gven n te appendx. Te onstants n manufatured solutons for p,, u, v are dretly taken from te work of Roy[4]. However, te forms of expressons for ndvdual spees are reated ourselves. Tese onstants are arefully osen troug numbers of numeral tests. Ill-defned onstants may requre more grds to perform order of auray test. Te solutons for O and N are presented n fg.. Te analytal soure terms are generated wt te elp of Matlab software. Dstrbutons of soure terms for x-dreton momentum equaton and ontnuty equaton for O are sown grapally n fg. 3. (6) Fg. Manufatured solutons for O(left) and N(rgt) Fg. 3 Soure terms for x-momentum equaton(left) and O ontnuty equaton(rgt)

6 6 L Wang, We-jang Zou, Cu-qun J / Proeda Engneerng 00 (014) Te boundary ondtons are enfored by dretly spefyng te boundary values wt Drlet values from manufatured solutons. Te reason of usng su type of boundary ondtons s tat te man goal of urrent work s te verfaton of nteror equatons. Te way of boundary ondtons enforement as no adverse nfluene on te order of auray test[7]. Fg. 4 sows te beavour of densty dsretzaton error norm as te mes s refned. Te oter varables error norms exbt smlar beavor. As an be seen, all te onservatve varables error norms are redued at te teoretal order n te vnty of fnest grd. Fg. 5 plots te observed order of auray of all te onservatve varables, w onfrms te prevous observatons n fg. 4. Fg. 4. Beavor of L norm of densty error as te mes s refned Fg. 5. Observed order of auray 4.. Euler Equatons Wt Cemal Soure Terms Te results obtaned n last seton establs great onfdene n te ode resolvng Euler equatons wtout emal soure terms. Based on te prevous work, te verfaton of Euler equatons wt emal soure terms s performed n ts exerse. Beause only 4 spees are nluded n te governng equatons, a sub set of Gupta s emal reaton model[8] s employed. Te sub reaton meansm takes nto aount only dssoaton proesses of O and N. Te emal reaton rate oeffents are gven n te appendx. Note tat te pre-exponental fator A b n te bakward rate onstants s spefed 3 orders less tan te orgnal value n Gupta s model. Te emal soure terms omputed wt te orgnal reaton rate onstants are so large tat te smulaton wll not be suessfully ompleted. Fg. 6 gves te numeral soluton and manufatured soluton for pressure on grd In ts fgure, we an see tat te numeral soluton sows slgt osllatons near te top rgt orner. Furter examnaton of te observed order of auray s onduted and te result s sown n fg. 7. Te order of auray of te alulatons s approxmate te same as formal order of auray on te grd 57 57, but te overall onvergene of te alulatons exbts large osllatons. It s not defendable to draw te onluson tat te ode s verfed based on tese results. Note tat te only dfferene between te two tests s te spees soure terms alulaton. It s reasonable to suspet tat some odng mstakes may resde n tat porton of te ode. Te emal soure terms are a funton of, T,and and all te operatons are algebra. For larty sake, t s mportant to dstngus two types of emal soure terms n te ode. One s omputed n te ode to be verfed and te oter s alulated n te ode generated by Matlab. We wll refer to te two soure terms as numeral emal soure terms and manufatured emal soure terms. If,T,and takes te same value as te manufatured solutons and te ode ontans no odng mstakes, te numeral emal soure terms wll be dental wt te manufatured emal soure terms. To onfrm ts asserton, rater tan spefyng values from atual alulaton, we dretly set te values of,t, and n te emal soure terms alulaton from manufatured solutons. Fg. 8 presents te observed order of auray. Ts fgure sows no dfferene from fg. 5, w ndates tere s no odng mstakes n te emal soure terms omputaton.

7 L Wang, We-jang Zou, Cu-qun J/ Proeda Engneerng 00 (014) Fg. 6. Comparson of numeral soluton and manufatured soluton for pressure Fg. 7. Observed order of auray Fg. 8. Observed order of auray (usng manufatured soluton n te emal soure alulaton) It s very onfusng tat te emal soure terms alulaton affets te beavour of dsretzaton error. Some possbltes exst to explan te observatons. In urrent ase, we solve te total densty and ns 1 spees equatons. Ts wll ause all te numeral errors n ns 1 spees aumulate to te ns-t spees. Ts error agan wll affet te spees soure terms alulaton. Ts nteraton between te relatve large spees densty error and emal soure terms leads to te osllaton of flow feld. Anoter possblty s tat a sub set of emstry model maybe not approprate. Consderng te maxmum temperature of te manufatured solutons s no more tan 400K,te dssoaton rates of O and N are almost neglgble. Te emal soure terms are mostly resulted from te reombnaton of O and N, w makes tem more senstve to te mxture omposton. In order to nvestgate te frst guess, we re-mplement te solver to be apable of solvng te two sets of governng equatons,.e., te governng equatons ontanng total mass onservaton equaton plus ns 1 spees onservaton equatons (set 1) and te governng equatons nludng ns spees equatons (set ). Te new solver s verfed wt prevous proposed manufatured solutons for solvng te frst set of Euler equatons and dental results were obtaned as before. Ten te seond set of governng equatons s solved usng te solver. Te observed

8 8 L Wang, We-jang Zou, Cu-qun J / Proeda Engneerng 00 (014) order of auray obtaned by te new solver s presented n fg. 9. Ts result s very smlar wt fg.7. Dfferent sets of equatons mplemented n te solver ave lttle nfluene on te observed order of auray wen same numeral tenque s employed. Fg. 9. Observed order of auray by solvng all ns spees equatons After elmnaton of te frst possblty, we onstrut new manufatured solutons. In ts ase, te gas mxture s onsst of 5 spees of O, N, O, N, and NO. Te analytal expressons for tem are gven n te appendx. Te 5 spees ar reaton model of Gupta s used and all te reaton rate oeffents are kept unanged. Fg. 10 and fg. 11 gve te observed order of auray by solvng equatons set 1 and respetvely. It an be seen tat, te order of auray of bot alulatons mates te teoretal order of auray and te onvergene urves are smoot. Fg. 10. Observed order of auray by solvng governng equatons set 1 5. Conlusons Fg. 11. Observed order of auray by solvng governng equatons set Te MMS was used n te ode verfaton of an nvsd, nonequlbrum flows solver. Wen te same manufatured solutons are used to verfy Euler equatons wt or wtout emal soure terms, sgnfant dfferene s observed n te beavor of te observed order of auray as te mes s refned. Formal order of

9 L Wang, We-jang Zou, Cu-qun J/ Proeda Engneerng 00 (014) auray s easly obtaned wt te emal soure terms dsabled, wle observed order of auray sows large osllaton wt te emal soure terms atvated. In order to fnd te reasons, te solver s re-mplemented to be apable of solvng two sets of ontnuty equatons: total densty equaton plus ns-1 spees equatons and all ns spees equatons. Dfferent emstry models, one s omposed of only dssoatons of O and N wle te oter s onsst of bot dssoaton and reombnaton reatons, are also studed. Results sow tat te form of ontnuty equatons as no nfluene on te overall order of auray test. Comparng to a sub set of emstry model, te omplete emstry model s more approprate to use n te order of auray test. Appendx A. Manufatured solutons for Euler equatons nludng 4 spees equatons A.1. Form of manufatured solutons a x x a y y ( xy, ) 0 x sn y os L L aux x auy y u( x, y) u0 uxsn u y os L L a a vx x vy y v( x, y) v0 vx os vysn L L apxx apyy p( x, y) p0 px os pysn L L a x a y x y ( x, y) 0 x sn y os L L (7) A.. Constants for manufatured solutons Table 1.Constants for manufatured solutons expressed by Eq. (7). Flow varable 0 x y a a x y ( kg / m 3 ) u( m / s ) v( m / s ) p( pa ) 1.e5 0.e5 0.5e5 /3 1.0 O N O A.3. Reaton oeffents Table.Reaton oeffents. Reaton A f f N M N M 1.9E E O M O M 3.61E E Trd body effenes for reaton 1 O /1/, N /1/, O /1/, N /.5 / Trd body effenes for reaton O / 5/, N /1/, O / 9/, N / / Te forward and bakward reaton rates are omputed troug B f A b b B b

10 10 L Wang, We-jang Zou, Cu-qun J / Proeda Engneerng 00 (014) k j B j j AjT exp. T Appendx B. Manufatured solutons for Euler equatons nludng 5 spees equatons Table 3. Constants for manufatured solutons expressed by Eq. (7). Flow varable 0 x y a a x y ( kg / m 3 ) u( m / s ) v( m / s ) p( pa ) 1.e5 0.e5 0.5e5 /3 1.0 O N O N Referenes [1]W.L. Oberkampf, T.G. Truano, Verfaton and Valdaton n Computatonal Flud Dynams, SAND []P.J. Roae, Verfaton of Codes and Calulatons, AIAA Journal, 1998, 36(5) [3] P.J. Roae, S. Stenberg, Symbol manpulaton and omputatonal flud dynams, AIAA Journal, 1984, (10) [4] C.J. Roy, T.M. Smt, C.C. Ober, Verfaton of a Compressble CFD Code Usng te Metod of Manufatured Solutons, AIAA [5]L. Ea, M. Hoekstra, A. Hay, D. Pelleter. Verfaton of RANS Solvers wt Manufatured Solutons. Engneerng wt Computers, 007, 3:53~70 [6]S.P. Velur, C.J. Roy, E.A. Luke, Compreensve Code Verfaton for an Unstrutured Fnte Volume CFD Code, AIAA [7]W. L. Oberkampf, C. J. Roy, Verfaton and Valdaton n Sentf Computng, Cambrdge Unversty Press, 010. [8] R.N. Gupta, J.M. Yos, R.A. Tompson, K.-P. Lee, A Revew of Reaton Rates and Termodynam and Transport Propertes for an 11- Spees Ar Model for Cemal and Termal Nonequlbrum Calulatons to 30000K, NASA-RP-13.