Compues and Fluids, 0 Compeensive Code Veificaion Tecniques fo Finie Volume CFD Codes Subamanya P. Velui and Cisope J. Roy Aeospace and Ocean Engineeing Depamen Viginia Tec, Blacksbug, Viginia 406 Edwad A. Luke 3 Compue Science and Engineeing Depamen Mississippi Sae Univesiy, Sakville, MS 3976 A deailed code veificaion sudy of a finie volume Compuaional Fluid Dynamics (CFD) code using e Meod of Manufacued Soluions is pesened. Te coecness of e code is veified oug ode of accuacy esing. Sysemaic mes efinemen equied fo ode of accuacy esing and e way i is acieved paiculaly fo unsucued meses is discussed. Te veificaion esing is pefomed on diffeen mes ypes wic include iangula and quadilaeal elemens in D and eaedal, pismaic, and eaedal elemens in 3D. Te sensiiviy of e ode of accuacy o bo mes qualiy and mes opology is eamined. Along wi e baseline seady-sae govening equaions, anspo models, ubulence models, bounday condiions, and unseady flows ae veified. A new appoac fo e combined veificaion of spaial and empoal ems in e govening equaions is developed and assessed. Keywods: Code Veificaion, Ode of Accuacy, Manufacued Soluions Nomenclaue C p, C v = specific eas, J/Kg K DE = disceizaion eo e = enegy, J Gaduae Reseac Assisan, Cuenly Tes Enginee a ANSYS Inc. Associae Pofesso. 3 Associae Pofesso.
g p = coefficien of e leading eo em = nomalized gid spacing; enalpy, J/Kg K f = ea of fomaion, J/Kg K F = blending funcion f = geneal soluion vaiable k = ubulen kineic enegy p = spaial ode of accuacy; pessue, N/m q L = lamina ea flu q T = ubulen ea flu R = gas consan, J/Kg K = efinemen faco T = empeaue, K = ime, sec V i = Volume of cell i u, v, w = Caesian velociy componens, m/s, y, z = Caesian coodinaes, m Geek Symbols = eo μ = viscosiy, N sec/m μ T = ubulen viscosiy, N sec/m ρ = densiy, Kg/m 3 ω = ubulen dissipaion ae Subscips eac = eac coninuum value k = mes level,,, 3, ec.; fine o coase ef = efeence value
. Inoducion VERIFICATION addesses e maemaical coecness of e simulaions. Tee ae wo fundamenal aspecs o veificaion: code veificaion and soluion veificaion [-3]. Code veificaion is e pocess of ensuing, o e degee possible, a ee ae no misakes (bugs) in a compue code o inconsisencies in e soluion algoim. Soluion veificaion is e pocess of esimaing e ee ypes of numeical eo a can occu in numeical simulaions: ound-off eo, ieaive eo, and disceizaion eo. Tis pape focuses mainly on code veificaion. Cuen pacices in code veificaion veify a e obseved ode of accuacy of e disceizaion eo asympoically appoaces e fomal ode of accuacy of e disceizaion sceme as e mes is sysemaically efined. One of e main difficulies in veifying a code is idenifying eac soluions o e govening equaions wic eecise all ems. Tadiional eac soluions eis only wen e govening equaions ae faily simple, wic is ceainly no e case fo moden CFD codes wic ae epeced o andle comple pysics (ubulence, combusion, eal gas effecs, ec.), comple geomeies, and significan nonlineaiies. Te Meod of Manufacued Soluions, o MMS, is a geneal appoac fo obaining eac soluions [-3]. Rae an ying o find an eac soluion o a sysem of paial diffeenial equaions, e goal is o manufacue an eac soluion o a sligly modified se of equaions. Ode of accuacy veificaion is a igoous code veificaion assessmen wic involves compaing e obseved ode of accuacy of e disceizaion eo o e fomal ode of accuacy of e cosen numeical meods. Te obseved ode of accuacy can fail o mac e fomal ode due o misakes in e compue code, defecive numeical algoims, soluions wic ae no sufficienly smoo, numeical soluions wic ae no in e asympoic mes convegence ange [], and lage ound-off o ieaive eo. Te asympoic ange is defined as e ange of disceizaion sizes (, y,, ec.) wee e lowes-ode ems in e uncaion eo dominae. Te fis applicaion of MMS fo code veificaion was by Roace and Seinbeg in 984 [4]. In ei pioneeing wok, ey used e MMS appoac o veify a code fo geneaing ee-dimensional ansfomaions fo ellipic paial diffeenial equaions. Addiional discussions of e MMS pocedue fo code veificaion ave been pesened by Roace [, 5]. Te book by Knupp and Salai [6] is a compeensive discussion of code veificaion, MMS, and ode of accuacy veificaion. Moe ecenly, Obekampf and Roy [3] discussed e use of MMS fo geneaing eac soluions along wi e ode of accuacy esing fo code veificaion. 3
In pio wok, MMS as been used o veify wo compessible CFD codes [7]: Pemo [8] (developed by Sandia Naional Laboaoies) and WIND [9] (developed by e NPARC alliance). In Ref. 7, e auos successfully veified bo e inviscid Eule equaions and e lamina Navie-Sokes equaions; oweve, is sudy employed only Caesian meses. An alenaive saisical appoac o MMS was poposed by Hebe and Luke [0] fo e Loci-CHEM combusing CFD code []. In ei appoac, ey employ a single mes level wic is sunk down (us poviding a locally efined mes) and used o saisically sample e disceizaion eo in diffeen egions of e domain of inees. Tei wok successfully veified e Loci-CHEM CFD code fo e 3D, muli-species, lamina Navie-Sokes equaions using bo saisical and adiional MMS. Anoe simila appoac o saisical MMS is e downscaling appoac o ode veificaion [, 3] wic also employs a single mes wic is scaled down abou a single poin in e domain insead of saisically sampling e smalle meses in e domain of inees and ence eliminaing e saisical convegence issues associaed wi e saisical MMS appoac. Bo e saisical MMS and e downscaling appoaces ave an advanage of being elaively inepensive because ey do no equie mes efinemen. Bu one of e disadvanages of using ese meods is a ey neglec e possibiliy of disceizaion eo anspo ino e scaled-down domain. Tus, i is possible o pass a code ode of accuacy veificaion es using saisical MMS o e downscaling appoac fo a case a would fail e es using adiional MMS. Tee ave been some coodinaed effos o apply MMS o ubulen flows. Pelleie and co-wokes ave summaized ei wok on D incompessible ubulen sea layes using a finie elemen code wi a focus on a logaimic fom of e k- wo-equaion RANS model in Refs. 4 and5. Tey employed Manufacued Soluions wic mimic ubulen sea flows, wi e ubulen kineic enegy and e ubulen eddy viscosiy as e wo quaniies specified in e Manufacued Soluion. Fo e cases eamined, ey wee able o veify e code by epoducing e fomal ode of accuacy of e code. Moe ecenly, Eca and co-wokes ave publised a seies of papes on Manufacued Soluions fo e D incompessible ubulen Navie-Sokes equaions [6-8]. Tey also employed pysically-based Manufacued Soluions, in is case mimicking wall-bounded ubulen flow. Tis goup looked a bo finie-diffeence and finie-volume disceizaions, and eamined a numbe of ubulence models including e Spala-Allmaas one-equaion model [9] and wo wo-equaion models: Mene s baseline (BSL) vesion k-ω model [0] and Kok s ubulen/non-ubulen k-ω model []. Wile successful in some cases, ei pysically-elevan Manufacued Soluion ofen led o numeical insabiliies, a educion in e obseved 4
mes convegence ae, o even inconsisency of e numeical sceme (i.e., e disceizaion eo did no decease as e mes was efined). In ode o independenly es diffeen aspecs of e govening equaions, in some cases ey eplaced ceain disceized ems (o even wole equaions) wi e analyic counepa fom e Manufacued Soluion. Fo e Spala-Allmaas model ey specified e woking vaiable ~ ν, wile fo e wo equaion models ey specified bo e ubulen eddy viscosiy and e ubulen kineic enegy. Te cases ey eamined employed a Reynolds numbe of 0 6 and used Caesian meses wic wee cluseed in e y-diecion owads e wall. Ou appoac o code veificaion fo RANS models diffes fom e pevious wok in a numbe of ways. Wile e above wok focused on pysically-based eac soluions wi comple eponenials o mimic e ubulence quaniies found in eal ubulen flows, we simply use sinusoidal funcions. Ou agumen fo aking is appoac is a e goal of code veificaion is o pefom maemaical ess o ensue e disceizaion appoac and e implemenaions ino a code do in fac mac e oiginal govening paial diffeenial equaions and ei soluion [, 3]. In ode o veify e implemenaion of bounday condiions, e manufacued soluion mus be ailoed o eacly saisfy a given bounday condiion on a domain bounday and Bond e al. [4] developed a geneal appoac fo doing e same.. Govening Equaions Te veificaion conceps fom is wok can be applied o any finie volume code, bu e CFD code on wic veificaion is pefomed in e cuen wok is Loci-CHEM [, 5]. Loci-CHEM was developed a Mississippi Sae Univesiy using e Loci famewok [6, 7] and can simulae ee-dimensional flows of ubulen, cemically-eacing miues of emally pefec gases. Te Loci famewok povides a ig-level pogamming envionmen fo numeical meods a is auomaically paallel and uilizes a logic-based saegy o deec o peven common sofwae fauls (suc as eos in loop bounds o eos caused by subouine calling sequences being inconsisen wi daa dependencies). Te baseline govening equaions include e 3D Eule and Navie- Sokes equaions [8]. 5
3. Meod of Manufacued Soluions 3. Meodology Te mos igoous appoac o code veificaion is e ode of accuacy es [, 6], wic deemines wee o no e disceizaion eo is educed a e epeced ae. Tis es is equivalen o deemining wee e obseved ode of accuacy maces e fomal ode of accuacy. Fo all disceizaion appoaces (finie diffeence, finie volume, finie elemen, ec.) e fomal ode of accuacy is usually obained fom a uncaion eo analysis of e discee algoim. Te uncaion eo analysis appeas o be sufficien fo egula, sucued gids; oweve, i as been sown o unde-pedic e fomal ode of soluion convegence fo finie volume meods on unsucued meses [9]. Te obseved ode of accuacy is diecly compued fom code oupu fo a given se of simulaions on sysemaically efined gids. Te disceizaion eo is fomally defined as e diffeence beween e eac soluion o e discee equaions and e eac soluion o e govening paial diffeenial equaions. Since e eac soluion o e discee equaions (wic will be diffeen on diffeen mes levels) is geneally no known, e numeical soluion is obained by solving e discee esidual equaions o macine zeo levels neglecing ieaive and ound-off eo. Te obseved ode of accuacy can be evaluaed eie locally wiin e soluion domain o globally by employing a nom of e disceizaion eo. Wile we ave eamined L, L, L noms of e cuen code veificaion sudy, ee we epo only L noms fo beviy. Te volume weiged L nom fo mes level k is defined as L N [ Vi ( DEk, i ) ] N, k = Vi i= i= () wee e i inde denoes a cell cene value of one of e conseved vaiables and V i is e volume of cell i. Conside a seies epansion of e disceizaion eo in ems of k, a measue of e elemen size on e mes level k, DE f f = g p k = k eac p k + HOT () wee f k is e numeical soluion on mes k, g p is e coefficien of e leading eo em, and p is e fomal ode of accuacy. Neglecing e ige ode ems, we can wie e disceizaion eo equaion fo a fine mes (k=) and a coase mes (k=) in ems of e obseved ode of accuacy pp as DE = g and f feac = pˆ p DE = f f eac = g pˆ p (3) 6
Since e eac soluion is known, ese wo equaions can be solved fo e obseved ode of accuacy pp. Inoducing, e aio of coase o fine mes elemen spacing (= / ), e obseved ode of accuacy becomes DE pˆ = ln DE ln ( ) (4) Tus, wen e eac soluion is known, only wo soluions ae equied o obain e obseved ode of accuacy. Wen evaluaing e obseved ode of accuacy, ound-off and ieaive convegence eo can advesely affec e esuls. Round-off eo occus due o finie digi soage on digial compues. Ieaive eo occus any ime an ieaive meod is used, as is geneally e case fo nonlinea sysems and lage, spase linea sysems. Te disceized fom of nonlinea equaions can necessaily be solved o wiin macine ound-off eo; oweve, in pacice, e ieaive pocedue is usually eminaed ealie o educe compuaional effo. In ode o ensue a ese souce of eo do no advesely impac e obseved ode of accuacy calculaion [], bo ound-off and ieaive eo sould be a leas 00 imes smalle an e nom of e fine mes disceizaion eo (i.e., <0.0 DE ). Fo all cases pesened eein, double pecision compuaions ae used and e ieaive esiduals ae educed down o macine zeo. Fo e cuen compuaions, is coesponds o an ieaive esidual educion of appoimaely 4 odes of magniude. Te pocedue fo applying MMS wi ode of accuacy veificaion is demonsaed below fo a simple eample poblem:. Coose e specific fom of govening equaions; ee we eamine e D unseady ea equaion T T α = 0 (5). Coose e manufacued soluion T ( ) = T ep( ) sin( π L), 0 0 (6) 3. Opeae e govening equaions on e cosen soluion, esuling in analyic souce ems T = T0 sin T = T ep 0 ( π L) ep( ) ( )( π L) sin( π L) 0 0 0 (7) 4. Solve e modified govening equaion (oiginal equaion plus souce ems) on vaious mes levels 7
T T α = 0 π + α L T0 ep ( ) sin( π L) 0 (8) 5. Compue e obseved ode of accuacy of e disceizaion eo and compae i wi e fomal ode of accuacy 3. Baseline Manufacued Soluions In ou cuen wok, we adee o e pilosopy a code veificaion is simply a maemaical es o ensue e numeical soluion uly epesens e soluion o e coninuum maemaical equaions a ae being solved. As suc, we ave specifically cosen Manufacued Soluions wic ae no pysically ealisic, bu wic ae simple, smoo, and eecise all ems in e govening equaions. Te seady Manufacued Soluions employed ake e following fom aφπ φ(, y, z) = φ 0 + φ f s + φ y f L aφ yzπ yz aφzπ z + φ f + f yz s φ z s L L s aφ yπ y + φz f L s aφzπ z L + a φy f s π y L φy (9) wee φ = [ρ, u, v, w, p, k, ω] T epesens any of e pimiive vaiables and e f s( ) funcions epesen sine o cosine funcions. A Manufacued Soluion fo e -componen of velociy in a 3D domain is sown in Figue (a) and a smoo analyic mass souce em in a 3D domain is sown in Figue (b). Te specific values fo e consans in e above equaion ae given in Appendi B. (a) (b) Figue (a) Manufacued soluion of u-velociy, and (b) Mass souce em 8
Te Manufacued Soluions need o be modified fo e veificaion of e bounday condiions, and is is eplained fue in Secion 5. Fo veificaion of e ime accuacy of unseady flows, e Manufacued Soluions will include a ime em as well. Fo eample, e Manufacued Soluion used fo a D unseady poblem is of e fom sown in Eq.0. aφ π aφπ aφ yπ y aφyπ y φ(, y, ) = φ + + + + 0 φ f s φ f s φ y f s φy f s T L L L (0) 3.3 Raio of Souce ems In e pocess of selecing e Manufacued Soluion fo code veificaion puposes, i is equied a diffeen ems in e govening equaions ae ougly e same ode of magniude suc a conibuion fom eac em in e govening equaion is of same ode of magniude. Tis pevens e lage magniude ems fom masking eos in oe ems of smalle magniude. In e cuen wok, duing e veificaion of e lamina Navie-Sokes equaions, a consan viscosiy value of 0 Nm/s is used suc a ee is an appoimaely equal conibuion fom e inviscid and viscous ems. Similaly, duing e veificaion of ubulence models, e Manufacued Soluions ae geneaed suc a all e ems in e ubulence models ae ougly e same ode of magniude ove e domain consideed fo veificaion. Fo eample, in e souce ems geneaed fom e ubulen kineic enegy equaion, e aios of e convecion, diffusion, and poducion ems o e desucion ems ae calculaed o ceck a all e ems in e ubulen kineic enegy equaion ae of simila odes of magniude. As an eample, e aio of e poducion em o e desucion ems in e ubulen kineic enegy equaion in a D ecangula domain is sown in Figue. Simila souce em aios fo e ubulen dissipaion ae equaion ae calculaed o ceck a all e ems ae of simila odes of magniude [3]. Figue Raio of e poducion em o e desucion em in ubulen kineic enegy equaion [Ref. 3] 9
4. Sysemaic Mes Refinemen and Mes Geneaion Te Loci-CHEM finie volume CFD code is veified on diffeen mes ypes. In ode o veify all mes ansfomaions ae coded coecly, e code is un on e mos geneal mes ypes [3, 30] wic include meses wi mild skewness, aspec aio, cuvaue, and secing. A D ybid mes wic includes quadilaeal and iangula cells wi cuvilinea boundaies, skewed cells, and seced cells is consideed as e mos geneal mes opology fo D veificaion. Similaly a 3D ybid mes wic include eaedal cells, pismaic cells, and eaedal cells wi cuvilinea boundaies, skewed, and seced cells is consideed as e mos geneal mes opology fo 3D veificaion. 4. Sysemaic Mes Refinemen Sysemaic mes efinemen [3] is defined as unifom and consisen efinemen ove e spaial domain. A mes is said o be unifomly efined wen e mes is efined by e same faco ove e enie domain and in all e coodinae diecions. A mes is said o be consisenly efined if e mes qualiy says consan o impoves wi mes efinemen. (Noe a e ode of accuacy es will be easie fo e code o pass if e mes qualiy impoves wi efinemen.) As discussed in Ref. 3, code ode of accuacy veificaion equies sysemaic mes efinemen. In e case of sucued meses, coasening of e meses fo e veificaion pupose is saigfowad. A coase mes is geneaed fom a fine mes by emoving evey alenae mes poin o mes line o poduce mes levels wi a efinemen faco of wo. In e pocess, e mes qualiy can be mainained o impoved fo e sucued meses. Bu in e case of unsucued meses efinemen/coasening of meses wi a unifom efinemen faco ougou e domain wile peseving e mes qualiy is moe callenging, paiculaly in 3D. Fo D unsucued meses, sysemaic mes efinemen can be acieved by geneaing an unsucued mes fom a sucued mes by spliing quadilaeals ino iangles using diagonals [30]. To ou knowledge, geneaion of 3D unsucued meses wi unifom efinemen wile peseving e mes qualiy as no ye been acieved using commecial sofwae. Teefoe, fo code veificaion puposes, a mes geneaion code was developed o geneae an unsucued eaedal mes fom a 3D sucued mes wi eaedal elemens. In e pocess, a cube will be spli ino five eaedal cells as sown in Figue 3(a). In e case of spliing a cube, e cenal eaedal cell, wic does no sae a suface bounday wi e paen cube, is isoopic and e oe fou 0
eaedal cells ave e same opology wi good cell qualiy. An unsucued mes wi eaedal elemens geneaed using e mes geneaion code is sown in Figue 3(b). By geneaing unsucued meses in is fasion, a unifom and consisen efinemen can be acieved by a unifom efinemen faco and mainaining e cell qualiy consan beween e mes levels. Based on is concep of geneaing an unsucued mes fom a sucued mes, anoe code fo geneaing 3D ybid meses wic conain eaedal, eaedal, and pismaic cells wi pope conneciviy beween e diffeen ypes of cells was also developed. Te pismaic cells in e 3D ybid mes ae geneaed by diagonally spliing a eaedal cell ino wo pismaic cells. To obain a ybid mes fom a sucued mes of eaedal cells, 5 pecen of e eaedal cells ae spli ino pismaic cells, 50 pecen of e eaedal cells ae spli ino eaedal cells and oe 5 pecen ae lef as eaedal cells. Te 3D ybid meses ae geneaed o saisfy e unifom and consisen efinemen cieia beween e mes levels (Fig. 4a). (a) (b) Figue 3 (a) Heaedal cell spli ino five eaedal cells, (b) 3D unsucued mes wi eaedal cells geneaed fom a Caesian mes 4. Mes Topologies Te code is esed on comple ybid meses and if e veificaion ess fail on e comple ybid meses, en e code is esed on simple meses wic isolae e effec of mes secing, aspec aio, skewness, cuvaue, and cell opology. A D ybid mes conaining quadilaeal cells and iangula cells and a 3D ybid mes conaining eaedal cells, eaedal cells, and pismaic cells ae given in Figue 4. Te 3D ybid mes as ee diffeen layes of cells, eac laye consising of cells of a paicula cell opology. In Figue 4(a), e boom laye consiss of eaedal cells, e op laye consiss of pismaic cells, and e cene laye consiss of eaedal cells.
(a) Figue 4 (a) 3D ybid mes, (b) D ybid mes (b) Wen a veificaion es fails on a ybid mes, esing is en pefomed on simple sucued and unsucued mes opologies. In D, e simple opologies include a cuvilinea mes wi quadilaeal cells and an unsucued mes wi iangula cells. Te mes wi iangula cells is geneaed by saing wi a sucued mes and adding diagonals o quadilaeal cells o make e mes unsucued. Using is pocedue, unifom and consisen mes efinemen can be acieved fo unsucued meses beween diffeen mes levels used fo veificaion puposes. Tesing e code sepaaely on e D sucued and D unsucued meses can find e code sensiiviies owads e cell opologies. Te D skewed cuvilinea sucued mes and e D unsucued mes used fo esing e finie volume code ae sown in Figue 5. Te code can also be esed on even simple meses [30] like Caesian, seced Caesian, and non-skewed cuvilinea meses o fue isolae e effecs of mes secing, aspec aio, skewness, and cuvaue. Similaly in 3D, wen e veificaion es fails on e 3D ybid mes, e code is en esed on meses wi a paicula cell ype o deemine e code sensiiviies o e cell opology. Figue 6 sows e 3D cuvilinea mes wi igly skewed eaedal cells, e 3D unsucued mes wi eaedal cells, and e 3D unsucued mes wi pismaic cells. Te 3D unsucued mes wi eaedal cells is obained by saing fom a sucued mes and en spliing eac eaedal cell ino five eaedal cells. Te 3D unsucued mes wi pismaic cells is obained by saing wi a geneal D unsucued iegula mes and pojecing e mes in e id diecion nomal o e D suface. Te code can be esed on even simple meses like e 3D Caesian mes if equied.
(a) (b) Figue 5 (a) D skewed cuvilinea mes wi quadilaeal cells and, (b) D unsucued mes wi iangula cells (a) (b) (c) Figue 6 (a) Higly skewed 3D cuvilinea sucued mes wi eaedal cells, (b) 3D unsucued mes wi eaedal cells, and (c) 3D unsucued mes wi pismaic cells 3
5. Code Veificaion Diffeen opions esed in e Loci-CHEM CFD code include Baseline seady-sae govening equaions Sueland s law fo viscosiy Temally pefec emodynamic model Bounday condiions (Adiabaic no-slip wall, Isoemal no-slip wall, Slip wall, Supesonic Inflow, Fafield, Isenopic Inflow, Ouflow, Eapolae) Tubulence models (Spala-Almaas one equaion model [3], k-ω ubulence model, k- ubulence model) Time accuacy fo unseady flows Te opions in e finie volume CFD code ae veified by compaing e obseved ode of accuacy of e disceizaion eo calculaed fo e CFD soluions fom muliple sysemaically-efined meses o e fomal ode of accuacy of e numeical meod. An opion is consideed fully veified if i passes is ode of accuacy es on e 3D ybid mes wic as all cell opologies and includes skewness, aspec aio, cuvaue, and mes secing. Duing e pocess of code veificaion, e code opions ae esed on diffeen mes ypes and e veificaion esuls ae sown only on D and 3D ybid meses. Wen a veificaion es fails on e ybid meses, en e govening equaions ae esed on oe simple meses o find wee e discee fomulaion of e govening equaions is inconsisen on a paicula mes opology o due o e cell qualiy aibues. Mos of e code opions in Loci-CHEM ae veified, bu ee we focus on ineesing cases a ad poblems duing veificaion. A summay of e opions veified in e finie volume Loci-CHEM CFD code is sown in Appendi A. An opion can be assumed veified on a paicula mes even oug i is no acually un on a mes, if a opion is aleady veified on a moe comple mes ype. 5. Baseline Govening Equaions Veificaion of e baseline govening equaions includes e esing of e Eule equaions and e Navie- Sokes equaions on e D ybid and 3D ybid meses. Successful veificaion of e Eule equaions on a paicula mes and failue of e ode of accuacy es fo e Navie-Sokes equaions means a ee is a 4
poblem wi e fomulaion of e diffusion opeao on a paicula mes, i.e., an algoim inconsisency, a coding misake, o simply mes qualiy sensiiviy. Te Navie-Sokes equaions ae veified on e D and 3D ybid meses and e obseved ode of accuacy appoaces wo fo e numeical sceme wi mes efinemen on bo D and 3D ybid meses. Te Navie- Sokes equaions ae un on si mes levels of D ybid mes and fou mes levels of 3D ybid mes. Te obseved ode of accuacy is calculaed using L, L and L noms of e disceizaion eo. Duing e code veificaion pocess, nomally e obseved ode of accuacy calculaed using L and L noms sowed simila beavio, bu asympoed o e fomal ode a a slowe ae using L nom, equiing moe mes levels o see e asympoic beavio. In is pape, e obseved ode of accuacy esuls sown ae calculaed using only e L noms fo claiy, bu second ode beavio is obseved using e L noms as well. Te obseved ode of accuacy esuls calculaed using e L nom disceizaion eo fo e 3D ybid mes (Fig. 4a) and e D ybid mes (Fig. 4b) ae sown in Figue 7(a) and Figue 7(b), especively. In ese plos e obseved ode of accuacy p is on e y-ais and efeence cell leng is on e -ais. Te value of is abiaily se o uniy on fine gid and ence in ese plos e obseved ode of accuacy appoaces wo as e value appoaces one. Te Eule equaions wee also veified successfully on e meses consideed in is sudy, bu e esuls ae no sown ee. 4 Ode of Accuacy, p 3 o o*u o*v o*e Ode of Accuacy, p 4 3 o o*u o*v o*w o*e 5 0 5 0 3 4 (a) (b) Figue 7 Obseved ode of accuacy calculaed fo e Navie-Sokes equaions using L nom of e disceizaion eo on e (a) D ybid mes and (b) 3D ybid mes Te Navie-Sokes equaions wee also esed on a igly skewed 3D ybid mes sown in Figue 8(a). Te Navie-Sokes equaions wee successfully veified o be second ode accuae on e igly skewed 3D cuvilinea mes wi eaedal cells sown in Figue 6(a), bu e veificaion es failed on e igly skewed 3D ybid 5
mes. Te obseved ode of accuacy esuls ae sown in Figue 8(b). Fom e plo, e obseved ode of accuacy appoaces a value less an one wi mes efinemen. Te diffeence beween e 3D ybid mes (Fig. 4a) and e igly skewed 3D ybid mes (Fig. 8a) is only e qualiy of e cells in e mes; oewise bo e meses ave e same mes opology and conneciviy. Tis indicaes a poblem in e discee fomulaion of e govening equaions wen e cells ave a compaaively lowe qualiy. Te L nom of e disceizaion eo fo bo e 3D ybid mes and e igly skewed 3D ybid mes is compaed and i is obseved a e eo is ige fo e skewed ybid meses elaive o e ybid meses fo e same numbe of cells and simila mes sucue. Te compaison of e L nom of e disceizaion eo is sown in Figue 9. In e plo, eos sown in e solid lines coespond o e 3D ybid mes and e eos sown in dased line coesponds o e igly skewed 3D ybid mes. Tis plo eplains e effec of cell qualiy on e eo in e soluion. Te eo in e soluion eie deceases slowly o does no decease wi mes efinemen fo lowe qualiy meses. Ode of Accuacy, p 3 0 o o*u o*v o*w o*e 3 4 (a) (b) Figue 8 (a) Higly skewed 3D ybid mes, and (b) obseved ode of accuacy calculaed fo e Navie- Sokes equaions using L nom of e disceizaion eo - LNom 0 4 o 0 3 o*u o*v o*w 0 o*e o - skew o*u - skew 0 o*v - skew o*w - skew o*e 0 0 - skew s ode-slope nd ode-slope 0-0 - 0-3 0-4 0-5 4 6 8 0 6
Figue 9 Compaison of disceizaion eo on 3D ybid mes (Fig. 4a) and igly skewed 3D ybid mes (Fig. 8a) To fue sudy e discee fomulaions of e inviscid and viscous ems in e govening equaions, e Eule and Navie-Sokes equaions ae esed sepaaely on meses wi a paicula mes opology, i.e., eie igly skewed eaedal cells o igly skewed pismaic cells alone. As an illusaion of ow e Meod of Manufacued Soluions along wi e ode of accuacy es can be used o find misakes in e code o inconsisencies in e discee fomulaions is eplained by esing on diffeen meses wi diffeen cell opologies and diffeen cell qualiy. Te eason fo failue of code veificaion on e igly skewed 3D ybid mes is because of e insabiliy in e inviscid opeao. Tesing only e ea equaion (simila o esing e diffusion opeao) on e igly skewed 3D ybid mes poduced second ode accuacy. Te Navie-Sokes equaions ae also esed on e igly skewed D ybid meses o look a e effec of cell qualiy of quadilaeal cells and iangula cells on e discee fomulaion of e govening equaions. A igly skewed D ybid mes is sown in Figue 0(a). On is mes, e code is successfully veified and e obseved ode of accuacy appoaces wo wi mes efinemen wic is sown in Figue 0(b). Tis sows a e finie volume code woks fine on igly skewed quadilaeal and iangula cells in D. 4 Ode of Accuacy, p 3 o o*u o*v o*e 5 0 5 0 (a) (b) Figue 0 (a) Higly skewed D ybid mes and (b) obseved ode of accuacy calculaed fo e Navie- Sokes equaions using L nom of e disceizaion eo Oe equaions veified in e finie volume code ae e equaion of sae, emally pefec emodynamic model, and Sueland s law fo viscosiy. Veifying e Navie-Sokes equaions in e Loci-CHEM code 7
auomaically veifies equaion of sae. Fo veifying e anspo models, emal conduciviy and viscosiy ae defined as funcions of empeaue. Tey wee esed o be second ode accuae on 3D ybid mes wi e obseved ode of accuacy appoacing wo wi mes efinemen. 5. Bounday Condiions In ode o veify e implemenaion of a bounday condiion in a code, e Manufacued Soluion can be ailoed o eacly saisfy a given bounday condiion on a domain bounday. A geneal appoac fo ailoing Manufacued Soluions o ensue a a given bounday condiion is saisfied along a domain bounday was developed by Bond e al. [6, 4] Te appoac is eplained wi a simple eample in D. Te sandad fom of e Manufacued Soluion fo e D seady-sae soluion can be wien as φ (, y) = φ0 + φ(, y) () wee, φ aφπ + φ y f L aφ yπ y + φy f L a φy (, y) = φ f s s s π y L () A bounday in D can be epesened by a geneal cuve F(,y) = C, wee C is a consan. Te new Manufacued Soluion fo veifying bounday condiions can be found by muliplying φ (, ) em wi e funcion [ C F(, y) ] m as sown in Eq. 3. y φ BC (, y) = φ0 + φ(, y) [ C F( y) ] m, (3) Tis pocedue will ensue a e Manufacued Soluion is equal o consan φ 0 saisfying Diicle bounday condiion along e specified bounday fo m =. Fo m =, i ensues a e Manufacued Soluion will saisfy bo Diicle and Neumann (zeo gadien) bounday condiions along e specified bounday. Te bounday condiion opions in e Loci-CHEM code ae esed on D ybid meses and 3D ybid meses. In e case of veifying e no-slip wall bounday condiions fo e D ybid mes, a well defined cuved bounday on one of e fou sides is consideed as a no-slip wall and fo e 3D ybid mes a well defined wavy suface on one side of e domain is consideed as no-slip wall bounday. Te analyic definiion of e cuved boundaies consideed fo veificaion in bo D and 3D ae defined as sown in Eq. 4. 8
5π F(, y) = y Sin 0.05Sin( π) = 0 80 5 6 (,, ) = π π F y z y Sin zsin 0.06Sin 80 80 ( π) 0.05Sin( πz ) = 0 (4) Tese boundaies ae used bo in e Manufacued Soluion and e gid geneaion. Te no-slip wall bounday is esed as an adiabaic bounday and an isoemal bounday. Te empeaue conous fo e case of a no-slip wall defined as an isoemal bounday and an adiabaic bounday in a 3D domain ae sown in Figue (a) and Figue (b), especively. Te boom wavy suface sown in e figue is e no-slip wall bounday. (a) (b) Figue (a) Tempeaue conous fo e isoemal wall fo e boom bounday and (b) empeaue conous fo e adiabaic wall fo e boom bounday By esing e no-slip wall as an adiabaic bounday, a Neumann bounday condiion fo empeaue (dt/dn = 0) is veified along wi e no-slip condiion (V = 0) on a paicula bounday. Te no-slip wall is esed as an adiabaic bounday on bo D ybid and 3D ybid meses and e obseved ode of accuacy calculaed fom e numeical soluions appoaces wo wi mes efinemen. Te obseved ode of accuacy calculaed using e L nom of e disceizaion eo wen e no-slip wall esed as adiabaic bounday on 3D ybid mes is sown in Figue. Similaly, e no-slip wall is esed as an isoemal bounday, a Diicle bounday condiion fo empeaue (T = consan) is veified along wi e no-slip condiion (V = 0) on a paicula bounday. Te obseved ode of accuacy fo is case appoaces wo wi mes efinemen fo bo e D ybid and 3D ybid meses (no sown). Duing e veificaion of no-slip wall bounday condiion i was found a e mes sould be nomal o e wall. Wen e mes was no nomal o e wall, e ode of accuacy es failed fo bo D and 3D 9
meses. Tis indicaed a e numeical fomulaion is second ode accuae only wen e mes is nomal o e wall and also e impoance of aving mes nomal o e wall fo acieving e equied accuacy. Ode of Accuacy, p 4 3 o o*u o*v o*w o*e 3 4 Figue Obseved ode of accuacy calculaed using L nom of e disceizaion eo fo adiabaic noslip wall bounday on 3D ybid mes Tesing e slip wall bounday is veifying e slip condiion V n = 0 on a paicula bounday wee V n is e velociy componen nomal o e suface. Also, on a slip wall bounday, e viscous sess ems need o be zeo. Te slip wall bounday condiion opion was esed wi bo Eule equaions and Navie-Sokes equaions and found o be second ode accuae on 3D ybid meses. Oe bounday condiion opions esed in e finie volume code include e fafield bounday condiion wic is an inflow-ouflow caaceisic based bounday condiion suiable fo fafield condiions in eenal flow fo bo supesonic and subsonic flow, isenopic bounday condiion wic peseved oal condiions on e bounday, ouflow bounday condiion wic is a caaceisic based bounday condiion used fo bo subsonic and supesonic flow, and eapolaed bounday condiion wic is useful fo supesonic ouflow condiions. All e above bounday condiion opions wee esed on e 3D ybid mes wi saig boundaies and wee successfully veified o be second ode accuae. 5.3 Tubulence Models Veificaion of RANS ubulence models povides addiional callenges fo MMS fo diffeen easons []. One of e easons is a e ubulence models ofen employ min o ma funcions o swic fom one beavio o anoe, us causing e souce ems o no longe be coninuously diffeeniable. Ou appoac [] selecs Manufacued Soluions suc a ey will only acivae one banc of e min and ma funcions fo a given Manufacued Soluion. Te ubulence models esed in e finie volume code ae e basic k-ω ubulence model 0
and e k- ubulence model wic ae pa of e baseline vesion of Mene s k-ω model [0] and e Mene s Sea Sess Tanspo k-ω model [0]. In bo e Mene s ubulence models, e k-ω ubulence model ges acivaed in e bounday laye egion and e k- ubulence model ges acivaed away fom e wall boundaies in fee sea layes. To implemen is, e k- ubulence model is ansfomed ino a k-ω fomulaion, and an addiional coss diffusion em is added. Te k-ω ubulence model and e k- ubulence model ae blended ogee using a blending funcion, F. By seing e blending funcion F o zeo, e coss diffusion em in e ubulen dissipaion ae equaion is acivaed and e k- ubulence model is esed. By seing e blending funcion F o uniy, e coss diffusion em in e ubulen dissipaion ae equaion is deacivaed and e k-ω ubulence model is esed. Te k-ω ubulence model is esed on e 3D ybid mes and e obseved ode of accuacy appoaced wo wi mes efinemen. Te obseved ode of accuacy calculaed fo e k-ω ubulence model on e 3D ybid mes is sown in Figue 3(a). Duing e esing of k- ubulence model, a poblem wi e ubulen dissipaion ae equaion was found and e disceizaion eo fo a equaion did no decease a e epeced ae. Te obseved ode of accuacy of e ρω disceizaion eo noms dopped o zeo wi mes efinemen, bu all e oe conseved vaiable disceizaion eo noms appoaced wo wi mes efinemen. See Figue 3(b). Ode of Accuacy, p 4 3 o o*u o*v o*w o*e o*k o*ω Ode of Accuacy, p 4 3 o o*u o*v o*w o*e o*k o*ω 0.5.5 3 3.5 4.5.5 3 3.5 4 (a) (b) Figue 3 Obseved ode of accuacy calculaed on e 3D ybid mes fo e (a) e k-ω ubulence model (F = ) and (b) e k- ubulence model (F = 0) 0 To eploe e eason fo e failue of e veificaion es fo e k- ubulence model on e 3D ybid mes, i is esed on simple meses. Iniially e k- ubulence model is esed on e D ybid mes and i is obseved a e veificaion is successful wi all e noms of e disceizaion eos appoacing wo wi mes
efinemen. Te ode of accuacy esuls ae sown in Figue 4(a). Te above es is also done on a igly skewed 3D cuvilinea mes wi eaedal cells and e k- ubulence model is successfully veified wi all e noms of e disceizaion eo appoacing wo wi mes efinemen and e calculaed obseved ode of accuacy is sown in Figue 4(b). Ode of Accuacy, p 3 o o*u o*v o*e.5 o*k o*ω.5 Ode of Accuacy, p 4 3.5 3.5.5 o o*u o*v o*w o*e o*k o*ω 0.5 5 0 5.5.5 3 3.5 4 (a) (b) Figue 4 Obseved ode of accuacy calculaed fo e k- ubulence model (F = 0) on (a) e D ybid mes and (b) e igly skewed 3D cuvilinea (i.e., sucued) mes wi eaedal cells 0.5 In addiion, e k- ubulence model is esed on a 3D unsucued mes wi eaedal cells as sown in Figue 3(b) and i is successfully veified wi all e noms of e disceizaion eo appoacing wo wi mes efinemen. Te obseved ode of accuacy esuls ae sown in Figue 5. By esing on diffeen meses, i can be concluded a ee is an issue in e discee fomulaion of some of e ems in e ubulen dissipaion ae equaion as i woks coecly only on D mes opologies, 3D sucued mes opologies, and 3D unsucued mes wi eaedal cells bu i fails on 3D unsucued mes opologies wi skewed cells. Pefoming ese ess, e eason fo e failue of veificaion ess fo e k- ubulence model on e 3D ybid mes is isolaed o e coss-diffusion em in e ubulen dissipaion ae equaion and o a paicula cell opology.
Ode of Accuacy, p 4.5 4 3.5 3.5.5 o o*u o*v o*w o*e o*k o* 0.5 4 6 8 Figue 5 Obseved ode of accuacy fo k- ubulence model Te Spala-Almaas ubulence model as also been veified on 3D unsucued and 3D sucued cuvilinea meses using a saisical appoac o MMS [0]. In is appoac, a single mes is consideed and sunk down poviding a locally efined mes and is pocedue is used o saisically sample e disceizaion eo in diffeen egions of e domain of inees. Using e saisical appoac, e Spala-Almaas ubulence model was successfully veified o be second ode accuae (see Ref. 0 fo deails). 5.4 Time Accuacy of Unseady Flows I is moe difficul o apply e veificaion pocedue using e ode of accuacy es o poblems a involve bo spaial and empoal disceizaion, especially wen e spaial ode of accuacy is diffeen fom e empoal ode. A combined spaial and empoal ode veificaion meod as been developed by Kamm e al. [3] In ei appoac, ey use a Newon-ype ieaive pocedue o solve a coupled, nonlinea se of algebaic equaions o calculae e coefficiens and obseved ode of accuacies fo e spaial and empoal ems in e disceizaion eo epansion. In is wok, we popose a simple appoac fo spaial and empoal ode veificaion. Neglecing e ige ode ems, e disceizaion eo fo a sceme wi spaial and ime ems can be wien as = g p ˆ + g qˆ (5) wee pp and qq ae e obseved odes of accuacy in space and ime, especively, and g and g ae e coefficiens of spaial and ime ems, especively. Similaly, e nom of e disceizaion eo can be found as = g p ˆ + g qˆ (6) 3
4 Iniially, a spaial mes efinemen sudy is pefomed wi a fied ime sep o calculae pp and g using ee mes levels wic makes e disceizaion eo equaion +φ = p g ˆ (7) wee q = g ˆ φ is e fied empoal eo em. Using ee mes soluions, efined by e faco, coase ( ), medium ( ), and fine ( ), e obseved ode of accuacy pp can be calculaed (Ref. 3) as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ φ φ φ = p p p p g g g g ˆ ˆ ˆ ˆ ( ) [ ] ( ) [ ] p p p p g g = ˆ ˆ ˆ ˆ ( ) p ln ln ˆ = (8) wee is e spaial efinemen faco beween wo mes levels and e coefficien of e spaial em g can be calculaed as ( ) = p p g (9) Similaly, a empoal efinemen sudy is pefomed on a fied mes o calculae qq and g using ee empoal disceizaions, coase ( ), medium ( ), and fine ( ). Wi all e coefficiens calculaed, e spaial sep size and e empoal sep size can be cosen suc a e spaial disceizaion eo em as e same ode of magniude as e empoal disceizaion eo em. Once ese wo ems ae e same ode of magniude, combined spaial and empoal ode veificaion is conduced by coosing e empoal efinemen faco suc a e empoal eo em dops by e same ode of magniude as e spaial eo em wi efinemen, i.e., q p ˆ ˆ =. Hee is e empoal efinemen faco, is e spaial efinemen faco, pp is e spaial ode and qq is e empoal ode. In ou case, e fomal ode is in bo space and ime, i.e., pp = qq =. Using is pocedue, e unseady ime em is veified on e 3D ybid mes (Fig.
4a) and e D ybid mes (Fig. 4b) fo Navie-Sokes equaions. Te obseved ode of accuacy on bo e meses appoaced wo wi mes efinemen. Te esuls ae sown in Figue 6. Ode of Accuacy, p 3 o o*u o*v o*e Ode of Accuacy, p 4 3 o o*u o*v o*w o*e 0 4 6 8 0 3 4 (a) (b) Figue 6 Obseved ode of accuacy calculaed fo e unseady ime macing em on (a) e D ybid mes and (b) e 3D ybid mes 6. Issues Uncoveed Duing Code Veificaion Duing code veificaion sudies, some code opions failed e ode of accuacy es fo e fis ime and e veificaion sudies elped in leaning moe abou e code o e algoim and ulimaely emove some of e misakes in e code o algoim. Because e veificaion pocedue is so sensiive o mino issues like e mes opology o mes qualiy, ey can ofen uncove sensiiviies o ese seemingly mino issues. Issues uncoveed duing e code veificaion ae documened below. 6. Coding Misakes/Algoim Inconsisencies. A coding misake was found and coeced in e fomulaion of e diffusion opeao wile esing e Navie-Sokes equaions in e finie volume code on e D skewed cuvilinea mes. Iniially, e disceizaion eo did no decease wi mes efinemen and a modificaion was done o e diffusion opeao by Luke [33]. Te new diffusion opeao ecified e poblem and an obseved ode of accuacy of wo was aained wi mes efinemen.. A coding misake was found and coeced in e fomulaion of e diffusion opeao wile esing e Navie-Sokes equaions on D unsucued mes wi iangula cells. Iniially, e diffusion opeao was 5
found o be only fis ode accuae on D unsucued meses and afe modifying e diffusion opeao fomulaion fo unsucued gids, e diffusion opeao esed was found o be second ode accuae. 3. Tesing e k- ubulence model on diffeen mes opologies, i was found a ee was an issue in e discee fomulaion of e coss-diffusion em in e ubulen dissipaion ae equaion. Te pesen fomulaion woked fine on D mes opologies, 3D sucued mes opologies, and 3D unsucued mes wi eaedal cells (Figue 5(a)) bu failed on 3D unsucued mes opologies wi skewed cells. Tis issue is unde invesigaion. 6. Gid Sensiiviies. Duing e veificaion of e no-slip wall bounday condiions i was found a e mes sould be nomal o e wall and e ode of accuacy es failed if e mes was no nomal o e wall.. Sysemaic mes efinemen was found o be impoan fo code veificaion puposes. Failue of code veificaion ess does no indicae coding eos wen meses ae no efined sysemaically. 3. Te discee fomulaion of e govening equaions was found o be sensiive o igly skewed eaedal and pismaic cells. Te baseline govening equaions wee successfully veified on e igly skewed D ybid mes and igly skewed 3D cuvilinea mes wi eaedal cells, bu failed e veificaion es on a igly skewed 3D ybid mes. 7. Conclusions Compeensive code veificaion of an unsucued finie volume code was pesened. Te Meod of Manufacued Soluions was used o geneae eac soluions. Diffeen opions in e finie volume CFD code wee veified wic included e baseline govening equaions, diffeen bounday condiion opions, ubulence models, and ime accuacy of unseady flows. All e opions wee deemined o be veified wen e obseved ode of accuacy maced e fomal ode on e D ybid and 3D ybid mes wic conained all cell opologies (iangula, quadilaeal, eaedal, eaedal, and pismaic cells). Wen e veificaion pocess failed on any one of ese comple ybid meses, en simple meses wee consideed o isolae e poblem. In addiion, a meod fo geneaing sysemaically-efined unsucued meses fo code veificaion was developed wic 6
involved e coasening of an undelying sucued mes followed by spliing of e eaedal elemens ino eaedal o pisms. Coding misakes, algoim inconsisencies, and mes qualiy sensiiviies uncoveed duing code veificaion wee pesened. Te equiemen of mes nomal o e no-slip wall boundaies o acieve accuae esul was also uncoveed duing code veificaion. By esing e finie volume code on diffeen meses, e effec of cell qualiy and cell opology on e accuacy of e code was assessed. Finally, a new code veificaion ecnique was developed and esed fo e simulaneous veificaion of spaial and empoal disceizaions wic can be applied even wen e fomal ode of accuacy in space and ime is no e same. Acknowledgemens Tis wok is suppoed by e Naional Aeonauic and Space Adminisaion s Consellaion Univesiy Insiues Pogam (CUIP) wi Claudia Meye and Jeffey Ryback of NASA Glenn Reseac Cene seving as pogam manages and Kevin Tucke and Jeffey Wes of NASA Masall Space Flig Cene seving as ecnical monios. 7
Appendi A A summay of e seveal opions veified in e finie volume Loci-CHEM CFD code is sown in Figue A. Figue A. Veificaion of diffeen opions in e finie volume Loci-CHEM CFD code 8
Appendi B Te consans and e igonomeic funcions used in e Manufacued Soluion ae given in Table B. Table B. Consans fo Manufacued Soluion Equaion, φ φ 0 φ φ y φ z φ y φ yz φ z ρ (kg/m 3 ) 0.5-0. 0. 0.08 0.05 0. u (m/s) 70 7-5 -0 7 4-4 v (m/s) 90-5 0 5 - -5 5 w (m/s) 80-0 0-5 p (N/m ) 0 5 0. 0 5 0.5 0 5 0. 0 5-0.5 0 5-0. 0 5 0. 0 5 k (m /s ) 780 60-0 80 80 60-70 ω (/s) 50-30.5 0 40-5 5 Equaion, φ aφ aφy aφz aφy aφyz aφz ρ (kg/m 3 ) 0.75 0.45 0.8 0.65 0.75 0.5 u (m/s) 0.5 0.85 0.4 0.6 0.8 0.9 v (m/s) 0.8 0.8 0.5 0.9 0.4 0.6 w (m/s) 0.85 0.9 0.5 0.4 0.8 0.75 p (N/m ) 0.4 0.45 0.85 0.75 0.7 0.8 k (m /s ) 0.65 0.7 0.8 0.8 0.85 0.6 ω (/s) 0.75 0.875 0.65 0.6 0.75 0.8 Equaion, φ f s (-em) f s (y-em) f s (z-em) f s (y-em) f s (yz-em) f s (z-em) ρ (kg/m 3 ) cos sin sin cos sin cos u (m/s) sin cos cos cos sin cos v (m/s) sin cos cos cos sin cos w (m/s) cos sin cos sin sin cos p (N/m ) cos cos sin cos sin cos k (m /s ) cos cos sin cos cos sin ω (/s) cos cos sin cos cos sin 9
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