Spurious oscillations and conservation errors in interface-capturing schemes

Transcription

1 Cener for Turbulence Research Annual Research Brefs Spurous oscllaons and conservaon errors n nerface-capurng schemes By E. Johnsen Movaon and objecves When shock-capurng schemes are appled o flows of mulple componens, spurous oscllaons usually develop a nerfaces (e.g., n Mulder e al. 199) and propagae hrough he flow, hus conamnang he soluon. These perurbaons are parcularly problemac when small flucuaons are mporan flow feaures (e.g., n acouscs or urbulence), and n general mulcomponen mul-dmensonal flows where hese errors may rgger undesrable nsables a nerfaces. The presen work focuses on nerface-capurng schemes, n whch nerfaces are no resolved on he grd bu are regularzed over a few compuaonal cells by addng numercal dsspaon. In he cone of nerface capurng, Abgrall (1996) raced such errors o he dsconnuy n he rao of specfc heas n he energy equaon and suggesed a means o overcome hs drawback for frs- and second-order accurae quas-conservave schemes. In he proposed quas-conservave formulaon, he Euler equaons are sll wren n conservave form, so ha correc shock speeds are acheved, bu he ranspor equaons used o descrbe he nerface are non-conservave. Hence, he mass of each speces s no eacly conserved. The correcon proposed by Abgrall (1996) mus be modfed for hgher-order accurae spaal dscrezaons, such as Weghed Essenally Non-Oscllaory (WENO) mehods (Lu e al. 1994): Johnsen & Colonus (6) showed ha a reconsrucon of he averaged prmve varables s requred n a fne volume nerface-capurng mplemenaon of WENO o mulcomponen flows. In relevan sudes, Larrouurou (1991) observed ha negave values of he mass fracon may be acheved. The objecve of he presen paper s o assess he level a whch conservaon losses, spurous oscllaons and negave mass fracons affec he soluons o mulcomponen flow problems n an nerface-capurng framework. Secon descrbes he governng equaons and he numercal mehods and Secon 3 presens resuls for wo es problems; n parcular, he behavor of he error s suded for dfferen formulaons of he governng equaons and under grd refnemen. Fnally, he fndngs are summarzed and an oulook for fuure work s oulned.. Numercal mehods For smplcy, he 1-D Euler equaons are consdered: ρ ρu ρu + ρu + p =, (.1) E (E + p)u In nerface-rackng mehods, such as he Ghos Flud Mehod (Fedkw e al. 1999) or pressure evoluon algorhms (Karn 1994), oscllaons are usually prevened by solvng a dfferen form of he equaons a nerfaces, so ha errors n he conservaon of oal energy may occur.

2 116 E. Johnsen where ρ s he densy, u s he velocy, E s he oal energy, p s he pressure, denoes me and denoes poson. For deal gases, he followng equaon of sae closes he sysem: E = p γ 1 + ρu, (.) where γ s he rao of specfc heas. When solvng he sysem of Eqs. (.1) and (.) for mulple fluds, an addonal equaon mus be specfed n order o deermne he approprae value of γ. In he presen problems, he dfferen flud componens are assumed mmscble. An nerface beween wo flud componens can be represened by a dsconnuy n γ. Snce maeral nerfaces are adveced by he flow, φ + uφ =, (.3) where φ could be any funcon of he maeral properes,.e., φ = φ(γ). Thus, Eq. (.3) specfes he locaon of he nerface. Because Eq. (.3) s no n conservave form, he sysem of equaons s ermed quas-conservave. Usng he connuy equaon, he ranspor equaon for maeral properes can be wren n conservave form: (ρφ) + (ρuφ) = (.4) In nerface-capurng schemes, φ s usually a feld varable. Numercal dsspaon s nroduced, such ha nerfaces dffuse over a few grd pons, n analogy o shock capurng. Thus, even hough he fluds are assumed mmscble, here ess a hn regon beween wo fluds made up parly of one flud, parly of anoher one. Ths dffuse nerface s a purely numercal arfac and s epeced o reduce o a sharp one as As a resul, an approprae defnon of γ mus be specfed. Based on he defnon of he mass fracon, Y = ρ /ρ m, where he subscrps and m denoe one componen and he mure, respecvely, and ha of he gas consan, R m = R u /M m, where R u s he unversal gas consan and M s he molar mass, he gas consan for a mure of wo componens s gven by: γ 1 1 M γ 1 γ µ = γ M 1Y 1 + γ Y 1 M Y 1 + Y 1 γ 1 1 M γ 1, (.5) wh Y = 1 Y 1. Thus, φ could be a funcon of Y 1. Though desrable, a physcal defnon of γ whn hs dffuse regon (.e., usng Eq. (.5)) s no srcly requred, snce hs regon vanshes as he grd s suffcenly refned. In fac, n order o preven spurous nerface oscllaons n frs- and second-order accurae schemes, Abgrall (1996) uses he funcon, φ = 1 γ = γ 1 φ, (.6) o defne γ drecly n Eq. (.). Oher funcons of γ lead o nerface oscllaons (Karn 1994). A drawback of Eq. (.6) s ha addonal equaons mus be added f he evoluon of dfferen gases of he same γ s consdered. As an alernave, Shyue (1998) uses he mass fracon and defnes 1 φ = Y 1, wh γ 1 = Y 1 γ Y 1 γ 1. (.7) Snce C v = R/(γ 1), Eq..7 mples ha he followng assumpon s made: R m = R 1 = R, or, alernaely, M m = M 1 = M. R m s no requred o solve he mulcomponen Euler equaons supplemened by a ranspor equaon for whch γ s epressed

3 Oscllaons and conservaon errors n nerface-capurng schemes 117 by Eq. (.6) or Eq. (.7), bu s needed only n he pos-processng of he resuls, e.g., f he emperaure s of neres. On he oher hand, f Eq. (.5) s used, R m mus be nally specfed n each flud. In he formulaons of Abgrall and Shyue, an nal dsrbuon of 1/(γ 1) or Y 1 s specfed and evolved accordng o he advecon Eq. (.3), wh approprae modfcaons o he Remann solver. Thus, he equaon of sae vares smoohly across he nerface. A fne volume dscrezaon of he equaons of moon s currenly employed. The governng equaons can be wren n sem-dscree form: dq = f +1/ f 1/, (.8) d where q s he vecor of conserved varables and f +1/ s he numercal flu vecor. A hrd-order accurae Toal Varaon Dmnshng Runge-Kua scheme s used for me marchng (Goleb & Shu 1998). The spaal reconsrucon s carred ou n physcal space and s based on WENO (ffh-order accurae, unless oherwse menoned); he La-Fredrchs (LF) solver s used for upwndng. A dffculy wh he non-conservave form of he ranspor Eq. (.3) s ha he appromae Remann solvers mus be modfed accordngly (Saurel & Abgrall 1999). For nsance, he sem-dscree verson of Eq. (.3) usng LF s: [ φ u R +1/ +φl +1/ α(φr +1/ φl +1/ ) ] [ φ u R 1/ +φl 1/ α(φr 1/ φl 1/ ) dφ =, d (.9) where he superscrps L and R denoe he value of he funcon on he lef and on he rgh of a cell edge, and α s he larges absolue value of he egenvalue over he doman. Several ypes of errors are hghlghed n he followng secon. The frs s he generaon of spurous pressure oscllaons due o he dsconnuy n γ n he energy equaon (Abgrall 1996). Ths error propagaes o he momenum hrough he pressure graden and hus alers he velocy, whch n urn affecs he densy. A second error s he occurrence of mass fracon values ousde of he allowed bounds because of an nconssen couplng of he addonal ranspor equaon o he Euler sysem (Larrouurou 1991; Abgrall & Karn 1). Fnally, he hrd error consdered n he presen work perans o he mass conservaon of each speces. The behavor of each of hese errors depends on he sysem of equaons ha s solved. Table summarzes he dfferen formulaons. Y sands for mass fracon, γ sands for he ranspor varable based on he rao of specfc heas, FC sands for fully conservave, QC for quas-conservave (QCC mples ha he conservave varables are reconsruced n he Euler equaons, whle QCP mples ha he average prmve varables are reconsruced), M denoes he physcal epresson for γ,.e., Eq. (.5). The WENO varables refer o he varables ha are reconsruced usng he WENO procedure. ] 3. Resuls In order o undersand fundamenal problems n nerface-capurng schemes, wo 1-D problems are consdered: he advecon of a maeral nerface, whch s suded n grea deal, and he neracon of a shock wh an nerface, whch s of praccal relevance. A reconsrucon n characersc space may lead o smaller oscllaons a nerfaces, bu wll no remove hem.

4 118 E. Johnsen Table Dfferen schemes consdered n he presen sudy. Scheme Transpor varable Advecon equaon Equaon for γ WENO varables Y -FC-M ρy 1 Eq. (.4) Eq. (.5) Conservave Y -FC ρy 1 Eq. (.4) Eq. (.7) Conservave Y -QCC Y 1 Eq. (.3) Eq. (.7) Conservave Y -QCP Y 1 Eq. (.3) Eq. (.7) Prmve Y -QCP-M Y 1 Eq. (.3) Eq..5 Prmve γ-qcp 1/(γ 1) Eq. (.3) Eq. (.6) Prmve 3. Advecon of a maeral nerface The smples possble case of he advecon of a maeral nerface beween wo dfferen gases s frs suded n deal. A op-ha dsrbuon of helum n nrogen s movng a a consan velocy (equal o he sound speed n helum) wh unform pressure. The doman s perodc and has 11 pons (unless oherwse menoned), wh grd spacng, =. The varables are non-dmensonalzed by he densy, ρ He, and sound speed, c He, of helum, and he doman lengh, L. A consan / = 4 s used and he fnal me s f c He /L = 4. (.e., he soluon s ploed afer he nerface has raveled wo perods). The nal condons are as follows: ρ/ρ He = p a/rt a p a /R He T a, u/c He = (1,, ), p/ρ He c He = 1/γ He, { 1, f 5 5, Y He =, oherwse, (3.1) where he subscrp a refers o amben condons and Y N = 1 Y He. Frs, he overall behavor of a fully conservave (Y -FC), and a quas-conservave (Y -QCP) scheme, boh of whch employ he mass fracon as he relevan varable n he relevan ranspor equaon. In he quas-conservave scheme, he average prmve varables are reconsruced. Fgure 1 shows profles a f for boh schemes. Oscllaons are observed n all felds for he fully conservave scheme, parcularly n he pressure and velocy; he densy and γ acheve values ousde of he allowed bounds. Somewha surprsngly, he γ profle s more dffuse han n he quas-conservave case. Because of he dsconnuy n γ (Abgrall 1996) and because of he nconssen couplng beween he energy and ranspor equaons (Johnsen & Colonus 6), he fully conservave scheme leads o an error n pressure a he end of he frs me sep. Due o he couplng of he energy, momenum and connuy equaons, he pressure oscllaons generae errors n he velocy and densy felds as well. No oscllaons are vsble n he quas- As shown n Johnsen & Colonus (6), he advecon form of he ranspor equaon alone s no suffcen o preven spurous nerface oscllaons for WENO schemes. If he radonal reconsrucon of he conservave varables s carred ou (Y -QCC scheme), oscllaons are generaed.

5 Oscllaons and conservaon errors n nerface-capurng schemes ρ/ρhe 4 u/che (a) Densy (b) Velocy. 5 γ (c) Rao of specfc heas. p/ρhec He (d) Pressure. Fgure Profles a f for he advecon of a dsrbuon of helum n nrogen. Dashed: fully conservave (Y -FC); sold: quas-conservave (Y -QCP). ma p pa /pa (a) Normalzed L error n pressure.. mn(y (1), Y () ) (b) Mnmum mass fracon undershoo. Fgure. Advecon of a op-ha dsrbuon of helum n nrogen. Dashed: fully conservave (Y -FC); sold: quas-conservave (Y -QCP). conservave scheme; n fac he only error n he pressure and velocy are a he round-off level (Johnsen & Colonus 6). Ne, he pressure oscllaons and undershoos n he mass fracon are quanfed. Fgure shows he normalzed L error n pressure and he mnmum value of he mass fracon a every me sep as a funcon of me. When usng he fully conservave formulaon (Y -FC), he pressure error can be large over he course of he smulaon (up o %). In addon, he mass fracon may ake a negave value, hough he error s small (up o %); hough no shown here, he upper bound s also oversho (.e., Y > 1). Ths error n he mass fracon s mporan, because affecs he value of γ n he Euler

6 1 E. Johnsen e + e + M()/M() 1 -e -14-4e -14 E()/E() 1 - e e (a) Normalzed error n oal mass (b) Normalzed error n oal energy. MHe()/MHe() (c) Normalzed error n mass of helum. KE()/KE() (d) Normalzed error n knec energy. Fgure 3. Advecon of a dsrbuon of helum n nrogen. Dashed: fully conservave (Y -FC); sold: quas-conservave (Y -QCP). equaons. I should be noed ha negave mass fracons are even observed n sngleflud problems (Abgrall & Karn 1), n whch case hey do no affec he flow. On he oher hand, he quas-conservave scheme (Y -QCP) does no ehb such problems, as he error s observed a he round-off level. Ne, he conservaon errors are quanfed. Fgure 3 plos he normalzed error n oal mass, oal energy, mass of helum, and oal knec energy as a funcon of me. As epeced from he conservave form of he Euler equaons, he oal mass and energy are conserved o he round-off level wh boh schemes. However, when usng he quas-conservave scheme (Y -QCP), he mass of helum ncreases over he course of he smulaon, due o numercal dsspaon; hough no shown here, he mass of nrogen follows a smlar behavor, bu decreases by mass conservaon. The knec energy ehbs small oscllaons n he fully conservave scheme (Y -FC) due o he flucuang velocy feld nduced by he pressure oscllaons. In order o assess he effec of conservaon errors and pressure oscllaons, her dependence on he grd sze s consdered. Fgure 4 shows he normalzed L error n pressure and he mnmum mass fracon for he fully conservave scheme (Y -FC), and he error n he mass of helum for he quas-conservave scheme (Y -QCP) as a funcon of me for N = 5, 1, and 4 When usng he fully conservave scheme (Y -FC), he amplude of he pressure oscllaons decreases slghly wh addonal grd pons; s no clear wheher he oscllaons vansh or wheher hey converge o some value. When consderng he quas-conservave scheme (Y -QCP), he error n he oal mass clearly decreases wh addonal grd pons.

7 Oscllaons and conservaon errors n nerface-capurng schemes 11 ma p pa /pa (a) Normalzed L error n pressure (fully conservave scheme: Y -FC). mn(y (1), Y () ) (b) Mnmum mass fracon undershoo (fully conservave scheme: Y -FC). MHe()/MHe() (c) Error n he mass of helum (quasconservave scheme: Y -QCP). Fgure 4. Advecon of a dsrbuon of helum n nrogen. Dashed: N = 5; sold: N = 1; doed: N = ; dashed-doed: N = 4 The pressure oscllaons and conservaon errors are epeced o be affeced by he amoun of dsspaon of he scheme. Ths propery s esed by consderng dfferen orders of accuracy of he WENO scheme. Fgure 5 shows he normalzed L error n pressure and he mnmum mass fracon for he fully conservave scheme (Y -FC), and he oal mass of helum for he quas-conservave scheme (Y -QCP) as a funcon of me for dfferen orders of accuracy of WENO. When usng he fully conservave scheme, he pressure oscllaons and undershoos n he mass fracon have smlar amplude bu a hgher frequency as he order of accuracy s ncreased. In he very dsspave frs-order scheme, so much dsspaon has been nroduced ha he mass fracon no longer acheves a value of zero afer a gven me. When usng he quas-conservave scheme (QCP), he error n he oal mass of helum decreases as he order of accuracy s ncreased. Ths behavor s epeced because a hgher-order accurae reconsrucon mples less dsspaon. Ne, he dfference beween mass fracon-based (Y -QCP) and γ-based (γ-qcp) algorhms s consdered. Fgure 6 shows he γ profle a he end of he run and he error n mass of helum as a funcon of me. The γ profles are vrually dencal, hus mplyng lle dfference n he calculaons of he Euler equaons. However, he γ-based mehod shows slghly less dsspaon n ha a smaller mass of helum s los. Fnally, he effec of he defnon of γ s consdered. Fgure 7 shows he normalzed L error n pressure, he mnmum value of he mass fracon and he error n mass of helum as a funcon of me for he dfferen defnons of γ based on Eqs. (.5) and (.7). Pressure oscllaons are generaed for he fully conservave schemes and when he

8 1 E. Johnsen 3 3 ma p pa /pa (a) Normalzed L error n pressure (fully conservave scheme: Y -FC). MHe()/MHe() 1 4 mn(y (1), Y () ) (c) Error n he mass of helum (quasconservave scheme: Y -QCP) (b) Mnmum mass fracon undershoo (fully conservave scheme: Y -FC). Fgure 5. Advecon of a dsrbuon of helum n nrogen. Dashed: WENO1; sold: WENO3; doed: WENO5; dashed-doed: WENO7. γ (a) Specfc hea of helum a f. MHe()/MHe() (b) Mass helum as a funcon of me. Fgure 6. Advecon of a dsrbuon of helum n nrogen wh quas-conservave schemes. Dashed: γ; sold: mass fracon. physcal mure defnon s used (even n quas-conservave form), whle he quasconservave scheme wh γ defned n Eq. (.7) s he only scheme ha does no generae pressure oscllaons. Surprsngly, he fully conservave scheme wh he physcal mure defnon (Y -FC-M) shows smaller oscllaons han he quas-conservave scheme wh he physcal mure defnon (Y -QCP-M). For he mass fracon, he fully conservave scheme (Y -FC) leads o he larges errors, bu hs error decreases when he physcal mure defnon s used. The quas-conservave schemes hardly show any error on hs scale. As epeced, he fully conservave schemes lead o no errors n he mass of

9 Oscllaons and conservaon errors n nerface-capurng schemes (a) Normalzed L error n pressure. ma p pa /pa, Y () ) ma(y (1) (b) Mnmum mass fracon undershoo. MHe()/MHe() (c) Error n he mass of helum. 4. Fgure 7. Advecon of a dsrbuon of helum n nrogen. Dashed: fully conservave (Y -FC); sold: fully conservave wh mure (Y -FC-M); doed: quas-conservave (Y -QCP); dashed doed: quas-conservave wh mure (Y -QCP-M). helum, whle he quas-conservave schemes show a dscrepancy; he scheme based on he physcal defnon of γ shows slghly less dsspaon. 3.. Shock-nerface neracon The mos srngen 1-D es case s he neracon beween a srong shock and an nerface, as consdered by Lu e al. (3). A Mach 9 shock propagang n helum hs a nrogen nerface. The doman has 1 pons, wh grd spacng = 1; non-reflecng boundary condons are used. The varables are non-dmensonalzed by he densy and sound speed of helum, and he doman lengh, L. A consan CFL of / = 5 s used and he fnal me s f c He /L =. The nal condons are as follows: ρ ρ He = u c He = p ρ He c = He Y He = { (γhe+1)m s (γ He 1)Ms p a/rt a +, 1 < 8, p a/r HeT a, 8 1, { ( ) ( 5) + M s 1 γ He+1 M s, 1 < 8, 5, 8 1, p a ρ Hec He p a ρ Hec He [ ] 1 + γhe γ He+1 (M s 1), 1 < 8,, 8 1, { 1, 1 <,, 1, (3.)

10 14 E. Johnsen ρ/ρhe (a) Densy. 8 u/che (b) Velocy. 15 γ (c) Rao of specfc heas. p/ρhec He (d) Pressure. Fgure 8. Profles a f for he shock-nerface problem. Dashed: fully conservave scheme (Y -FC); sold: quas-conservave scheme (Y -QCP). MHe()/MHe() (a) Normalzed mass of helum. mn(y (1), Y () ) (b) Mnmum mass fracon undershoo. Fgure 9. Shock-nerface neracon. Dashed: fully conservave scheme (Y -FC); sold: quas-conservave scheme (Y -QCP). where Y N = 1 Y He, R = R u /M and M s = 9. Clearly, he problem could have been suded as a Remann problem sarng a he me when he shock and nerface concde, bu he presen problem s more relevan o praccal applcaons. Fgure 8 shows he densy, velocy, γ and pressure profles for he fully conservave (Y -FC) and quas-conservave (Y -QCP) schemes usng he mass fracon formulaon. The soluon corresponds ha of a Remann problem wh a lef-movng shock, and a rgh-movng maeral nerface and shock. Oscllaons are observed n he velocy and n he pressure wh he fully conservave scheme. Because such large pressures are acheved, even small oscllaons are large compared o oher flow feaures. The locaon

11 Oscllaons and conservaon errors n nerface-capurng schemes 15 of he maeral nerface s slghly dfferen; however he locaons converge as he grd s refned. I should be noed ha pos-shock oscllaons (Robers 199) are observed downsream of he lef-movng shock, because of s slow speed relave o he grd. Fgure 9 shows he error n he mass of helum and he mnmum value of he mass fracon as a funcon of me. As epeced, he quas-conservave scheme ehbs an error n he mass of he lef gas, due o he non-conservave form of he ranspor equaon. The abrup change n he slope occurs jus as he shock hs he nerface, hus hghlghng he fac ha hs conservaon error may depend on he velocy of he nerface. The fully conservave scheme shows a large undershoo n he mass fracon (over 5%). 4. Conclusons and fuure work In he presen paper, he errors generaed by dfferen formulaons of he governng equaons for nerface-capurng mehods have been characerzed. The problem of he advecon of a maeral was nvesgaed n deal, n order o undersand he dependence of he errors on varous properes; he neracon beween a shock and an nerface llusraed he effecs of such errors n flows of praccal neres. On one hand, spurous pressure oscllaons, whch hen affec he momenum and densy, and negave values of he mass fracon are observed when usng a fully conservave scheme. The amplude of he oscllaons does no seem o decrease rapdly wh grd refnemen and he wavelengh decreases wh he order of accuracy of he scheme. On he oher hand, quas-conservave schemes do no conserve he mass of each speces; hough hs error ncreases wh me, decreases as he grd s refned. The long-erm neres for he curren work les n mul-maeral mng n Raylegh- Taylor and Rchmyer-Meshkov nsably. Therefore, s absoluely necessary o have a basc undersandng of he errors generaed by he reamen of nerfaces n compressble flows n order o correc such effecs when more comple flows are consdered. In order o run reasonable mul-dmensonal problems, hese schemes mus be mplemened n fne dfference form, because of he cos of appromang he negral of he flu n he ransverse drecon. However, addonal modfcaons are requred for he quas-conservave scheme n order o preven he generaon of spurous oscllaons a nerfaces. Acknowledgemens The auhor s graeful for neresng dscussons wh Dr. Sosh Kawa and for he commens by Dr. Mehd Raess on he manuscrp. REFERENCES Abgrall, R How o preven pressure oscllaons n mulcomponen flow calculaons: a quas conservave approach. J. Comp. Phys. 15 (1), Abgrall, R., & Karn, S. 1 Compuaons of compressble mulfluds. J. Comp. Phys. 169 (), Fedkw, R. P., Aslam, T., Merrman, B., & Osher, S A non-oscllaory Euleran approach o nerfaces n mulmaeral flows (he Ghos Flud Mehod). J. Comp. Phys. 15 (), The mass of helum enerng he doman from he lef s subraced.

12 16 E. Johnsen Goleb, S., & Shu, C. W Toal varaon dmnshng Runge-Kua schemes. Mah. Comp. 67, Johnsen, E. & Colonus, T. 6 Implemenaon of WENO schemes n compressble mulcomponen flows. J. Comp. Phys. 19 (), Karn, S Mulcomponen flow calculaons by a conssen prmve algorhm. J. Comp. Phys. 11 (1), Larrouurou, B How o preserve he mass fracons posvy when compung compressble mul-componen flows. J. Comp. Phys. 95 (1), Lu, X. D., Osher, S., & Chan, T Weghed essenally non-oscllaory schemes. J. Comp. Phys. 115 (1), 1. Lu, T. G., Khoo, B. C., & Yeo, K. S. 3 Ghos flud mehod for srong shock mpacng on maeral nerface. J. Comp. Phys. 19 (), Mulder, W., Osher, S., & Sehan, J. A. 199 Compung nerface moon n compressble gas dynamcs. J. Comp. Phys. 1 (), 9 8. Robers, T. W. 199 The behavor of flu dfference splng schemes near slowly movng shock waves. J. Compu. Phys. 9 (1), Saurel, R A smple mehod for compressble mulflud flows. SIAM J. Compu. Phys. 1 (), Shyue, K. M An effcen shock-capurng algorhm for compressble mulcomponen problems. J. Comp. Phys. 14 (1), 8 4. Toro, E., Spruce, M., & Speares, W Resoraon of he conac surface n he HLL-Remann solver. Shock Waves 4 (1), 5 34.