A STUDY ON NUMERICAL INSTABILITY OF INVISCID TWO-FLUID MODEL NEAR ILL-POSEDNESS CONDITION. Jun Liao Renwei Mei James F. Klausner

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1 rocds of T5 5 ASME Smmr a Trasfr Cofrc Jy 7-, 5, Sa Fracsco, Cafora, USA T5-765 A STUDY ON NUMERCAL NSTABLTY OF NVSCD TWO-FLUD MODEL NEAR LL-OSEDNESS CONDTON J Lao R M Jams F. Kasr Darm of Mchaca ad Arosac Er, Uvrsy of Forda, asv, FL 36, USA ABSTRACT Th o-fd mod s dy sd sdy as-qd fo sd s bcas ca qaavy rdc h fo fd a o comaoa. ovr, h o-fd mod bcoms -osd h h s vocy cds a crca va, ad comaos ca b q sab bfor fo rachs h sab codo. hs sdy comaoa saby of varos covco schms for h o-fd mod s aayzd. A rssr corrco aorhm for vscd fo s carfy mmd o mmz s ffc o mrca saby. Vo Nma saby aayss for h av roh ras by s h s ordr d, d ordr d, QUCK, ad h cra dffrc schms shos ha h cra dffrc schm s mor accra ad mor sab ha h ohr schms. Th d ordr d schm s mch mor sscb o saby a o avs ha h s ordr d ad accra for shor avs. Th saby assocad h -osdss of h o-fd mod s sfcay dffr from h saby of h dscrzd o-fd mod. Ec arm s obad b h comd ad rdcd av roh ras. Th coco b h -osdss of h o-fd mod ad h mrca saby of h aorhm sd o mm h vscd o-fd mod s cdad. NTRODUCTON as-qd fo sd a s rva h had ad rasorao of fds. A rab fo mod s ssa o h rdco of h fo fd sd h. To fy sma h sysm, Navr Sos qaos hrdmsos ar rqrd. ovr, s vry sv o sma fos a o h oday s comr maos. To rdc h comaoa ad oba basc ad ssa fo rors, sch as as vom fraco, qd ad as vocy, rssr, a o-dmsoa mod s cssary. Th o-fd mod s drd o v a rasc rdco for h as-qd fo sd a. Th o-fd mod, aso o as h sarad fo mod, ss of o ss of rvao qaos for mass, momm ad ry for h as has ad h qd has. as roosd by Was [], ad frhr rfd by sh []. Ahoh has sccss sma o-has fo a, h o-fd mod sffrs a -osdss robm. Wh rav vocy b qd ad as cds a crca va, h ovr qaos do o ossss ra characrscs [3] [4] [5]. Ths -osdss codo sss ha h rss of h o-fd mod a ha codo do o rfc h ra fo sao sd h. Th o-fd mod oy vs maf rss h h rav vocy b as ad qd has s ss ha a crca va, hch dds o ravy ad qd v, amo ohr fo rors. ovr, hs crca va cocds h h saby codo of vscd Kv-mhoz saby (K) aayss [6]. Bcas h saby of K aayss rss h fo rm raso from srafd fo o s fo or aar fo [7], -osdss of o-fd mod s rrd as o rr h fo rm raso [7] [8]. Th comaoa mhods for sov h o-fd mod hav b vsad by may rsarchrs. hs sdy, s frhr assmd ha boh qd ad as hass ar comrssb bcas mos of srafd fos ar a ravy o sd comard h h sd of sod. To sov h comrssb o-fd qaos, o aroach s o smfy h ovr sysm o oy o qaos for qd has vom fraco ad qd vocy ad c h ras rms h as mass ad momm qaos [7] [9]. A mor ffcv mhod s o s a rssr corrco schm []. ssa ad Kmf [6], ad ssa ad Woodbr [] ad h rssr corrco schm for h o-fd mod ad smad srafd fo ad s fo sd a. Coyrh 5 by ASME

2 Wh o-fd mod bcoms -osd, h soo bcoms sab. A ood dscrzd mod shod b caab of car h cc of h saby o. ovr, mrca saby may o b h sam as h saby casd by h -osdss. Lyczos a. [] sd vo Nma saby aayss o sdy a comrssb o-fd mod h hr mrca schm ad fod ha mrca saby ad -osdss may o b dca. ovr, hr o-fd mod acd h ravaoa rm ad h sdy focsd o o dscrzao schm ad s hs com. Sar [3], Ohaa ad Tomyama [4] amd o aayz h mrca saby of a comrssb o-fd mod h a smfd mod qao as a arav. Thr sdy shod ha hhr ordr d schms yd a mor sab mrca soo ha h s ordr d schm. hs sdy, a rssr corrco schm s rsd ha s dsd o cras h saby of h mrca schm h h fo s ar h -osdss codo. Th vo Nma saby aayss s moyd o sdy h saby of h dscrzd o-fd mod h dffr roao schms for h covco rm. For h av amfcao facor s h s ordr d, d ordr d, QUCK, ad cra dffrc schms, h cra dffrc schm s mor accra ad mor sab. Ec arm for h roh of av amd s obad b h aayss ad h aca comao dr varos cofraos. NOMENCLATURE c av sd E commo amd facor amfcao facor ravy avmbr hydrac dh rssr N rds mbr m vocy sac coorda vom fraco a of cao from h horzoa amd facor has a λ characrsc roo dsy Sbcrs as fac of coro vom as qd rfac, rd d cr of ma coro vom s fac of coro vom NUMERCAL METOD ovr Eqaos Th bass of h o-fd mod s a s of odmsoa rvao qaos for h baac of mass, momm ad ry for ach has. Th o-dmsoa rvao qaos ar obad by ra h fo rors ovr h cross-scoa ara of h fo. as has Lqd has as vocy Lqd vocy rfac ravy as vom faco Lqd vom faco cross sco F. Schmac dco of o-has fo a horzoa. Bcas h -osdss oras from h hydrodyamc saby of h o-fd mod, oy coy ad momm qaos ar drd h vscd o-fd mod. Srfac so s aso cd sc oy acs o sma scas, h h avs drm h fo srcr fos ar say of o avh. Th as has s assmd comrssb, as h Mach mbr of h as has s say vry o for srafd fo. c, h ovr qaos ar as foos: ( ) ( ), () ( ) ( ), () ( ) ( ) s, (3) ( ) ( ) s, (4) hr ad ar h rscv m ad aa coordas, s h vom fraco, s h vocy, s h rssr, s h dsy, s h hydrac dh, s h ravaoa accrao, s h a of cao of h ; h sbscrs ad do h qd ad as, rscvy, ad h sbscr dos h rfac. Comaoa rocdr Th ovr qaos (-4) ar sovd ravy. Th basc rocdr s o sov h coy qao of qd for h qd vom fraco, ad h qd ad as has momm qaos ar sd o oba h qd ad as has vocs. To oba a ovr qao for h rssr, Eq. () ad Eq. () ar frs combd o form a oa mass rvao, ( ) ( ). (5) Sbs h qd ad as momm qaos o (5) yds a ra o rssr. SMLE y of rssr corrco schm [] s h sd. A f vom mhod s moyd o dscrz h ovr qaos. A sard rd (F. ) s adod o oba a comac sc for rssr [5]. O h sard rds, h fo rors sch as vom fraco, dsy ad rssr ar ocad a h cr of h ma coro vom, Coyrh 5 by ASME

3 ad h qd ad as vocs ar ocad a h c fac of h ma coro vom. Ma coro vom W E W E W E Vocy coro vom F.. Sard rd arram o-fd mod. Th Er bacard schm s moyd for h ras rm. Th dscrzd qd coy qao bcoms ( ) ) ( ) ( ), (6) hr h srscr rrss h vas a h as m s. Th sbscr dos h cr of h ma coro vom, h sbscrs ad do h as ad s facs of ma coro vom, rscvy. Th qd vocy a h c fac s o, ad h vom fraco a h c fac ca b vaad s varos roao schms. Amo hm, cra dffrc (CDS), s ordr d (FOU), d ordr d (SOU) ad QUCK schms ar commoy sd. Eq. () for h as has s smary dscrzd. Th qd momm qao s rad o h vocy coro vom. Us smar oaos, o obas ( ) ( ) ) ( ) ( ) ( ) ( ) (7) ( ) ( ) (( ) ( ) ) ( ) s. s mora o o ha h roao schms sd Eq. (7) ms b acy h sam as hos Eq. (6) ordr o rdc h dssao ad dsrso rrors. Th as has momm qao s smary rad, ( ) ( ) ) ( ) ( ) ( ) ( ) (8) ( ) ( ) (( ) ( ) ) ( ) s. For h rssr corrco schm, Eq. (5) s rad across h ma coro vom. Th dscrzd qao s ( ) ( ) ( ) ( ). (9) Bcas Eq. (5) s obad by comb Eq. () ad Eq. (), h dscrzao of Eq. (9) shod b acy h sam as ha of Eqs. (, 6). Th fa rssr qao s obad by sbs o momm qaos, Eqs. (7, 8) o Eq. (9). Characrscs ad -osdss Eqs. (-4) form a sysm of s ordr DEs ad characrsc roos, λ, of h sysm ca b fod. f λ s ar ra, h sysm s hyrboc. Com roos my a c sysm hch cass h o-fd mod sysm o bcom osd bcas oy a codos ca b scfd h mora drco. Ay fsma dsrbac cas h avs o ro oay ho bod. Th characrsc roos of Eqs. (-4) ar ± s ( ), () λ Wh, Eq. () ca hav ra roos oy f λ. Ohrs, h o-fd mod s -osd (daso, 974). f, h ra roos (or -osdss) rqrm vs ( ) < s. () Eq. () vs h crca va for h o has s vocy byod hch h sysm bcoms -osd. Th o-fd mod -osdss codo s acy h sam as from h K aayss o h o-fd mod [7]. Vo Nma Aayss for Varos Dscrzao Schms Vo Nma saby aayss s commoy sd for aayz h saby of a f dffrc schm F.3. rd d sard rd for vo Nma saby aayss. hs drvao, h s ordr d (FOU) schm s sd as a am, ad boh qd ad as vocs ar assmd osv. Dscrzao of Eq. (6) s FOU ads o ( ) ( ) ( ) ). () S h varabs o bas va ad dsrbacs, h arzd qao for h dsrbac s ( ) ( ) ( ( ) ( ) ) ( ) ( ) ),(3) hr ^ dos dsrbac vas. Erss dsrbacs as, (4) ( ) E ( ) E ( ) E, (5), (6) v v hr E s a commo amd facor, s h avmbr, ad rrss maary. Eq. (3) s smfd o ( ) ( ), (7) hr s h amfcao facor: E E, (8) 3 Coyrh 5 by ASME

4 4 Coyrh 5 by ASME ad s h has a:, (9) dfd ovr [, π] hch rrss a h rsovab av comos h comaoa doma for h v rd. Shor avs corrsod o h ro ar π. Th av roh qao for h as has mass rvao qao s smary obad: z. () For h qd momm qao, Eq. () s dscrzd h h FOU schm o, s () hch s sbsqy arzd ad smfd h h ad of h qd mass rvao qao,. () For h as has, h vocy dsrbac s ovrd by, (3) Th rssr rm ca b cacd by comb Eqs. (-3),. z z (4) Eqs (7,, 4) ca b r h form of a amfcao mar. No-rva soos for ( ) T,, s oy h h drma of h mar s zro. c, c b a, (5) hr a, (6a) b, (6b) 4s c, (6c) ad s h Cora mbr, ad. (7) Th vas of ar v Tab. From Eq. (5), h amfcao facor ca b asy fod, ac b b a 4 ± (8) Saby rqrs for a. Tab.. for dffr dscrzao schms. Schm s ordr d Cra dffrc d ordr d 4 3 QUCK vscd Kv-mhoz (K) Aayss K aayss rovds a saby codo for h o-fd mod as as sf formao o h roh ra of dsrbac h o-fd mod. Eqs. (-4) ar arzd ad sbsd for h rrbd qd vom fraco, qd ad as has vocs, ad rssr h form of [(-)] hch s h amd, s h aar frqcy, ad s h avmbr. Th foo sysm s obad for h dsrbac amd:. (9) Th dsrso rao s obad as: c ± s, (3) hr c s h av sd. Th av maary ar of drms h roh ra of dsrbac. Eq. (3) s dca o Eq. (), oy h λ b racd by c. Das of K aayss ca b fod [7]. a ad Bodary Codos for Nmrca Soo vo Nma saby aayss, a rodc bodary codo s mcy assmd. comaos, sch rodc bodary codos ar aso moyd ordr o rovd a vad comarso. Th vo Nma saby aayss s for h roh of a fsma rrbao. comaos, a sma a rrbao ms b rory rodcd ho ra addoa, hhr harmoc os. hs sdy, h soo of h vscd Kv-mhoz aayss s sd as h a codo. Ths, f ad ar scfd a, corrsod

5 vas for, (9-3)., ad ms b s h Eqs..5 RESULTS AND DSCUSSONS Comaoa Saby Assssm basd o vo Nma Saby Aayss Comarso of saby of h s ordr d, d ordr d, cra dffrc, ad QUCK schms ar codcd frs for fo codos bfor ar, ad afr h saby. s o ha for ordary covco-dffso qaos h s ordr d schm s ss accra h hh mrca dffso, ad hh ordr schms, sch as SOU, CDS, ad QUCK, hav or mrca dffso. hs sdy, for srao ross, ar ad ar ar drd ad h damr s a o b.78m. Th comaoa doma s m o, h rd mbr s N. Th c a s. Th rrbd vas ar:.5, m/s, 7 m/s ad of qd s.. Saby codo basd o Eq. () or (3) for h abov aramrs s U < 6.768m / s. Ths, h o-fd mod for hs codo s -osd. srvs as a da s cas o assss h rformacs of varos schms sc h sysm s q o b -osd. Thr ar o vas of v by Eq. (8) ad h arr o drms h saby; so ha oy h arr roh ra s sd hr. F. 4 comars h amfcao facors of for dscrzao schms. Th sod s h amfcao facor by K aayss (). Th dod s for h CDS schm. s shy or ha o b q o o h a sh dffso rror a hh avmbr ra. Ths ms h CDS s a da schm o com h o-fd mod. Th dashd s for h FOU schm hch osssss cssv mrca dffso a hh. Frhrmor, > a o. Ths h comao s FOU s sab dr hs fo codo. Th dashd ad dod s for h SOU schm. Ahoh SOU s rardd as a br schm ha FOU h ss mrca dffso, s rformac h o-fd mod s vry oor. For ar, h mrca dffso of SOU s mch arr ha ha of FOU. For sma, h amfcao facor of SOU s mch arr ha ha of FOU. Dashd dob dod s h amfcao facor of h QUCK schm. s mrca dffso a hh s or ha ha of FOU ad SOU, b s s mch arr ha ha of CDS. A sma, s shy arr ha dca ha QUCK s sab as. Th raso ha h amfcao facor of CDS s o h aayca amfcao facor s robaby d o a ac of d ordr dffso rror ad o dsrso rror. Comard h SOU, mrca dffso of FOU schm s mch hhr ad dsrso s shy or. Ovra rformac of FOU s br ha ha of SOU hch sss ha h dsrso rror h o-fd mod s mch mor mora ha s h sm covco-dffso qao. Th roao of QUCK s ssay ar roao h h d corrco. Thrfor s mrca dffso ad saby s ors ha ha of CDS, b br ha ha of FOU ad SOU. N, h ffc of h s vocy U o h mrca saby s dscssd. F. 5 shos h amfcao facors of h CDS schm for a ra of vas of U FOU SOU CDS QUCK Ra roh ra F. 4 Comarsos of amfcao facors of varos mrca schms. N, a. 5, m 7 m / s ad.. Wh U s smar ha ha v by h K saby, h amfcao facors of a h harmocs h comaoa doma ar ss ha o. ovr, f U s hhr ha h K saby crra, h o-fd mod s aaycay -osd, ad for sma cds o, as sho by h crv for U 6.m / s F. 5. From mrca rss, a ra saby codo of CDS s fod o b ar U 6.773m / s for h codo sd F. 5, hch s q o K saby codo of m/s. As U frhr crass, crass oo. Th ra of sab harmoc avmbr bcoms d. Th amfcao facor of CDS schm machs ha of K oy a vry o avmbr. h hh ra, mrca dam cass o b mch or ha o U , aayca 6.5, aayca 7, aayca F.5. Amfcao facors of CDS for dffr vas of U. N, a. 5, m ad.. F. 6 shos h amfcao facors of h FOU schm a dffr vas of U. U h CDS schm, hr s o sfca cha of h U vars. Nmrca rss dca ha h ra saby for h codo sho F. 6 s U 4.77m hch s mch or ha h aayca va of / m / s. Th bhavor of SOU ad QUCK 5 Coyrh 5 by ASME

6 schm s smar o h FOU. Th saby codo for SOU s U 3.73m / s ad for QUCK s U 6.3m / s U , aayca 8,aayca.6.4. Lqd vocy F. 6 roh ra of FOU schm for dffr vas of N, a. 5, m ad.. U. F. 7 shos h ffc of h qd vocy o h amfcao facors of CDS h U 6m / s ad.. For.m / s ad.m dcrass mooocay h h has a. Dam aars a hh. Wh crass, a hh ra rss sfcay, av a hh dam sadd a h rmda ra. O h ohr had, f U s a, s mch arr ha o h s sma so ha s hard o boh ad h modra ra, hch s ssa o h comaoa saby. F. 8 shos h ffc of o for h FOU schm h U 6m.. Th bhavor of FOU s mch dffr h ha of CDS. Wh s sma, mos harmocs ar sab. For a arr, cssv mrca dffso mas h comaos sab Lqd vocy F. 7 Amfcao facors of CDS schm a dffr vas of. N, U 6m a. 5, ad () F. 8. Amfcao facors of FOU schm for dffr vas of. N, U 6m a. 5, ad / F. 9 Amfcao facors of CDS schm for dffr va of. N, m U 6m ad a / F.. Amfcao facors of FOU schm for dffr vas of. N, m U 6m ad a.5. Fs. 9- sho h ffc of o for h CDS ad FOU schms. Boh sho cras mrca dffso h cras rs a dcras. Ths ca b ( / ) ( / ) 6 Coyrh 5 by ASME

7 ad by am Eq. (6c), hr h as rm s rad o h ravaoa ffc. s o ha ravy sabzs h fo. Ths cras comaoay hs h saby. Schm Cosscy Ts hs schm scy s, roh ras of harmocs a dffr rd dss ar comard. Th comaoa doma s m o. A, a fsma ssoda dsrbac h π s rodcd. a codos of vom fraco, qd ad as vocs ad rssr ar comab h h rss of K aayss, Eq. (9). F. shos h comarso of av roh a dffr rds. Th c fac roao schm s CDS, m 7.5m,., ad a. 5. Th rd mbrs ar N, N, N4. Bcas, ad ar a hs comarso, s a a. Ths srs ha os o zro as aroachs zro. A aayca soo for av roh by K aayss s od F. for comarso h h mrca rss. Wh N cras from o 4, h rror b h ac ad mrca soos dcrass as rqrd by scy. Ahoh h rror of soo a rds N s shy arr ha ha a N ad N 4, h rror of soo a N s q o ha a N 4. Ths sss ha N s ar oh for π ; hc N s sd ss ohrs mod. Dsrbac (m/s) 6.E-6 4.E-6.E-6.E E-6-4.E-6-6.E-6 K N N4 N (m).5s F.. Comarso of û roh for dffr N by CDS schm. m 7.5m., ad a. 5. Comaoa Assssm basd o h roh of Dsrbac To vada h rssr corrco schm, comarsos b h comd av roh h h aayca roh from h vo Nma saby aayss ar rsd. Frs dr π, N, m 5m a.5,. 5, ad h comaoa m s 4s. Basd o K aayss, h dsrbac dos o ro. Th comaoa schm sd s CDS. F. shos ha a 4s, h amd of h comd av s shy or ha ha of h aayca soo. Th hass of h aayca ad mrca soos ar amos dca. Ths dmosras c rformac of CDS for h o-fd mod. F. 3 shos h masrd dcay of h amd of h qd vocy dsrbac. Th amfcao facor of CDS h π s s h vo Nma saby aayss. Sc as 6 ss o rach 4s, h rao of h amd a 4s o ha s ( ) Th aca ra s CDS s.9687, h a rror of.6%. Carf amao of F. 3 rvas sma amd rs h av amd. Th raso s ha h a codo s h aayca soo of K aayss, hch s shy dffr from h soo by h CDS dsrso qao. Ths msmach of h a codos ads o h rao of a a av ad vry o mrca dffso of CDS srs ha hs a av ss for a o m. Fr 4 shos av roh for a -osd codo, h m 7.5m a. 5,.. Th rav vocy crro s arr ha 6.768m/s so ha ay rrbao ro h m. Th a dsrbac s rodcd h π. F. 4, h comaoa rs s obad afr 399 m ss. Th ora o av h π s ovrhmd by a mch sror shor av. F. 5, h roh hsory of h amd s rsd. Th a roh sa, from o 4s, corrsods o h roh of h a o av h π. Ths s frhr cofrmd by comar h h aayca for π. Th rdcd oa roh s.84 for h amd rao from o 4s, h h comd amd rao s.89. Afr h a roh sa a shor av h hhr as ovr ad bcoms doma h comd soo. Ths s h sa of fas roh F.5. Ahoh h amd of h shor av s o sma, h avmbr of h shor av machs rdcd vas basd o vo Nma aayss. For U 6.5m / s h rs comao, h av h h hhs occrs a ma f h m doma s occd by hs av, h oa mbr of avs s N ma /(π ) 9, hch s acy h mbr of avs F. 4. Dsrbac (m/s) 4.E-6 3.E-6.E-6.E-6.E E-6 -.E-6-3.E-6-4.E-6 Nmrca rs Aayca rs (m) 4s a av F.. roaao of û. CDS schm, N, m U 4m. 5, a Coyrh 5 by ASME

8 Amd (m/s) 3.64E-6 3.6E-6 3.6E E E E-6 3.5E-6 3.5E (s) F. 3. Amd of qd vocy dsrbac. CDS schm, N, m U 4m. 5, a E-4.5E-4 5.s.E-4.5E-4 h vo Nma saby aayss. Th roh ra of FOU s sho F. 6. Th harmoc h ma. s ma s acad ha hs harmoc ro from h rod-off rror ad vay doma h comao. h comao, a sma amd ssoda av h π s rodcd a. F. 7 shos h qd vocy varao afr 8 m ss. Obvosy, a shor av has ovrhmd h a o av. Bcas h shor avs ora from mach v rror hch has a broad scra dsrbo, h amd ad frqcy of h avs ar o form. ovr, h doma av como F. 8 s 9 by co mbr of avs h m comaoa doma. Sc ma for h crr codo, ms ma 9. Ths ars vry h h comaoa obsrvao. Frhrmor, for. ma, h amd ca ro by a facor of.9 8 ss. Sc h a amd of mach v os s of O ( 6 ), s rasoab o c h amd of h doma shor av o b o h ordr of O ( 6 ) afr 8 m ss, hch s s h h rss sho F. 7. Dsrbac(m/s).E-4 4s 5.E-5.E E-5 -.E E E-4 (m) F. 4. Lqd vocy dsrbac afr 399 m ss s CDS. N, m U 6.5m Amd(m/s)., a. 5, ad 5. s..e.e-.e-.e-3.e-4.e-5.e (s) F.5. roh hsory of qd vocy dsrbac. N, a comarso b rss of h FOU schm ad rd from h vo Nma aayss s rsd. Th aramrs of comao ar: N,.5m U 6m,., ad a. 5. Th fo s sab basd o K saby aayss, b sab basd o Dsrbac (m/s) F. 6. roh facor of FOU schm. N,.5m U 6m., ad a. 5..E-5.5E-5.E-5 5.E-6.E E-6 -.E-5 -.5E-5 -.E-5 (m) F. 7. Lqd vocy dsrbac afr 8 ss h FOU schm. N,.5m U 6m., ad a Coyrh 5 by ASME

9 CONCLUDN REMAKRKS Nmrca saby for h comrssb o-fd mod ar h -osd codo s vsad for varos dscrzao schms, h h rssr corrco mhod s sd o oba h rssr. Th vo Nma saby aayss s carrd o o oba h amfcao facor of a sma dsrbac h dscrzd sysm. Th cra dffrc schm has h bs saby characrscs had h o-fd mod, food by h QUCK schm. s q rs o o ha h cssv mrca dffso h s ordr d schm sms o romo h mrca saby comarso h h cra dffrc schm. Ds s oma d ordr accracy ad oary, h d ordr d schm s mch mor sab ha h s ordr d schm for sov o-fd mod qaos. Dffr dscrzao schms for h covco rm h vary drs of h mrca dffso ad dsrso cao cas a day h saby; hy of romo saby h o-fd mod. Th aaycay rdcd av amd roh s aso comard h ha obad from carfy mmd comaos s varos dscrzao schms for h covco rm. Ec arm b h mrca rss ad h rdcd rss s obad for h roh of h av amd ad h doma avmbr h h comao bcoms sab. Th rao b h comaoa saby ad h -osdss s dscssd. h rsc of h smaamd o-av dsrbac, hos amd s mch arr ha h mach rod-off rror, h roh of h dsrbac acy machs h rdco of h vo Nma saby aayss h h comaoa saby codo s voad. h mam, a shorr av mrs from h mach rod-off rror, ad vay domas h r dsrbac, hch cass h comao o bo. Ths comaoa saby s dy rrd as h rs of -osdss of h o-fd mod. Th rss of h rs sdy ss ha h comaoa saby s ary h rory of h dscrzd o-fd mod ad s sroy affcd by h hr -osdss of h o-fd mod dffra qaos. rodco of mrca dffso ad/or dsrso ca sfcay cha h saby of h dscrzd sysm; hovr, sch ss of yd favorab comaoa rss. For sov o-fd mods, cra dffrc s rcommdd sc s mch mor accra ad ddab ha ohr schms vsad. Wh rsc h shar srss rms o-fd mod, h -osdss codo s o affcd, b h fo saby codo s chad. Major cocsos hs ar ovra s s vad [7]. ACKNOWLEDMENTS Ths or as sord by NASA Rsarch Cr dr corac NA3-75 ad NASA Kdy Sac Cr. REFERENCES. Was,.B., 969, O-Dmsoa To-has Fo, Mcra-, N Yor.. sh, M., 975, Thrmo-Fd Dyamc Thory of To has Fo, Eyros, ars. 3. daso, D., 974, Mod of To has Fo, rocds of h Ffh raoa a Trasfr Cofrc, V, Jos, A.V., ad rosr, A., 985, O h Saby of Frs-Ordr Dffra Mods for To-has Fo rdco, raoa Jora of Mhas Fo,, So, J.., ad sh, M.,, Th W-osdss of comrssb O-dmsoa To-Fd Mod, raoa Jora of a ad Mass Trasfr, 43, ssa, R.., ad Kmf, M..W., 3, Smao of S Fo orzoa ad Nary orzoa s h h To-Fd Mod, raoa Jora of Mhas Fo, 9, Bara, D., ad Ta, Y., 994, rfaca ad Srcra Saby of Sarad Fo, raoa Jora of Mhas Fo,, S., Brar, N., ad Maro, D.M., 99, Saby Aayss of Srafd Lqd-Lqd Fo, raoa Jora of Mhas Fo, 8, Cha, A. M. C., ad Barj, S., 98, Rf ad R of a o orzoa Tb ar : Srcr of a To-Fd Mod, Jora of a Trasfr, 3, aaar, S. V., 98, Nmrca a Trasfr ad Fd Fo, Mcra-, N Yor.. ssa, R.., ad Woodbr,.J., 998, Nmrca rdco of sabs ad S Formao orzoa To-has Fos, 3rd raoa Cofrc o Mhas Fo, CMF98, Lyo, Frac.. Lyczos, R.W., daso, D., Sobr, C.W., ad hs, E.D., 978, Characrscs ad Saby Aayss of Tras O-Dmsoa To-has Fo Eqaos ad Thr F Dffrc Aromaos, Ncar Scc ad Er, 66, Sar, B.., 979, Saby of To-has Fo Cacao Us To-Fd Mods, Jora of Comaoa hyscs, 33, Ohaa, T., ad Tomyama, A., 995, Acaby of h-ordr Ud Dffrc Mhods o h To-Fd Mod, Advacs Mhas Fo, Esvr Scc, Frzr, J.., ad rc, M., 996, Comaoa Mhods for Fd Dyamcs, Srr, Br. 6. rsch, C., 988, Nmrca Comao of ra ad Era Fos, vom : Fdamas of Nmrca Dscrzao, Joh Wy & Sos, N Yor. 7. Lao, J., 5, Mod To-has Trasor dr Cryoc Chdo a, h.d Dssrao, Uvrsy of Forda. 9 Coyrh 5 by ASME