Continuum Breakdown Parameter Based on Entropy Generation Rates

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1 4st Arospac cincs Mting and Ehibit 6-9 Janar 00, Rno, Nada AIAA Continm Bradown Paramtr Basd on Entrop Gnration Rats José A. Cambros nitd tats Air Forc Rsarch Laborator Wright Pattrson AFB, OH 454 Po Hng Chn Dpartmnt o Arospac Enginring, nirsit o Michigan Ann Arbor, MI 4809 ABRAC h dimnsionlss ratio o th man r path to a rrnc lngth scal tpicall antiis th tnt to which th continm assmption in lid low dnamics bras down. Howr, ambigit ists in th choic o an appropriat lngth scal and som rsarchrs ha sggstd a local paramtr, li th Kndsn nmbr, or antiing continm bradown. h athors riw a w slctd paramtrs and sggst a continm bradown paramtr basd on local ntrop prodction rats. h slctd paramtr was chosn b obsring and comparing altrnatis and abstracting common lmnts and rslts. Rslts prsntd incld laminar bondar low and on dimnsional shoc wa strctr. Althogh both cass ar wll within th continm dscription, th proid an idal stting or comparison sinc both ha smi analtical soltions. INRODCION Intrst in rarid gas dnamics contins to b stimlatd b th possibilit o high spd light at r high altitds. Rarid gas dnamics dscribs low in which th lngth o th molclar man r path is comparabl to som macroscopic rrnc dimnsion o th low ild. ndr ths conditions th gas dos not bha ntirl as a continos lid bt rathr hibits som o th proprtis o its coars molclar strctr. Rarid gas cts incld rgions o sharp gradints in locit, prssr, and/or tmpratr. In som cass, on or mor low rgims, inclding r molcl, transition, slip, and continm low, ma b important. Bcas o th dirnt phsical procsss inold in ach low rgim, dirnt mathmatical modls mst b sd, ach o which rirs altrnati nmrical tchnis or comptational simlation (.g., Dirct imlation Mont Carlo rss Comptational Flid Dnamics). h dimnsionlss ratio o th man r path to a rrnc lngth scal tpicall antiis th tnt to which th continm assmption bras down. Howr, ambigit ists in th choic o an appropriat lngth scal and som rsarchrs ha sggstd a local paramtr, li th Kndsn nmbr, or antiing continm bradown (Bird, [], []). In act, as discssd in Chaptr o Bird [], th limitations o th consration ations (Nair tos) appars to b dirctl rlatd to a mathmatical rirmnt or dtrminat st o ations. h phsical procsss acting ths rlations ar classiid as non ilibrim cts and not limitd to rarid gas dnamics. h transport trms li shar strsss and hat ls do not orm a closd st whn gradints in th macroscopic ariabls bcom so stp that thir scal lngth is o th sam ordr as th molclar man r path. h Kndsn nmbr Kn λ (.) L Arospac Enginr and AIAA nior Mmbr. Gradat tdnt, mmr 00 Rsarch Associat at WPAFB, AIAA tdnt Mmbr (chnph@mich.d). his matrial is dclard a wor o th.. Gornmnt and is not sbct to copright protction in th nitd tats.

2 antiis th dgr to which this condition is mt. h traditional rirmnt stiplats that th Nair tos ations ar alid p to a Kndsn nmbr o 0.. A mor prcis limit spciis a local Kndsn nmbr whr th charactristic lngth ma b rplacd b th local lngth scal o th macroscopic gradints. h macroscopic gradints ma b dnsit, locit, tmpratr, or prssr and ariations o ths ha appard in th litratr (s Wang & Bod [4] and Wang, n, Bod [5] or mor dtail). h accrac o th gorning Nair tos ations bgins to dgrad arond a Kndsn nmbr o 0. whil a sggstd ppr limit or th continm modl ma b tan as 0. (Bird [], pp. ). ransitional low is otn catgorizd as alling btwn a Kndsn nmbr al o 0.0 to 0., althogh ths als ar not nirsall accptd. wo othr dimnsionlss paramtrs that dpnd on local low gradints proid som masr o continm bradown. hs wr discssd b P. Canpp [] and dind b: K τ τ p, K. (.) pc Hr w riw ths w slctd paramtrs and sggst a continm bradown paramtr basd on local ntrop prodction rats. his approach was inspird b th wor prsntd in Wang and Bod [4], whr this particlar iss was raisd. First, w riw th ntrop balanc ations rom th continm standpoint, basd on th tnsion o th scond law o thrmodnamics or nstad lid low. cond, w propos to din a continm bradown paramtr basd on a dimnsionlss ormla or th ntrop ration rat. hird, w prsnt a comparison or a w slctd cass btwn bradown paramtrs. Frthr wor is ndrwa tilizing ll CFD soltions o th Nair tos ations and will b prsntd in a ollow on papr. h ocs o th prsnt papr is on dloping th appropriat ormlas or calclation and comparison. Entrop Balanc Eations h gorning ations o lid dnamics can b writtn in compact notation as Q t 0, (.) whr Q rprsnts th stat ctor and th algbraic l ctor o stat antitis. hs can rprsnt ithr th Elr or Nair tos ations. For an idal gas th ntrop ormla is radil aailabl rom thrmodnamics, sch that (Q) with rspct to th stat ariabls h spciic ntrop ormla, rom thrmodnamics, is h total spciic nrg is so that th tmpratr can b writtn as Q (ρ,ρ,ρ,ρw,ρ). (4.) s( ρ, ) s 0 c ln ln ρ V R. (5.) 0 ρ0 ( w ) ρr ρ, (6.) ρ γ 5 4 R ( Q ) ( γ ). (7.) tting th constant rrnc stat s 0 al to zro and writing in trms o th stat ariabls gis ( γ ) ( ) 5 4 ( Q ) ρs ln (8.) γ γ

3 whr th gas constant proids a connint wa to rndr th ntrop dimnsionlss. h balanc o ntrop mrgs b sbstitting () into th prssion: Q 0 t Q t 0 t Q I th ls contain onl th concti transport o mass, momntm, and nrg as rprsntd in th Elr ations, thn it is possibl to show that Q whr th ntrop ls ar F ρs and th ntrop ration rat F (9.) (0.) F 0 (.) t dmonstrats that ntrop ration is zro or adiabatic, iniscid (rrsibl) gas dnamics. I th ls contain trms that modl iscosit and hat condction, thn Q F τ i whr rprsnts th hating rat pr nit ara (hat l) and τ i th iscos strss tnsor. On sbstittion in th ntrop balanc ation, w gt two prssions that rprsnt ntrop ration: i (.) F (.a) t τ. (.b) i i W idnti th scond prssion b collcting all trms that ha a dinit sign. Eation (.a) ma b obtaind b ralizing th scond law o thrmodnamics or an opn, nstad sstm. On can dri (.b) dirctl rom (.a) b sing all th constitti rlations and consration ations. h procdr is tdios and not straight orward. h procdr otlind abo is atpical bt ors a mthodological adantag (Cambros, [6]). W propos a continm bradown paramtr basd on ntrop ration rat dind b th dimnsionlss paramtr: K (4.) ρr R Comparing this prssion with Eation (), w s that th ntrop ration rat in dimnsionlss orm inclds portions o both K τ and K and thror rprsnts th cts o both locit and tmpratr gradints. In particlar, it is hopd that this prssion ma sr at last as a thortical bnchmar or antiing th cts o that hrald th onst o continm bradown. W considr in th sctions that ollow two ampls to dmonstrat th concpt or problms that ha a thortical or narl analtic soltion. Rgions o intrst in th ral simlation o lid low incld bondar lars and shoc was whr th transport proprtis ar most signiicant. Both rgions contain ntrop ration d to iscos dissipation and hating. hror, on wold pct ths rgions to sr as idal candidats or comparing bradown paramtrs. h accptd critrion is to dclar th onst o continm bradown whn th al o th rlant paramtr als 0.. With both th bondar lar and shoc proil, w will plor and compar th arios paramtrs ping this al in mind.

4 EXAMPLE : INCOMPREIBLE BONDARY LAYER Nglcting boanc, th stad D incomprssibl low (bondar lar) ations ar: Mass: 0 (5.) Momntm: ν (6.) with th bondar conditions (,0) (,0) 0 and (, ) (). Conditions in th r stram and otsid th bondar lar ar dnotd b th smbol. Blasis Eation L. Prandtl concid th concpt o a bondar lar and it was his irst stdnt, H. Blasis who introdcd similarit analsis and assmd small displacmnt thicnss so that const. and d/d 0 to obtain a soltion. I w s th dimnsionlss similarit ariabl and stram nction: ψ ν ψ ν ( ) (7.) ν his mas th locitis: ψ ψ ν '( ) ( ' h momntm ation rdcs to th Blasis ation or a lat plat: h bondar conditions ar: (,0) (,0) 0 ( 0 ) ( 0) 0 ) (8.) ''' '' 0. (9.) and (,) ( ) ( ) (0.). (.) h accptd al or th 0 ond b nmrical soltion o (9) and itration is At , so that a complt soltion proids a masr o th bondar lar thicnss, th displacmnt thicnss, th riction coicint, wall shar strss and riction drag. Flat Plat Hat ransr or Constant Wall mpratr o accont or th cts o tmpratr gradints, it is ncssar to incld th nrg ation, n thogh this is dcopld rom th Blasis momntm ation. Considr th cas whr hat transr occrs at constant wall Θ. Withot tmpratr w and din th dimnsionlss tmpratr dirnc as ( ) ( )( ) nglcting dissipation (contrar to what is assmd in ttboos), th nrg ation rdcs to Θ' ' Pr.. Θ ' Pr.Ec.( ) 0, (.) with bondar conditions Θ(0) and Θ( ) 0. Pr is th Prandtl nmbr and Ec is th Ecrt nmbr dind as: Ec /c p, whr als th tmpratr dirnc btwn th id wall tmpratr and th r stram. w 4

5 Flat Plat Laminar Bondar Lar Entrop Prodction h dtails prsntd abo ar dplicatd rom what on can ind in th litratr and ttboos on lid mchanics in particlar (.g., Whit [7], pp. pag 86 88). W rpat thm hr or conninc in driing th ollowing. h ormla or ntrop ration rat consistnt with th bondar lar ations can b writtn as Φ. (.) Whr Φ rprsnts th iscos dissipation nction. h complt ormla or th iscos dissipation nction in Cartsian coordinats, assming a Nwtonian lid, can b rdcd to D incomprssibl low as Φ i i µ τ. (4.) Forir s law or th hat condction (D) gis. (5.) o that th complt D incomprssibl ntrop ration rat is rprsntd b µ. (6.) tilizing th standard similarit transormation, w now ill in th driatis sing th similarit nctions and paramtrs: ( ), ( ), ( ) ν ν, ( ) ( ) w Θ, ( ) ( ) w Θ, ν. From th dinition o, w can sbstitt and simpli to obtain th ollowing, whr an ordr o magnitd inspction o trms gis th ntrop ration rat: ( ) ( )( ) c w p Pr Θ ρ ρ µ (7.) whr Pr µc p /κ (Prandtl nmbr). ral possibilitis ist or maing th prssion dimnsionlss. inc ntrop is most closl associatd with thrmodnamic cts, w can choos to diid b th thrmal nrg clstr ρc p / to obtain ( ) ( ) * Pr Θ Θ Θ w p p c c ρ. (8.) 5

6 Howr, it is tpicall mor connint to prss dimnsionlss antitis in trms o mor amiliar and asil spciid paramtrs li Mach and Rnolds nmbrs. sing th idal gas rlations gin b w rwrit th Kndsn li paramtrs as ollows. γr c p, γ M γ R, ρ R, (9.) µ M Entrop: K γ γ ( ) Θ ( Θ ) γ γ ( ) Θ γ M Pr (0.) Kndsn: λ µ Kn ρ π R γπ M R (.) K τ µ µ ρ γm R τ p ρr ρr µ R Viscos trss: ( ) ( ) ( ) (.) Hating: K γm R pc Pr γ ( ) R ( ) Θ (.) Rslts or Flat Plat Bondar Lar Flow It is a simpl mattr ths das with sophisticatd sotwar aailabl to obtain a nmrical soltion or ordinar dirntial ations. oltions or th Blasis momntm ation and th nrg ation ths proid th nmrical als or alating and comparing th arios paramtrs. Bcas o th wa th Kndsn li paramtrs ar scald, th ratio o (/L) appars in th dnominator. hror, nar th bondar lar dg, whr CFD tpicall rirs grid clstring to sol th bondar lar ations, gradints ar mch mor sr. In th calclations prsntd, w sd L.0 to ocs on th dirncs btwn th paramtrs at lngth scals comparabl to th macroscopic scal. A rasonabl application wold b to s ths paramtrs to dtct how clos to th bondar lar lading dg a continm mthod wold b alid. Howr, it is wll nown that or laminar bondar lar lows, grid conrc otn sols th problm long bor th continm assmption coms into stion. h Blasis and nrg bondar lar ations wr sold with th Mathmatica smbolic calclation sotwar. Figr shows th calclatd soltion or th locit and tmpratr proils as wll as thir gradints. It is idnt that indd th locit gradint is larg whil th tmpratr gradint is not as grat. Fr stram conditions wr st at Mach 0. and standard sa ll or prssr, tmpratr, and dnsit. Figr shows that th main ntrop rating mchanism is d to hat transr in th normal dirction. h mint contribtion d to iscos strsss alidats th small Ecrt nmbr approimation tpicall assmd. h Kndsn paramtrs ar shown in th scond hal o Figr and obiosl indicat that this problm is wll within th continm domain (th man r path ndr ths conditions is o th ordr o 0 8 ). Entrop ration (Figr ) is two ordrs o magnitd largr than n a local Kndsn nmbr with lngth scal basd on th locit gradint: K λ in Figr (right hand sid). his paramtr was introdcd to accommodat th prsnc o larg locit gradints. inc ntrop basd paramtr scals to / as compard to or th othr paramtrs, it is pctd that it wold hrald th onst o continm bradown mch soonr i lading dg conditions wr approachd. 6

7 Vlocit Vlocit mpratr mpratr FIGRE. Vlocit proil and tmpratr proils ratd with nmrical ODE soltion o Blasis ation or incomprssibl bondar lar low. Hating, dirction K Hating, dirction har trss K K τ Kn FIGRE. Hating and shar strss proils (lt) and Kndsn paramtr proils rslting rom a nmrical ODE soltion o Blasis ation or incomprssibl bondar lar low. * K FIGRE. Entrop Basd Kndsn li paramtr proils rslting rom a nmrical ODE soltion o Blasis ation or incomprssibl bondar lar low. 7

8 EXAMPLE : ONE DIMENIONAL PLANE HOCK WAVE hoc was rprsnt anothr sitation whr th transport proprtis and th closr conditions bcom important. For simplicit, w considr onl asi on dimnsional low gornd b th stad Nair tos ations: Mass: ( ) 0 d d Momntm: ( ρ τ ) 0 d ρ, (4.) d p, (5.) d Enrg: ( ρ p τ ) 0. (6.) d A ni soltion ists or ths ations proidd that th scond law o thrmodnamics is satisid. his soltion rprsnts a plan shoc wa with low dirction dtrmind b comprssion. hs soltions ha long bn stdid in discssing th alidit o th continm assmption (Gilbarg and Paolcci [8]). Intrst contins to this da bcas th or an idal cas or comparing continm and molclar basd modls rlant to th closr conditions or th transport proprtis. h constitti rlations or th transport proprtis applicabl in th continm rgim ar: 4 d τ µ, d d κ. (7.) d h two prssions or th ntrop balanc ations bcom rdndant or th stad stat and ar in act intrchangabl. It is thror onl ncssar to calclat onl on or th othr: d d t ( ρs) d d c 4 d d µ (8.) d d In nstad sitations, th two prssions shold ild th sam al in principl bt ma dir whn approimat and compl nmrical calclation is inold (as in sophisticatd CFD algorithms). With th right sotwar aailabl, it is st as as to obtain a nmrical soltion to a scond or highr ordr ordinar dirntial ation. Howr, a connint stratg is to sol th ntir problm as a sstm o copld irst ordr ODEs. B sing th ations o stat or an idal gas p ρr, ρr (9.) γ th sstm o ations can b rdcd to sol or th macroscopic low ariabls. Entrop ration rats and th othr Kndsn li paramtrs ar obtaind rom th nmrical soltion o th sstm o ations, which ar mad dimnsionlss b sing a rrnc man r path as th lngth scal. On caat: It is rall now nown that th propr approach to obtain a nmrical soltion to ths ations is to intgrat rom th down stram to th p stram conditions. his is bcas th down stram conditions rprsnt a nodal point in locit tmpratr spac, whil th p stram conditions ar a saddl point. Nmrical soltions that bgin p stram will soon diat and will most lil not approach th down stram conditions. In th igrs that ollow, th calclations wr obtaind sing a dimnsionlss orm o th ations with all macroscopic antitis normalizd b thir p stram als, a Mach nmbr o, and spac normalizd b th p stram man r path. Downstram initial conditions or nmrical calclation wr obtaind with th Ranin Hgoniot rlations or D shoc was. h ations wr sold with th Mathmatica smbolic calclation sotwar. 8

9 Rslts or On Dimnsional hoc Wa trctr oltions or th D Nair tos ations proid th nmrical als or alating and comparing th arios paramtrs. In th calclations prsntd, all bradown paramtrs tilizd a local al o th man r path as th lngth scal. hs wr dind as in Eations (.a) and (.b) or th ntrop, Eation () or K τ and K, and Eation () or th local Kndsn nmbr with rrnc lngth scal basd on th mass dnsit gradint. Figr 4 shows th calclatd soltion or th macroscopic ariabl proils as wll as th hating and shar strss proils. h proils ar cntrd at sonic conditions and th scal is in man r paths. Pa hating occrs st pstram o th sonic point whil pa sharing occrs st downstram. Figr 5 shows that th prssr gradint is largr and broadr than th gradints or th othr ariabls, maing a iabl paramtr on which to bas th lngth scal or a Kndsn li paramtr, althogh dnsit, locit, and tmpratr ha bn mostl tilizd in prior wor (Wang and Bod, [4]). In Figr 5, right hand sid, w prsnt th ntrop and ntrop ration proils. h non monotonic bhaior or th ntrop, which is thorticall corrct, is idnt both rom its proil and its gradint across th shoc wa. All th paramtrs ar prsntd in Figr 6 or comparison. h paramtr basd on hating rats across th shoc spans a broad ara whil th ntrop ration rat span is narrow. Basd on this inormation alon, a narrowr span might indicat an adantag sinc on wold li to s th most icint mthod (CFD) as ar as possibl into th shoc intrior. nortnatl, with strongr shocs othrs ha shown that continm bradown, as masrd b comparing th diation btwn a CFD soltion and a DMC soltion indicats that continm bradown ma occr or a broadr rang than indicatd b an o ths paramtrs (Wang, n, and Bod [5]). his rmains an iss o intrst and dpr instigation. REMARK: ENROPY PRODCION IN CFD Inspiration or this wor bgan with th sggstion that prhaps ntrop ration rats cold b tilizd to dtct rgions o trm locit and tmpratr gradints. W ha shown that this ind o paramtr compars wll with priosl proposd altrnatis. It has th adantag o inclding cts o locit and tmpratr gradints togthr as wll as th tmporal cts associatd with strongl nstad low, dpnding on how th ntrop balanc ation is ormlatd. In addition to its possibl s in continm bradown stdis, howr, th scond law o thrmodnamics has mch mor ral applications (Ban [9], cbbia [0]). W discss som o ths applications in bri and onl or th prpos o stting th stag or wor to ollow. A nmrical mthod applid to th soltion o th gorning ations aims at calclating th pdatd ariabls rom th prsntl nown distribtion at a prios tim stp. h spac and tim intgration o th balanc ations can b sparatd so that th rslting init olm, smi discrt ormla dqi dt m A V i ( nˆ ) 0 whr A /V i rprsnts th ratio o th cll ac ara to th discrt cll olm, can b intgratd in tim to obtain nw als o th stat ariabls. Whthr implicit or plicit tim intgration is sd, th calclation o ntrop ration rats can b don a postriori, whn all th othr stat ariabls ar nown rom th soltion o th lid low ations. For th Nair tos ations th ntrop transport ation (.a) holds and on thror ma s th discrt approimations (implicit or plicit tim intgration): n ( ρs) ( ρs) n m A nˆ ρs t Vi * (40.). (4.) An altrnati prssion ma b obtaind sing nmrical stimats or th locit and tmpratr gradints (tpicall alrad nown or highr ordr mthods) and soling or th ntrop ration rat with Eation (.b). A nmrical mthod that satisis th non ngati ntrop ration principl in discrt orm can b said to satis a local orm o th scond law o thrmodnamics. B calclating and monitoring th magnitd and sign o th ntrop ration locall, on can idnti rgions that iolat th scond law. his also allows an pandd capabilit or nsring that th comptational simlation satisis phsical thor, is nmricall stabl, and incrs a minimm o nmrical rror. Each o ths possibilitis ha not bn ll plord and ploitd. 9

10 CONCLION W ha prsntd a r basic ida and sicint dtail or th dinition o a continm bradown paramtr basd on local ntrop ration rats. h paramtr attmpts to addrsss th nd or dtcting rgions whr non ilibrim phsical procsss dominat so that appropriat mathmatical and nmrical modls ma b mplod towards th soltion o th gorning ations. A bradown paramtr basd on ntrop ration rats has th adantag o inclding cts o locit and tmpratr gradints togthr as wll as th tmporal cts associatd with strongl nstad low, dpnding on how th ntrop balanc ation is ormlatd. Althogh th lmntar rslts prsntd rl ntirl on th continm assmption and wa non ilibrim, b comparing th nw paramtr with thos priosl proposd and tilizd, w ar abl to sggst its iabilit as a thortical bnchmar, althogh its s in trm conditions has not bn pron. h ocs hr has bn in idntiing a paramtr or possibl s in dtcting rgions o trm gradints whr it ma b ncssar to switch to a molclar basd mthod whil sing a continm CFD solr in smoothr rgions. ch hbrid soltions ha bn plord in th past with aring dgrs o sccss. An ort in this dirction rirs a practical and robst continm bradown indicator. In ordr to ll alat th ntrop ration as a iabl paramtr, it will b ncssar to irst alidat CFD calclations with limiting cass li thos prsntd hr and thn s how wll th paramtr prorms ndr mor sr conditions (li strong shocs, trblnt bondar lars, and othr non ilibrim phnomna). In addition, th init grid cll dimnsion in CFD proids anothr locall alid rrnc lngth scal to plor. ACKNOWLEDGMEN his wor was sponsord b th Air Forc Oic o cintiic Rsarch (AFOR) grant 07NA to th Air Forc Rsarch Laborator at Wright Pattrson Air Forc Bas and b grant F to th nirsit o Michigan ndr Dr. John chmissr. h iws and conclsions containd hrin ar thos o th athors and shold not b intrprtd as ncssaril rprsnting th oicial policis or ndorsmnts, ithr prssd or implid, o th AFOR or th.. Gornmnt. REFERENCE. Bird, G. A., Bradown o ranslational and Rotational Eilibrim in Gasos Epansions, AIAA Jornal Vol. 8, No. 0, pp Bird, G. A., Molclar Gas Dnamics and th Dirct imlation o Gas Flows, Oord nirsit Prss, Nw Yor: NY, Canpp, P. W., h Inlnc o Magntic Filds or hoc Was and Hprsonic Flow, AIAA chnical Papr No , Dnr, CO, Amrican Institt o Aronatics & Astronatics, Jn 000, pp Wang, W. and Bod, I. D., Continm Bradown in Hprsonic Viscos Flows, AIAA chnical Papr No , Rno, NV, Amrican Institt o Aronatics & Astronatics, Janar 00, pp.. 5. Wang, W., n, Q., and Bod, I. D., owards Dlopmnt o a Hbrid DMC CFD Mthod or imlating Hprsonic Intracting Flows, AIAA chnical Papr Nmbr , t. Lois, MO, Amrican Institt o Aronatics & Astronatics, Jn Cambros, J. A. h Prodction o Entrop in Rlation to Nmrical Error in Comprssibl Viscos Flow, AIAA Papr No. 000, Dnr, CO, Amrican Institt o Aronatics & Astronatics, Jn Whit, F. M., Viscos Flid Flow, nd Ed., McGraw Hill, Boston: MA, 99, pp Gilbarg, D. and Paolcci, D., h trctr o hoc Was in th Continm hor o Flids, Jornal o Rational Mchanics & Analsis, Vol., pp , Ban, A., Entrop Gnration Minimization, CRC Prss, Boca Raton: FL, 996, pp cbbia, E. Calclating Entrop with CFD, Mchanical Enginring Magazin, Vol. 9, No. 0, AME Intrnational: Octobr 997, pp

11 Prssr haring Vlocit Dnsit mpratr Hating FIGRE 4. Vlocit, prssr, tmpratr, dnsit proils, (lt) and hating/shar strsss (right) ratd with nmrical ODE soltion o th D Nair tos ation or a stad normal shoc. Prssr Dnsit Entrop Vlocit mpratr Entrop Gnration FIGRE 5. Macroscopic gradints and ntrop proil proils rslting rom a nmrical ODE soltion or D shoc strctr. K Kn K τ K FIGRE 6. Kndsn li paramtr proils rslting rom a nmrical ODE soltion or normal shoc wa strctr.

Figure 1: Schematic of a fluid element used for deriving the energy equation.

Figure 1: Schematic of a fluid element used for deriving the energy equation. Driation of th Enrg Eation ME 7710 Enironmntal Flid Dnamics Spring 01 This driation follos closl from Bird, Start and Lightfoot (1960) bt has bn tndd to incld radiation and phas chang. W can rit th 1 st