JournalofEngineeringMathematics44:5782,2002.2002KluwerAcademicPublishers.PrintedintheNetherlands.Modellinggasmotioninarapid-compressionmachineM.G.MEERE1,B.GLEESON1andJ.M.SIMMIE2DepartmentofMathematicalPhysics,NUI,Galway,Ireland2DepartmentofChemistry,NUI,Galway,IrelandReceived25July2001；acceptedinrevisedform8May2002Abstract.Inthispaper,amodelwhichdescribesthebehaviourofthepressure,densityandtemperatureofagasmixtureinarapidcompressionmachineisdevelopedandanalyzed.Themodelconsistsofacoupledsystemofnonlinearpartialdifferentialequations,andbothformalasymptoticandnumericalsolutionsarepresented.Usingasymptotictechniques,asimplediscretealgorithmwhichtracksthetimeevolutionofthepressure,temperatureanddensityofthegasinthechambercoreisderived.Theresultswhichthisalgorithmpredictareingoodagreementwithexperimentaldata.Keywords:gasdynamics,rapid-compressionmachines,shock-waves,singularperturbationtheory1.Introduction1.1.RAPID-COMPRESSIONMACHINESArapid-compressionmachineisadeviceusedtostudytheauto-ignitionofgasmixturesathighpressuresandtemperatures,withparticularreferencetoauto-ignitionininternalcombus-tionengines；see13.Atypicalcombustionengineisaverydirtyandcomplexenvironment,andthishaspromptedthedevelopmentofrapid-compressionmachineswhichenablethescientificstudyofcompressionandignitioninenginesinacleanerandsimplersetting.InFigure1weschematicallyillustrateatwo-pistonrapid-compressionmachine,suchastheoneinthedepartmentofChemistryatNUI,Galway.However,single-pistonmachines,withapistonatoneendandastationarysolidwallattheother,aremoretypical.Theanalysisdevelopedinthispaperisappropriatetobothsingle-andtwo-pistonmachines.Theoperationofarapid-compressionmachineisverysimple-thepistonsaresimul-taneouslydriveninpneumatically,compressingtheenclosedgasmixture,therebycausingthegaspressure,temperatureanddensitytorisequickly.InFigures1(a),1(b)and1(c)weschematicallyrepresentarapid-compressionmachinepriorto,during,andaftercompression,respectively.TheratioofthefinalvolumetotheinitialvolumeofthecompressionchamberforthemachineatNUI,Galwayisabout1:12,thisvaluebeingtypicalofothermachines.Attheendofthecompression,thegasmixturewilltypicallyhavebeenpushedintoapressureandtemperatureregimewhereauto-ignitioncanoccur.InFigure2,wedisplayanexperimentalpressureprofileforaH2/O2/N2/ArmixturewhichhasbeentakenfromBrettetal.4,withthekindpermissionoftheauthors.Inthisgraph,thetimet=0correspondstotheendofthecompressiontime.Wenotethat,forthegreaterpartofthecompression,thepressureinthechamberisrisingquitegently,butthattowardstheendofthecompression(thatis,justbeforet=0),thereisasteepriseinthepressure.Aftercompression,thepressureprofilelevelsoffasexpected；theextremelysteepriseattheendof58M.G.Meereetal.Figure1.Schematicillustratingtheoperationofarapid-compressionmachine；wehaveshowntheconfiguration(a)priortocompression,(b)duringcompressionand(c)aftercompression.Figure2.AnexperimentalpressureprofileforagasmixtureH2/O2/N2/Ar=2/1/2/3,asmeasuredintherapid-compressionmachineatNUI,Galway.Itistakenfrom4,andhasaninitialpressureof005MPaandaninitialtemperatureof344K.Modellinggasmotioninarapid-compressionmachine59theprofilecorrespondstotheignitionofthemixture.WenotethatthecompressiontimeandthetimedelaytoignitionaftercompressionarebothO(10)ms.Thepressurehistoryistheonlyquantitywhichismeasuredinexperiments.However,thetemperatureinthecoreaftercompressionisthequantitywhichisofprimaryinteresttochemistssincereactionratesdependmainlyontemperatureforalmostallsystems,althoughtheremayalsobesomeweakerpressuredependence.Measuringtemperatureaccuratelyinthecorecanbeproblematicbecauseofthepresenceofathermalboundarylayer；seethecommentsbelowonroll-upvortices.However,withtheexperimentalpressuredatainhand,thecorrespondingtemperaturescanbeestimatedusingtheisentropicrelationln(p/pi)=integraldisplayTTi(s)s(s)1)ds,(1)where(Ti,pi)aretheinitialvaluesforthecoretemperatureandpressure,(T,p)arethesequantitiesatsomelatertime,and(s)isthespecificheatratioattemperatures.Inexper-iments,theinitialcoretemperatureistypicallyO(300K),whilethecoretemperatureaftercompressionisusuallyO(1000K).Inthispaper,weshallconsideronlythebehaviourofthegasmixtureduringcompression；thepost-compressionbehaviourisnotconsideredhere,butthiswillformthesubjectforfuturework.Nevertheless,themodelpresentedheredoesprovideareasonabledescriptionofthepost-compressionbehaviourofasinglespeciespuregas,oraninertgasmixture；seeSection.THEMODELWesupposethatthecompressionchamberislocatedalong0x0.Thisassumptionisactuallyquiteastrongoneinthiscontextsincehigher-dimensionaleffectsarefrequentlyobservedinexperiments,roll-upvorticesnearthecornerregionsdefinedbythepistonheadsandthechamberwallbeingparticularlynoteworthy；see,forexample,5.Thesevorticesariseduetothescrapingbythepistonsofthethermalboundarylayeratthechamberwall,andtheycan,andfrequentlydo,disturbthegasmotioninthecoreofthecompressionchamber.However,thejustificationfortheone-dimensionalmodelstudiedhereistwofold:(i)thecornervorticescanbesuccessfullysuppressedbyintroducingcrevicesatthepistonheadswhichswallowthethermalboundarylayerasthepistonsmovein(seeLee6),renderingthegasmotionawayfromthechamberwallsone-dimensionaltoagoodapproximation,and,(ii)thestudyoftheone-dimensionalmodelprovidesausefulpreliminarytothestudyofhigher-dimensionalmodels.Wenowgivethegoverningequationsforourone-dimensionalmodel.Areasonablycom-pletederivationofthegoverningequationsforamulti-componentreactinggascanbefoundintheappendicesof7；thesestandardderivationsarenotreproducedhere.Themodelwhichweshallstudyincludesanumberofsimplifyingassumptionsandthesewillbeclearlystatedastheyarise.Theequationexpressingconservationofmassisgivenbyt+x(v)=0,60M.G.Meereetal.where=(x,t)andv=v(x,t)arethedensityandthevelocityofthegas,respectively,atlocationxandtimet.Itshouldbeemphasizedthatthesequantitiesrefertoagasmixture,sothatifthereareNdifferentspeciesinthemixturethen=Nsummationdisplayi=1i,wherei=i(x,t)isthedensityofspeciesi.Also,thevelocityvabovereferstothemass-averagedvelocityofthemixture,thatis,v=Nsummationdisplayi=1Yivi,whereYi=i/andvi=vi(x,t)arethemassfractionandvelocity,respectively,ofspeciesi；see7.Neglectingbodyforcesandviscouseffects,theequationexpressingconservationofmo-mentumisgivenbyvt+vvx=1px,wherep=p(x,t)isthepressure.Weassumethatthegasmixtureisideal,sothattheequationofstateisgi