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AllofthefollowingdiscussionpertainstoaflamesheetmodelsolutionforacaseofaVgeometryflame

stabilizedonacenterbodywithnoradius,withaconstantaxialvelocityandnoradialvelocityeffects.

InmynotationIusethefollowingdefinitions:

(1)

and 1Firstconsiderthecaseinwhichthevelocityperturbationisacousticinnatureonly:

(2) , Theoscillatingflamesurfaceposition()isgivenbythefollowingequation(comparetoPreetham,2008forVflameswhereK>0):

(3)

1

Theheatreleaseisproportionaltotheflamesurfacearea. Onalocalbasiswecancalculatethe

linearizedflamesurfacearea:

(4)

1

Notethatforanaxisymmetriccase,thiswouldbesimilar,thedifferencebeingareadirectlyproportional

totheradius. Observethatoneofthetermsinthesurfaceequationisnotafunctionofradius:

(5)

Takingthemagnitudeofthisgives:

(6)

|| Thisisconstantwithrespecttoradius. Againrecallthatforanaxisymmetriccase,thedifferentialarea

hasadifferentformandthus:

(7)

||

Bothhavethesamevalueforphaseangle,indicatingconvectionalongtheinclinedflamesurface:

(8)

Sotosummarize,theequationfortheflamesurfacepositionisthedifferenceoftwooscillatingterms,

whoserelativephasedependsontheradialdistance,resultingininterferenceasafunctionofradius.

However,intermsoftheamplitudeofoscillatingheatrelease,theamplitudehasaconstant(orlinear)

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valueasafunctionofradius,similarlytotheamplitudeofoscillationsfortheconvectivedisturbance

modelpresentedattheconference.

Werewetodiscussvorticaldisturbancesinstead,theflamesurfacepositionsolutionis(againcompare

toPreetham,2008):

(9)

1 1

Inthiscasebothtermswillmakeacontributiontotheflamesurfacearea:

(10) 1 1 1

Ididnotdiscussthewaytodealwithmorecomplicatedexcitationconditionslikethisattheconference

duetotimeconstraints,butthisisessentiallyhandledinaconvectivedisturbancemodelbyadding

togethertheresponsetotwolinearlyindependentperturbationsmovingwithdifferentconvective

speeds( and ). Indeedinthiscase,thecombinedsolutionWILLhaveanonconstant||becauseofinterferencebythetwodisturbances. However,lookingattheequation,wecanseethat

whatshappeningiswehavetwoseparateperturbations,eachofwithhasconstantamplitudewith

respecttoradius,butdifferentvaluesforthephaseangle. Thuswhenweaddthetwotogether,weget

anonconstantvalue.