hrr vs radius
Post on 07-Apr-2018
212 Views
Preview:
TRANSCRIPT
-
8/6/2019 HRR vs Radius
1/2
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)
-
8/6/2019 HRR vs Radius
2/2
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.
top related