book-chapt4——Problem Closure Formulation.pdf

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1、Chapter4ProblemClosureFormulation4.1TurbulenceclosureInthissection,thesub-modelsformulatedneededtoevaluatethesourcetermsinboththeaveragegasphaseequationsandtheconservationlawsforthenumberofliquidglobules(2.1)-(2.3).Thesesourcetermsdependontheevolutionofthe

2、dropletstatevariableandthelocalgasphaseconditions.Variousmodelshavebeenproposedtoachieveclosureofthesystemofequationsforturbu-lenttwo-phasereactingow.IfthestandardFavre(seeAppendixI)averagedKmodelisadopted,theReynoldsstressandtherateofstrainaredescribed

3、byalinearrelationasfollowsg00002@u~k@u~i@u~juiuj=ij(Kt)t(+);(4.1)3@xk@xj@xiwheretheturbulentviscosityisgivenbyK2t=C(4.2)withthemodelconstantC=0.09.Theaveragingnotationisnowsuspendedwiththevariablesunderstoodtobeaveragedwhereappropriate.Thedeta

4、iledderivationoftheconservationequa-tionsoftheKturbulenceforthegasphasecanbyfoundin[15].TheprocedurestartswiththederivationofthetransportequationforReynoldsstresses,andthenbycontractionofthisequation,theK-equationisobtained.Underthepresentcircumstances,E

5、q.(2.28)reducesto@Kt2+r[K~urK]=Kr
~u+^:r~u+W_s;(4.3)@tPrk3wherePrkistheturbulentPrandtlenumberfortheturbulentkineticenergy.Tocalculateaturbulentuxofthescalarquantity,agradient-diffusionmodelisemployedug0000=t@~;(4.4)j@xtkwheretistheturbule

6、ntPrandtle(orSchmidt)numberforvariable.Derivationofthetransportequationfortheturbulentkineticenergydissipationrate,,inthecaseofcompressibleowsisacomplexprocedure.Forexample,theequationdeducedin[15]contains66terms.Afterapplyingtheorderofmagnitudeanalysis,

7、thenumberoftermsforinertowscanbereducedto8whichrequiremodeling.SinceforhighReynoldsnumbersturbulencetendstobeisotropic,themathematicaldenitionofsimpliestom@u00@u00=ii;@xk@xk17andtheconservationequationreducesto@+r[~utr]=(2CC)r
~u+[C^

8、:r~uC+CW_];(4.5)1312ss@tPr3kwherePristheturbulentPrandtlenumberfor.Thefollowingsetofthemodelparametersisrecommended:C1=1.44,C2=1.92,C3=-1,Cs=1.5,Prk=1.0,Pr=1.3.Thespray-relatedtermW_sinEq.(4.5)