Joint Scalar PDF Simulations of a Bluff-Body Stabilised Flame with the REDIM Approach

Transcription

1 Jont Scalar PDF Smulatons of a Bluff-Body Stablsed Flame wth the REDIM Approach B. Merc,1,2, B. Naud 3, D. Roekaerts 4 and U. Maas 5 1 Postdoctoral Fellow of the Fund of Scentfc Research Flanders (Belgum) (FWO-Vlaanderen) 2 Ghent Unversty UGent, Dept. of Flow, Heat and Combuston Mechancs, Ghent, Belgum 3 Modelng and Numercal Smulaton Group, Energy Dept., Cemat, Madrd, Span 4 Dept. of Mult-Scale Physcs, Delft Unversty of Technology, Delft, The Netherlands 5 Insttute for Techncal Thermodynamcs, Karlsruhe Unversty (TH), Germany Abstract Transported jont scalar probablty densty functon (PDF) results are presented for Sydney Flame HM3, a jet type turbulent flame wth strong turbulence chemstry nteracton, stablzed behnd a bluff body. We apply the novel Reacton-Dffuson Manfold (REDIM) technque, by whch a detaled chemstry mechansm s reduced, ncludng dffuson effects. Only N 2 and CO 2 mass fractons are used as reduced coordnates. A second-moment closure RANS turbulence model s appled. As mcro-mxng model, the modfed Curl s coalescence/dsperson (CD) and the Eucldean Mnmum Spannng Tree (EMST) models are used. In physcal space, agreement between expermental data and smulaton results s good up to the neck zone, for the uncondtonal mean values of velocty, mxture fracton, major and some mnor chemcal speces. Condtonal mean profles n mxture fracton space are also n reasonable agreement wth experments up to the neck zone, though condtonal fluctuatons tend to be underpredcted. CD generally yelds better predctons for the level of fluctuatons n mxture fracton space than EMST, but ths s partly due to unrealstc partcle evoluton n composton space. In general, smulatons usng the REDIM approach for reducton of detaled C 2 -chemstry confrm earler fndngs for mcro-mxng model behavour, obtaned wth C 1 -chemstry. Introducton Turbulence-chemstry nteracton s a central ssue n non-premxed turbulent flame smulatons. Wth the transported probablty densty functon (PDF) technque, an exact treatment of the chemcal reacton source term, gven a certan chemstry model, s acheved [1]. Yet, reducton of chemstry s typcally requred n order to keep computng tmes wthn acceptable lmts. In [2], where we present a comparatve study on transported jont velocty-scalar PDF (JVSPDF) and jont scalar PDF (JSPDF) smulatons, for the Sydney bluff-body stablzed flames HM1 and HM3 [3-6], we appled for the frst tme the novel Reacton-Dffuson Manfold (REDIM) technque [7] to reduce a detaled reacton mechansm, consstng of 34 speces and 302 reactons [8]. Ths s a step forward, compared to the study of [9], where the nfluence of mcro-mxng models was studed n the context of C 1 -chemstry skeletal mechansms. In [10] and [11], detaled and systematcally reduced C 2 - chemstry models are used, respectvely. In [12], the condtonal moment closure (CMC) s appled, wth the GRI 2.11 or 3.0 chemstry mechansm. In contrast to the study n [2], we focus here on the qualty of the results n composton space for the JSPDF method. Prevous studes [8-10] reveal that flame HM3, wth the strongest turbulence chemstry nteracton effects of the seres and whch s close to blow-off condtons, s the most challengng flame. In order to acheve good agreement to expermental data, up to the neck zone, for the turbulent flow feld, a second moment closure model s appled wth modfed model constants [13]. Smlarly, for good agreement n physcal space for mean mxture fracton and mxture fracton varance, the turbulent Schmdt number s set to Sc T = 0.85 [2]. Two mcro-mxng models are appled: the modfed Curl s coalescence/dsperson (CD) [14] and the Eucldean Mnmum Spannng Tree (EMST) model [15]. In [9], we showed that local extncton was underestmated wth EMST and CD n general performed better, but no statonary soluton could be obtaned for HM3 wth the CD mxng model. In the present study, applyng reduced chemstry, obtaned wth the REDIM technque, a statonary soluton s obtaned, also wth the CD mxng model. PDF Approach All calculatons are performed wth the same hybrd Fnte-Volume / partcle methods, as mplemented n the same n-house computer program PDFD [16]. The statstcal descrpton of the flow s n terms of the jont one-pont scalar PDF f φ : f φ (ψ;x,t).dψ s the probablty that the composton vector φ s n the nterval [ψ,ψ+dψ[ at pont (x,t). The jont PDF s defned as [1,17]: fφ ( ψ; x, t) = δ φ( x, t) ψ. For varable densty flows, t s useful to consder the jont scalar mass densty functon (MDF) F φ (ψ)=ρ(ψ)f φ (ψ). Densty weghted averages (Favre averages) can then be consdered. Fluctuatons wth respect to the Favre average are defned as: q ( x, t) = Q( x, t) Q ( x, t).the MDF transport equaton s modelled and solved [1]: Correspondng author: Bart.Merc@UGent.be European Combuston Meetng 2009

2 + + = Fφ U jfφ Sα ( ) Fφ t xj ψ ψ α (1) u ψ F θ ψ F x φ α φ ψ α wth the mxng term ( φ ) θ α = ρ α 1 j ( φ ) J x j, and where S α s the reacton source term for scalar φ α and J α ts molecular flux. The frst term on the rght hand sde s modeled usng a gradent dffuson assumpton: ( Fφ ρ ) u Fφ T x ψ = Γ, where Γ T s the x x turbulent dffusvty, modeled as Γ T =μ T /Sc T, wth μ T the eddy vscosty. Mean velocty, mean pressure gradent, Reynolds stresses and turbulent dsspaton rate are obtaned by a standard Fnte-Volume (FV) method based on a pressure-correcton algorthm [2]. A partcle method s appled for the soluton of the MDF transport equaton. A set of unformly dstrbuted computatonal partcles evolves accordng to stochastc dfferental equatons. Each partcle has a set of propertes {w,m,x,φ}, where w s a numercal weght, m s the partcle mass, X ts poston and φ the partcle s composton. Partcle mass m s constant n tme. Increments of partcle poston X and composton φ over a small tme step dt are gven by: 1 2 c 1 Γ T 2Γ T dx = U dt + U + dt + dw, (2) ρ x ρ dφα = θαdt+ Sα ( φ ) dt. (3) The correcton velocty U c, resultng from a poston correcton algorthm [18], ensures that the volume represented by the partcles n a computatonal cell, equals the cell geometrc volume. In the random walk model for partcle poston evoluton, dw s an ncrements over dt of the Wener process W. All quanttes between brackets [ ] are FV propertes nterpolated at the partcle locaton, usng blnear bass functons [19]. The method of fractonal steps [1] s used to ntegrate the system of equatons wth local tme-steppng [20]. More detals on numercal ssues are found n [2]. Chemstry reducton by REDIM A recent comprehensve revew on chemstry reducton s provded n [21,22], wth arguments n favour of the close-parallel assumpton for chemstry based slow manfolds. We reduce the detaled reacton mechansm of [8], wth 34 speces and 302 reactons, by means of the REDIM technque [7], for a mxture of CH 4 /H 2 and ar as n HM3 (see below). Ths concept leads to an nvarant low-dmensonal manfold, drectly accountng for the couplng of chemstry and dffuson (and convecton). It s based on a relaxaton process, where an ntal guess for a low-dmensonal manfold evolves n such a manner that an nvarant slow reacton/dffuson manfold s obtaned. Ths manfold, computed a pror, s steady and nvarant n space n the turbulent flame smulatons. One major advantage over ILDM, s the fact that a REDIM exsts n the entre accessed doman, even at low temperatures (close to the unburned mxture), where chemstry s slow and ILDM does not yeld an exstng manfold. Close to equlbrum, the REDIM s close to ILDM. If chemstry governs the overall process, the REDIM concept yelds slow manfolds or, equvalently, teratvely refned ILDMs as a lmtng case [23]. In REDIM, an m-dmensonal vector of reduced coordnates represents the composton vector φ. In order to smplfy the subsequent use of the tables, sutable smple progress varables are dentfed after the calculaton of the REDIM. They are used as the reduced coordnates. For the test case under study, we use m = 2 reduced coordnates: mxture fracton (or mass fracton of N 2 ) and mass fracton of CO 2 (the only reacton progress varable) [2]. Equal dffusvtes and unty Lews number are assumed n the present applcaton of REDIM, although t s possble to extend the REDIM concept to systems wth non-equal dffusvtes. For the test cases under study, no evdence has been provded yet that dfferental dffuson would play an mportant role. In the transported PDF equatons, t s also typcally gnored n the mcro-mxng modelng. In [2], we descrbed some specfc mplementaton detals for the consdered system. In equaton (1), the composton vector s thus reduced to φ = (ξ,y CO2 ) and the chemcal source term S CO2 (φ) s gven by the REDIM reduced chemstry. Note that n the expermental data below, the mxture fracton ξ s defned on the bass of Blger s formula [24]: 2( ZC ZC, o) ( ZH ZH, o) ZO ZO, o + WC 2WH WO ξ =. (4) 2( ZC, f ZC, o) ( ZH, f ZH, o) ZO, f ZO, o + WC 2WH WO where Z β s the total mass fracton of element β and W β s ts atomc mass. The subscrpts f and o refer to the fuel and oxdant streams. In the present numercal results, where no dfferental dffuson s consdered, YN2 mxture fracton s defned as: ξ = 1. YN2, o Mcro-Mxng Modellng As mxng model θ α, frst the modfed Curl s CD model [14] s used, whch prescrbes the evoluton of partcle composton as a seres of par-wse mxng events. The partcpatng mxng partcles are chosen at random from the set of partcles present n a fnte volume cell and ther compostons change n the drecton of the nteracton partner. The degree of mxng n a par s determned by a random varable, unformly dstrbuted between 0 (no mxng) and 1 (complete mxng). The model constant C φ s set to 2.0, as n [2,9]. Comparson s also made to applcaton of the EMST model [15], where partcles nteract preferably wth neghborng partcles n composton

3 space. As n [9], the model constant C φ s then set to C φ = 1.5, the default value for ths mxng model. For comparson reasons, we also nclude results wth C φ = 2.0 wth the EMST model. Test Case The gaseous fuel mxture of 50% H 2 and 50% CH 4 by volume, s njected from the central ppe, wth dameter D j = 3.6mm, of the bluff-body burner, wth outer dameter D b = 50mm. The burner s surrounded by an unconfned co-flow ar stream. Fuel and ar are mxed n the recrculaton zone behnd the bluff body where chemcal reactons take place. The hot products stablze the flame. In the expermental studes [3] and [4], the jet and co-flow bulk veloctes were 214m/s and 40m/s (HM3). The velocty measurements, on the other hand, were provded for slghtly reduced veloctes (195m/s and 35m/s (HM3e)) [5]. The reader s referred to [6] for a more complete descrpton. The numercal settngs are as descrbed n [2]. A 6D b long and 3D b wde two-dmensonal axsymmetrc structured grd s used, stretched n both axal and radal drectons. Free-slp boundary condtons are prescrbed on the bluff-body surface and on the lateral boundary. An average number of 100 partcles per cell s used. Iteraton averages are made over 500 teratons. As n [2], converged equlbrum chemstry assumedshape β-pdf results are used as ntal condtons. A statstcally statonary stuaton s reached after about 5000 partcle tme steps. Results obtaned after partcle tme steps are now dscussed. Results n Physcal Space We frst llustrate the qualty of the turbulent mxng felds. Due to space lmtaton, only two postons are consdered: x = 13mm, close to the burner; and x = 90mm, whch s n the neck zone behnd the recrculaton regon (x > D b ). For more results, ncludng the turbulent flow feld data,.e. radal profles of mean (axal) velocty and velocty fluctuatons, we refer to [2], where good agreement to expermental data was llustrated, except for a wellknown mean axal velocty under-predcton near the axs n the neck zone. We verfed that there are practcally no dfferences between CD and EMST results, confrmng earler fndngs reported n [9]. Fgure 1 shows radal profles of mean mxture fracton and mxture fracton varance. The profles beng slghtly wde, agreement for mean mxture fracton s good. As expected, there s hardly any dfference between the results, obtaned wth the dfferent mcromxng models. The mxture fracton varance s overpredcted near the burner. In the neck zone, agreement s qute good, although the profles are agan slghtly too wde. The mpact of the model constant C φ s apparent: n the modeled mxture fracton varance transport equaton, mpled by the transported MDF equaton [2], the scalar dsspaton rate s lower wth C φ = 1.5, so that mxture fracton fluctuatons are hgher. Dfferences are small between the results, obtaned wth the two mcromxng models, wth the same C φ value. Fgure 1. Radal profles of mean mxture fracton and mxture fracton varance. Fgures 2 and 3 show radal profles of mean temperature and some major and mnor chemcal speces. The test case s challengng, as there s much local extncton (see below). Consequently, agreement wth expermental data s not very good (and not as good as for HM1 [2], not shown here). One reason mght be that the REDIM wth only one reacton progress varable does not represent the thermochemcal states well enough. Ths problem could be overcome by usng a REDIM wth two reacton progress varables. Ths remans to be nvestgated. Yet, the global shapes are well reproduced. It s nstructve to observe that close to the burner (x = 13mm), where there s relatvely lttle local extncton, the smulaton results are more affected by the choce of C φ than by the choce of mcro-mxng model, whereas n the neck zone ths s less clear. Ths s also related to the dscusson, reported n [9]: close to the burner, the mcro-mxng tme scale for the partcles s relatvely large, compared to mean convectve and turbulent dffuson tme scales. Consequently, the effect of the mcro-mxng model s small n physcal space results. In the neck zone, ths s no longer true. Most nterestngly, the arguments, provded n [9], wth C 1 chemstry skeletal mechansms, stll hold wth the present REDIM applcaton wth reduced C 2 chemstry. Results n Composton Space For the profles of mean values and fluctuatons, condtoned on mxture fracton, the mxture fracton nterval [0,1] has been dvded nto 100 bns. For each bn, the arthmetc mean value and the fluctuatons around these mean values have been computed accordngly, takng the root mean square value.

4 Fgure 2. Radal profles of mean temperature and CO 2 mass fracton. Fgure 4. Profles of condtonal mean value and rms value of condtonal fluctuatons of temperature. In fgure 5, we present the profles for major speces and REDIM coordnate CO 2 mass fracton. There s a global over-predcton of condtonal mean CO 2 mass fracton. The level of condtonal fluctuatons s n general under-predcted. It only seems correct wth CD at x = 90mm. Yet, note that the condtonal mean values are stll over-predcted. Fgure 6 shows mnor speces OH, whch s partcular n the sense that all the acton takes place n the narrow regon [0; 0.02] n mxture fracton space. In general, the expermental condtonal mean values are over-predcted by a factor of two. Note the relatvely small dfferences between CD and EMST, ndcatng that chemstry does not strongly ntertwne wth the mcro-mxng modelng. Fgure 3. Radal profles of mean CO and OH mass fracton. Fgure 4 shows condtonal temperature profles. At x = 13mm, agreement for the condtonal mean temperature s qute good, but condtonal fluctuatons are under-estmated. Dfferences between the two mcro-mxng models are small at x = 13mm, most probably because processes n physcal space are more promnent than mxng/reacton n composton space [9]. At x = 90mm, dfferences between the mcromxng models are apparent. Whle CD s closer to the expermental data, none of the results are n good agreement wth expermental data: condtonal mean temperatures are over-estmated and condtonal fluctuatons are under-predcted. As mentoned, probably the REDIM wth only one reacton progress varable does not represent the thermo-chemcal state well enough. Ths problem could be overcome by usng a REDIM wth two reacton progress varables. Fgure 5. Profles of condtonal mean value and rms value of condtonal fluctuatons of mass fracton CO 2.

5 Fgure 6. Profles of condtonal mean value and rms value of condtonal fluctuatons of mass fracton OH. Fgure 7, showng scatter plots of temperature versus mxture fracton, reveals strong dfferences between CD and EMST at x = 90mm. Wth EMST, too lttle local extncton s seen. Agan, t s nterestng to note that these observatons are completely n lne wth what was reported n [9], where we appled a C 1 chemstry based skeletal mechansm. Thus, chemstry does not seem to ntertwne wth these fndngs. Dscusson In the present study, we appled the novel REDIM technque. Arguably, detaled comparsons to other technques, such as the FPI/FGM (e.g. [25]), wll be nterestng future work. In partcular, t wll be nterestng to nvestgate the mportance of ncluson of dffuson effects n the constructon of the manfold (as n REDIM) n lmtng cases. Ths s consdered beyond the scope of the present paper. Another ssue s that the expermental data mght not le close to the REDIM manfold. Ths remans to be nvestgated. Conclusons The novel REDIM technque for reducton of a detaled chemstry model, has been appled to the turbulent non-premxed bluff body stablzed jet type flame HM3, whch has strong turbulence chemstry nteracton and a substantal amount of local extncton. Dfferences between results, obtaned wth two mcro-mxng models on the one hand or wth two dfferent C φ model constants on the other hand, have been explaned. The nterplay wth phenomena n physcal space has been hghlghted. The most mportant observaton s that, n general, n composton space, earler fndngs for mcro-mxng model behavor, obtaned wth C 1 chemstry, have been confrmed wth the REDIM approach for reducton of detaled C 2 chemstry. Fgure 7. Scatter plots of temperature. Acknowledgements Ths collaboratve research s supported by the COMLIMANS program, by the Spansh MEC under Project ENE C04-04/CON and by the Fund of Scentfc Research Flanders (Belgum) (FWO- Vlaanderen) through FWO-project G References 1. S.B. Pope. Prog. En. Combust. Sc., 11: , B. Merc, B. Naud, D. Roekaerts and U. Maas, Flow Turbul. Combust. (n press). (DOI: /s ) 3. B.B. Dally and A.R. Masr. Combust. Theory Mod., 2: , B.B. Dally, A.R. Masr, R.S. Barlow, and G.J. Fechtner. Combust. Flame, 114: , ame.htm 6. R.S. Barlow. Internatonal Workshop on Measurement and Computaton of Turbulent Nonpremxed Flames V. Bykov and U. Maas, Combust. Theory Mod. 11(6): , 2007.

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