Conclusion g´en´erale et perspectives

Conclusion g´en´erale et perspectives

Synth`ese du manuscrit

Ce travail de th`ese r´epond `a une probl´ematique cruciale pour la conception des foyers a´eronautiques : ´elaborer un outil num´erique capable de pr´edire la dynamique instationnaire de la flamme diphasique au sein d’une chambre de combustion industrielle. En plus du r´egime ´etabli, deux r´egimes instationnaires essentielles pour les turbines sont ´etudi´es : la phase d’allumage et un r´egime puls´e. Cette derni`ere ´etude s’inscrit dans une approche plus globale de m´ethodes couplant m´ecanique des fluides et acoustique afin de mieux pr´edire les instabilit´es de combustion.

L’approche SGE est choisie pour ses capacit´es `a simuler pr´ecis´ement les ´ecoulements fortement ins-tationnaires. Les m´ethodes RANS sont moins bien adapt´ees `a l’´etude d’une phase d’allumage et les m´ethodes DNS ne sont pas applicables, `a l’heure actuelle, `a des g´eom´etries r´eelles. Cette constatation est le point de d´epart du d´eveloppement d’AVBP: code parall`ele, volumes finis, explicite, capable de r´ealiser

des simulations aux grandes ´echelles d’´ecoulements r´eactifs dans des g´eom´etries complexes. Cependant, cet outil ne permettait pas jusqu’alors de prendre en compte la phase dispers´ee et se limitait donc `a des applications o`u le carburant ´etait inject´e sous forme gazeuse ou dont le temps d’´evaporation du spray liquide ´etait tr`es court.

Partie I : d´eveloppement d’une extension aux probl´ematiques diphasiques

Apr`es un ´etat de l’art des ´etudes men´ees sur la combustion diphasique pr´esent´e dans le Chap. 1, de nombreux efforts restent `a faire pour comprendre et mod´eliser les ph´enom`enes instationnaires au sein d’une chambre de combustion r´eelle aliment´ee en carburant liquide. Mˆeme si la combustion turbulente est ´etudi´ee depuis de nombreuses d´ecennies, la combustion de spray n’a pas atteint une telle maturit´e et peu d’outils SGE sont d´edi´es `a ce type d’´ecoulement, notamment avec un formalisme eul´erien.

L’outil d´evelopp´e dans ce travail de th`ese constitue un ensemble complet de mod`eles pour la si-mulation aux grandes ´echelles d’´ecoulements r´eactifs au sein de g´eom´etries complexes. Cependant, ce formalisme fait apparaˆıtre des besoins en mod´elisation : atomisation primaire et secondaire, mod`ele de sous-maille pour la phase dispers´ee, mouvement d´ecorr´el´e, granulom´etrie multi-classes, interaction du spray avec les parois... En conclusion, l’outil d´evelopp´e et valid´e dans la partie I permet de r´ealiser une ´etude de faisabilit´e de la simulation d’´ecoulements fortement instationnaires tels que des s´equences d’allumage, en prenant en compte la phase dispers´ee.

CONCLUSION GEN´ ERALE ET PERSPECTIVES´

Partie II : application du solveur LES diphasique `a une chambre industrielle

Le Chap. 6 pr´esente une ´etude SGE `a froid et `a chaud d’un foyer r´eel. L’´etude `a froid porte sur la dis-persion et l’´evaporation d’un spray et met en ´evidence l’influence de la dynamique de la phase porteuse sur le mouvement des petites gouttes. Le processus d’´evaporation produit du carburant gazeux principa-lement dans les zones o`u la dynamique de la phase porteuse maintient les gouttes et augmente leur temps de r´esidence. Le lien entre dynamique des deux phases, ´evaporation et m´elange est ainsi d´etaill´e. L’´etude `a chaud permet de d´eterminer les m´ecanismes de stabilisation de la flamme diphasique. Les r´esultats ins-tantan´es montrent l’importance du processus d’´evaporation dans la structure partiellement pr´em´elang´ee de la flamme, tandis que les r´esultats moyenn´es montrent l’influence de l’´evaporation sur les diagrammes de fraction de m´elange et valident les r´esultats par comparaison avec le code RANS N3S-NATUR.

La phase d’allumage est un enjeu crucial pour la conception des foyers de demain et demeure une probl´ematique scientifique importante. Sa simulation num´erique pose de nombreux probl`emes notam-ment du fait des tr`es faibles ´echelles de temps. Le code AVBPest explicite et poss`ede un pas de temps li´e au CFL de l’ordre du dixi`eme de microseconde ce qui en fait un outil appropri´e pour cette probl´ematique. Le Chap. 7 pr´esente les premiers r´esultats d’allumage obtenus en g´eom´etrie r´eelle en prenant en compte la dispersion et l’´evaporation de la phase dispers´ee. Les ´etapes d’un allumage r´eussi sont pr´esent´ees : la naissance d’un noyau sph´erique empli de gaz chauds autour de la bougie et la propagation de cette flamme sph´erique `a l’ensemble du brˆuleur pour finalement atteindre une position stable proche de l’in-jection liquide. Ce r´esultat d´emontre la faisabilit´e de l’outil LES `a ce type d’applications et sert de base `a des ´etudes plus quantitatives.

Le Chap. 8 s’int´eresse `a une autre probl´ematique industrielle forte li´ee au d´eveloppement de mo-teurs moins polluants de type LPP : la pr´ediction pr´ecise des instabilit´es de combustion. Bien que de nombreuses publications exp´erimentales et num´eriques sur le sujet sont disponibles dans la litt´erature, son ´etude SGE en contexte diphasique est une ´etape suppl´ementaire dans la compr´ehension physique des interactions spray/flamme/acoustique. Les r´esultats obtenus dans cette th`ese montrent la relation flamme/acoustique mais ´egalement spray/acoustique. De plus, le m´elange est perturb´e `a la fois par la dynamique de la phase porteuse et par le processus d’´evaporation, tous deux influenc´es par la perturba-tion sinuso¨ıdale. Cette interacperturba-tion ´etait ´egalement visible dans l’´etude de l’allumage, le d´epˆot d’´energie produisant une onde acoustique se propageant au sein du foyer et influenc¸ant la dynamique de la flamme de spray.

Bilan

Ce travail de th`ese apporte des ´el´ements suppl´ementaires pour la compr´ehension des ph´enom`enes mis en jeu dans une flamme diphasique turbulente et leurs interactions. La simulation aux grandes ´echelles am´eliore la compr´ehension des ph´enom`enes instationnaires et permet d’appr´ehender les mod`eles `a mettre en oeuvre pour mieux capturer la physique. Les mod`eles existants, dont une validation a ´et´e pr´esent´ee, ont permis d’expliquer la naissance et la stabilisation d’une flamme de spray, ainsi que sa r´eponse `a une perturbation acoustique.

Conclusion g´en´erale et perspectives Perspectives

Dans le formalisme eul´erien pr´esent´e dans ce travail de th`ese, de nombreuses hypoth`eses simplifi-catrices ont ´et´e prises en compte et m´eritent des ´etudes plus approfondies. La prise en compte du ca-ract`ere polydisperse de la phase dispers´ee est une ´etape suppl´ementaire indispensable et a fait l’objet, en formalisme eul´erien, d’un travail de th`ese par Mossa [63] qui a conduit `a une ´equation de transport suppl´ementaire d’une variable li´ee au diam`etre moyen et `a la densit´e de surface de l’interface. Les inter-actions particule-particule et particule-paroi sont ´egalement `a prendre en compte et de nombreux mod`eles disponibles dans la litt´erature en approche lagrangienne doivent ˆetre adapt´es au formalisme eul´erien et pris en compte dans le solveur SGE. Les mod`eles d´evelopp´es pour mod´eliser le mouvement d´ecorr´el´e doivent ˆetre valid´es sur une configuration industrielle. Les mod`eles de sous-maille doivent ˆetre appro-fondis pour une application industrielle [81]. L’´etude pr´ecise de l’atomisation primaire a conduit, en formalisme lagrangien, `a des mod`eles qui doivent ˆetre adapt´es en formalisme eul´erien pour d´eterminer avec pr´ecision la granulom´etrie initiale du spray. La probl´ematique de l’allumage de la chambre de com-bustion annulaire compl`ete, c’est-`a-dire de la propagation du front de flamme du secteur pilote vers les autres secteurs, reste ´egalement `a ´etudier [7].

CONCLUSION GEN´ ERALE ET PERSPECTIVES´

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[93] SELLE, L., NICOUD, F., AND POINSOT, T. The actual impedance of non-reflecting boundary conditions : implications for the computation of resonators. AIAA Journal 42, 5 (2004), 958–964.

Bibliographie [94] SENGISSEN, A., POINSOT, T., VAN KAMPEN, J., AND KOK, J. Response of a swirled

non-premixed burner to fuel flow rate modulation. In The Cyprus International Symposium on Complex Effects in Large Eddy Simulation(2005).

[95] SIMONIN, O. Continuum modelling of dispersed two-phase flows. In Combustion and turbulence in two phase flows, VKI Lecture Series 1996-02(Von Karman Institute for Fluid Dynamics, 1996), Von Karman Institute for Fluid Dynamics.

[96] SIMONIN, O., FEVRIER, P., ANDLAVIEVILLE, J. On the spatial distribution of heavy-particle velocities in turbulent flow : from continuous field to particulate chaos. Journal of Turbulence 3, 040 (2002).

[97] SMAGORINSKY, J. General circulation experiments with the primitive equations : 1. the basic experiment. Monthly Weather Review 91, 3 (1963), 99–164.

[98] SOMMERFELD, M.,ANDQIU, H. H. Detailed measurements in a swirling particulate two-phase flow by a phase-doppler anemometer. International Journal of Heat and Fluid Flow 12, 1 (1991). [99] SOMMERFELD, M.,ANDQIU, H. H. Experimental studies of spray evaporation in turbulent flow.

International Journal of Heat and Fluid Flow 19(1998), 10–22.

[100] SPALDING, D. Mixing and chemical reaction in steady confined turbulent flames. In 13th Symp. (Int.) on Combustion(1971), The Combustion Institute, Pittsburgh, pp. 649–657.

[101] SPALDING, D. B. The combustion of liquid fuels. In 4th Symp. (Int.) on Combustion. The Combustion Institute, Pittsburgh, 1953, pp. 847–864.

[102] SPALDING, D. B. Development of the eddy-break-up model of turbulent combustion. In 16th Symp. (Int.) on Combustion(1976), The Combustion Institute, pp. 1657–1663.

[103] STULL, D., ANDPROPHET, H. Janaf thermochemical tables, 2nd edition. Tech. Rep. NSRDS-NBS 37, US National Bureau of Standards, 1971.

[104] SUZUKI, T., ANDCHIU, H. Multi droplet combustion on liquid propellants. In Proceedings of the Ninth International Symposium on Space and Technology and Science(1971), pp. 145–154. [105] TANGUY, S. D´eveloppement d’une m´ethode de suivi d’interface, applications aux ´ecoulements

diphasiques. PhD thesis, Universit´e de Rouen, 2004.

[106] THOMPSON, K. Time dependent boudary conditions for hyperbolic systems. Journal of Compu-tational Physics 68(1987), 1–24.

[107] WIDMANN, J.,ANDPRESSER, C. A benchmark experimental database for multiphase

combus-tion model input and validacombus-tion. Combuscombus-tion and Flame , 129 (2002), 47–86. [108] WILLIAMS, F. Combustion theory. Benjamin Cummings, Menlo Park, CA, 1985.

BIBLIOGRAPHIE

Troisi`eme partie

Table des Mati`eres

A Solution analytique d’une flamme diphasique 1D 137

B LES of turbulent spray combustion in aeronautical gas turbines [67] 147 C LES of steady spray flame and ignition sequences in aeronautical combustors [68] 169

TABLE DES MATI `ERES

Annexe A

Solution analytique d’une flamme

diphasique 1D

Reproduction du Chap. 2 du rapport de stage de C. Saulnier [85] relatif `a la d´etermination de la solu-tion analytique d’une flamme de pr´em´elange, laminaire, diphasique, homog`ene, monodimensionnelle.

SOLUTION ANALYTIQUE

Chapitre 2

Combustion diphasique stationnaire :

Cas de l’´

evaporation totale avant la

flamme

Nous nous placerons dans le cas d’une flamme de pr´em´elange diphasique1_{, mono-dimensionnelle}

et stationnaire. On supposera que la flamme se trouve derri`ere la zone d’´evaporation, c’est-`a-dire que lorsque le fuel brˆule, il est enti`erement sous forme gazeuse (voir Fig. 2.1). Le probl`eme peut alors ˆetre divis´e en deux parties avec d’une part l’´etude de la zone d’´evaporation et d’autre part l’´etude d’une flamme gazeuse de pr´em´elange.

d’evaporation zone gazeuse phase flamme de premelange gaz brules x_in x_ev x_out

Fig.2.1 – Evaporation et flamme de pr´em´elange

2.1

Zone d’´

evaporation

Les r´esultats pr´esent´es ci-dessous sont inspir´es des travaux de Lin et Law (cf. [11] et [12]).

2.1.1 Position du probl`eme

Nous ´etudierons la zone d’´evaporation dans la limite des hypoth`eses suivantes :

– L’´ecoulement est monodimensionnel, laminaire.

– L’´ecoulement est stationnaire, les d´eriv´ees temporelles seront donc consid´er´ees comme nulles. Cette hypoth`ese et la pr´ec´edente permettent de conclure que toutes les grandeurs ne d´ependent que d’une coordonn´ee d’espace x. Sous cette deuxi`eme hypoth`ese, nous remarquons

1_{Le fuel et l’oxydant pr´esent dans l’air sont inject´es ensemble sous forme gazeuse avec des inclusions de fuel liquide}

10

Rapport de stage de C. Saulnier

aussi que l’´equation de conservation de la quantit´e de mouvement Eq. 1.12 n’est pas n´ecessaire `a la r´esolution, puisque cette ´equation permet, dans le cas stationnaire, de r´esoudre le champ de pression, ce qui n’est pas l’objectif du travail effectu´e (voir [16]).

– La conduction dans le liquide est infinie, ce qui implique que la temp´erature `a l’int´erieur des gouttes est uniforme (cf. Uniform Temperature model [13]).

– Le liquide est `a la temp´erature d’´ebullition, la temp´erature du liquide n’´evolue pas pendant l’´ecoulement : Tl= Teb= constante. L’enthalpie du liquide est constante, on n’a plus

besoin de l’´equation de l’´energie Eq. 1.13 dans la phase liquide.

– Les gouttes sont suffisamment petites, de mani`ere `a ce que la force de traˆın´ee soit n´egligeable : le temps caract´eristique des gouttes est petit devant le temps caract´eristique du fluide. Ce qui implique que : Ul(x) = Ug(x) = Um(x). Par la suite, pour simplifier l’´ecriture,

on notera cette vitesse U (x).

– Le spray est tr`es dilu´e, ce qui signifie que αlest tr`es faible (de l’ordre de 10−5 ou moins).

L’hypoth`ese est r´ealiste : dans un injecteur typique, la fraction volumique du liquide est d’environ 2.10−5 _{lorsque le fuel et l’oxydant sont en proportion stochiom´etrique. Puisque}

αl+ αg = 1, on pourra consid´erer que αg ∼ 1 (cf. [11]). Avec cette hypoth`ese, on pourra

´ecrire ρm = ρg+ αlρl. Pour all´eger l’´ecriture, dans la suite du rapport, on notera ρ la densit´e

du m´elange.

– La phase gazeuse est multi-esp`ece, nous consid`ererons donc l’´equation 1.19.

– On consid`erera que le m´elange est pauvre en fuel, seule l’´equation de conservation du fuel sera prise en compte. La fraction massique d’oxydant peut ˆetre calcul´ee de la mˆeme fa¸con que la fraction massique de fuel, mais l’´evolution de YO n’est pas primordial dans cette

´etude. (voir [16])

– La phase gazeuse est consid´er´ee comme un gaz parfait.

– L’´ecoulement est isobare, ce qui signifie, avec l’hypoth`ese pr´ec´edente que le produit ρgTg

est constant.

– Le spalding de masse est ´egal au spalding de temp´erature(cf. [10]), on notera donc

B = BM = BT =

Cpg(Tev− Tζ)

∆H (2.1)

On sait aussi que Γl= −Γg(cf. Eq. 1.11). Dans la suite de ce rapport, on notera Γ = Γg= −Γl.

Sous les hypoth`eses ´enum´er´ees ci-dessus, les ´equations `a r´esoudre sont : d dx(ρgU ) = Γ (2.2) d dx(ρgU YF) − [ρD]g d2YF dx2 = Γ (2.3) d dx(ρgU CpgTg) − λ d2Tg dx2 = CplTebΓ (2.4) ρgTg = pWF R (2.5)

De plus, les ´equations 1.17 et 1.22 deviennent :

ρU = F = constante = ρinUin (2.6)

nU = N = constante = ninUin (2.7)

11

SOLUTION ANALYTIQUE

2.1.2 Mise en place du syst`eme `a r´esoudre

On d´efinit la variable Z = ρg

ρ (cf. [11] et [12]). Les ´equations 2.2, 2.3 et 2.4 deviennent : ZdZ dx = A 1 Tg (1 − Zin)2/3(1 − Z)1/3 (2.8) d dx(ZYF) − [ρD]g F d2YF dx2 = dZ dx (2.9) d dx(ZTg) − λ CpgF d2Tg dx2 = CplTeb Cpg dZ dx (2.10) avec A = 3Sh [ρλ]ζln (1 + B) pW 2ρlRr2inF (2.11)

On effectue ensuite le changement de variable β = 1 − Z

1 − Zin . Les ´equations 2.8, 2.9 et 2.10 deviennent (cf. annexe C) : −dβ dx+ (1 − Zin) β dβ dx = A Tgβ 1/3 _(2.12) dYF dx − (1 − Zin) d dx(βYF) − [ρD]g F d2YF dx2 = − (1 − Zin) dβ dx (2.13) dTg dx − (1 − Zin) d dx(βTg) − λ CpgF d2_T g dx2 = − CplTeb Cpg (1 − Zin) dβ dx (2.14)

avec les conditions limites suivantes :

β (x = 0) = 1 ; β (x = xev) = 0 (2.15) YF(x = 0) = YF,in ; dYF dx (x = xev) = 0 (2.16) Tg(x = 0) = Tg,in ; dTg dx (x = xev) = 0 (2.17)

2.1.3 D´eveloppement par rapport `a un petit param`etre

Puisque nous sommes dans l’hypoth`ese d’un spray tr`es dilu´e, on peut consid´erer que Zin est

proche de 1. On peut donc ´ecrire que Zin= 1 − εγ, avec εγ est tr`es petit devant 1. ε est le rapport

entre la temp´erature de flamme au carr´e et la temp´erature d’activation :

εγ = 1 − Zin ; ε =

T_f2 Ta

et γ = 1 − Zin

ε (2.18)

On d´eveloppe les variable β, YF et Tg par rapport `a εγ selon la m´ethode de Wentzel, Kramers

et Brillouin connue sous le nom de d´eveloppement de WKB (cf. [3] et [2]) :

β = β0+ εγβ1+ O ³ (εγ)2´ (2.19) YF = YF,0+ εγYF,1+ O ³ (εγ)2´ (2.20) Tg = Tg,0+ εγTg,1+ O ³ (εγ)2´ (2.21) 12

Rapport de stage de C. Saulnier

Ce d´eveloppement par rapport `a un petit param`etre permet de d´ecoupler les ´equations obtenues pr´ec´edemment. On d´eveloppe l’´equation 2.12 `a l’ordre 0, ce qui donne :

dβ0

dx +

A Tg,0

β₀1/3+ O (εγ) = 0 (2.22)

Le d´eveloppement des ´equations 2.13 et 2.14 `a l’ordre 1 (cf. annexe C.3) : · dYF,0 dx − [ρD]g F d2YF,0 dx2 ¸ + εγ· dYF,1 dx − d (β0YF,0) dx − [ρD]g F d2YF,1 dx2 + dβ0 dx ¸ +O³(εγ)2´ = 0 (2.23) · dTg,0 dx − λ CpgF d2Tg,0 dx2 ¸ + εγ· dTg,1 dx − d (β0Tg,0) dx − λ CpgF d2Tg,1 dx2 + CplTeb Cpg dβ0 dx ¸ +O³(εγ)2´ = 0 (2.24) 2.1.4 R´esolution `a l’ordre 0

On r´esoud les ´equations 2.23 et 2.24 `a l’ordre 0, ce qui revient `a r´esoudre : d2_Y F,0 dx2 − F [ρD]_g dYF,0 dx = 0 (2.25) d2Tg,0 dx2 − CpgF λ dTg,0 dx = 0 (2.26)

Etant donn´e que ces ´equations ne font pas intervenir des termes d’´evaporation, on a : YF,0(x = 0) = YF,0(x = xev) = YF,in

Tg,0(x = 0) = Tg,0(x = xev) = Tg,in

On en conclut que les ordres 0 de YF et Tgsont constants (inversion partielle du syst`eme compos´e

des ´equations 2.25 et 2.26) :

YF,0(x) = YF,in (2.27)

Tg,0(x) = Tg,in (2.28)

On peut alors r´esoudre l’´equation 2.22 `a l’ordre 0 qui devient : dβ0 dx + A Tg,in β<sub>0</sub>1/3 = 0 β0(x) = µ − 2A 3Tg,in x + 1 ¶3/2 (2.29) On en d´eduit une approximation `a l’ordre 0 de Z :

Z(x) = 1 − (1 − Zin) µ − 2A 3Tg,in x + 1 ¶3/2 (2.30) 13 SOLUTION ANALYTIQUE 142

2.1.5 R´esolution `a l’ordre 1

On peut alors r´esoudre les ´equations 2.23 et 2.24 `a l’ordre 1. Soit `a r´esoudre : d2_Y F,1 dx2 − F [ρD]_g dYF,1 dx = F [ρD]_g(1 − YF,in) dβ0 dx (2.31) d2Tg,1 dx2 − CpgF λ dTg,1 dx = (CplTeb− CpgTg,in) F λ dβ0 dx (2.32)

avec les conditions limites :

YF,1(x = 0) = 0 ; dYF,1 dx (x = xev) = 0 (2.33) Tg,1(x = 0) = 0 ; dTg,1 dx (x = xev) = 0 (2.34)

La r´esolution de ces ´equations donnent : YF,1(x) = F [ρD]_g(1 − YF,in) ·Z xev 0 e− F [ρD]gx0_β0(x0_)dx0_−e[ρD]gF Z xev x e− F [ρD]gx0_β0(x0_)dx0 ¸ (2.35) Tg,1(x) = CpgF λ µ Cpl Cpg Teb− Tg,in ¶·Z xev 0 e−Cpg F λ x0β₀(x0)dx0−e Cpg F λ Z xev x e−Cpg F λ x0β₀(x0)dx0 ¸ (2.36) On obtient ensuite les expressions de YF et Tg en faisant la somme de l’ordre 0 et de l’ordre 1

multipli´e par εγ = 1 − Zin(voir calculs `a la section C.4) :

YF(x) = YF,in (2.37) + F [ρD]_g(1 − YF,in) (1 − Zin) ·Z xev 0 e− F [ρD]gx0_β0(x0 )dx0− e F [ρD]g Z xev x e− F [ρD]gx0_β0(x0 )dx0 ¸ Tg(x) = Tg,in (2.38) + CpgF λ µCpl CpgTeb−Tg,in ¶ (1 − Zin) ·Z xev 0 e−Cpg Fλ x0β₀(x0)dx0−e Cpg F λ Z xev x e−Cpg Fλ x0β₀(x0)dx0 ¸

Remarque : En ayant r´esolu l’´equation 2.22 `a l’ordre 0, on peut r´esoudre analytiquement les ´equations 2.13 et 2.14. Cette r´esolution a ´et´e effectu´ee et cod´ee pendant le stage. Cependant elle se comportait tr`es mal num´eriquement `a cause de l’addition de grandes exponentielles (de l’ordre de 1060).

2.2

Solution analytique approch´

ee pour une flamme de pr´

em´

elange

En x = xev, la totalit´e du fuel liquide est ´evapor´ee (cf. Fig. 2.1). Ceci nous am`ene `a ´etudier

une flamme gazeuse de pr´em´elange stationnaire et laminaire. Tous les r´esultats pr´esent´es ci-dessous sont d´eriv´es de [16].

2.2.1 Hypoth`eses de travail

Cette flamme sera ´etudi´ee dans le cadre des hypoth`eses suivantes :

– On se placera dans le cadre d’une chimie simple avec une seule r´eaction irr´eversible : Fuel + Oxydant −→ Produits

14

Rapport de stage de C. Saulnier

– On supposera que toutes les esp`eces ont la mˆeme masse molaire, le mˆeme Cp et le mˆeme

coefficient de diffusion D, donc le mˆeme nombre de Lewis : on d´efinit le nombre de Lewis not´e Le comme le rapport entre la diffusion des esp`eces et de la chaleur, c’est-`a-dire : Le = λg

ρgCpgD.

– Le taux de r´eaction ˙ωk d´epend du taux de progression auquel la r´eaction simple progresse.

On peut alors ´ecrire que ˙ωT = −Q ˙ωF, o`u Q est la chaleur de r´eaction par unit´e de masse

(cf [16]).

– La flamme est pauvre.

Ces hypoth`eses paraissent tr`es fortes, mais elles sont raisonnables dans la plupart des cas.

2.2.2 Mise en ´equation

Les ´equations r´egissant une flamme de pr´em´elange stationnaire sont (cf. [16]), dans le cas isobare dilatable : ρgU = constante = ρg,evUev (2.39) ρg,evUev dYF,g dx = d dx µ [ρD]dYF,g dx ¶ + ˙ωF (2.40) ρg,evUevCpg dTg dx = d dx µ λdTg dx ¶ − Q ˙ωF (2.41)

o`u ˙ωF est le taux de r´eaction du fuel et Q est la chaleur de r´eaction par unit´e de masse.

2.2.3 Equivalence de la temp´erature et de la fraction massique de fuel

Une simple int´egration des ´equations 2.40 et 2.41 entre x = xev et x = ∞ permet d’´ecrire que

(cf. [16]) : Tg,out= Tg,ev+ QYF,ev/Cpg.

On d´efinit alors les variables r´eduites :

Y = YF YF,ev ; Θ = Cpg(Tg− Tg,ev) QYF,ev = Tg− Tg,ev Tg,out− Tg,ev (2.42) En supposant que le nombre de Lewis est ´egal `a 1, on montre que (cf. [16]) : Y + Θ = 1 . Il suffit donc de r´esoudre l’´equation de la temp´erature pour r´esoudre enti`erement le syst`eme.

2.2.4 Taux de r´eaction simplifi´e

Pour pouvoir r´esoudre analytiquement les ´equations d’une flamme gazeuse de pr´em´elange, Echekki et Ferziger ont propos´e un mod`ele de taux de r´eaction plus simple que le mod`ele d’Arrhe-nius (cf. [4]), not´e ˙ωEF. Leur mod`ele propose de consid´erer le taux de r´eaction comme une fonction

triangulaire en fonction de la temp´erature r´eduite qui s’´ecrit :

˙ωEF = ρg,evRr(1 − Θ) H (Θ − Θc) (2.43)

o`u Θc = 1 − 1_β est donn´e par l’´etude asymptotique (cf. [22]), H est la fonction d’Heaviside et Rr

est une constante du mod`ele. La r´esolution donne : Θ(x) =          (1 − β) exp µ _λ g ρg,evCpgsL ¶ si x < xf 1 −1 βexp à ρg,evCpgsL 2λg à 1 − µ 1 + 4 Rrλg ρg,evCpgs2L ¶1/2! x ! si x > xf (2.44)

sL est la vitesse de flamme calcul´ee `a la section 2.3.2.

15

SOLUTION ANALYTIQUE

2.3

Raccordement entre ´

evaporation et combustion

2.3.1 Distance caract´eristique d’´evaporation

A partir de l’´equation 2.30, on peut d´eterminer rapidement la distance d’´evaporation, c’est-`a-dire la distance `a laquelle la totalit´e du liquide est ´evapor´ee. En effet, on a la condition Z(x = xev) = 1,

ce qui revient `a (en utilisant Eq. 2.11) : xev=

ρlRr2inF Tg,in Sh[ρλ]ζln(1+B)pW

. Or la loi des gaz parfaits (Eq. 2.5) permet d’´ecrire que Tg,inR

pW = 1 ρg,in, donc : xev = ρinuin ρg,in ρld2in 4Sh [ρλ]_ζln(1 + B) (2.45)

On retrouve alors le temps d’´evaporation caract´eristique d´emontr´e par Kuo (cf. [10]), qui cor-respond `a la loi du d2 _:

tev =

ρld2in

4Sh [ρλ]_ζln(1 + B) (2.46)

Dans cette expression la vitesse d’entr´ee uinn’est pas connue a priori, elle est calcul´ee de mani`ere

`a ce que la zone d’´evaporation et la flamme soient stationnaires.

2.3.2 Calcul de la vitesse de flamme

La vitesse de flamme est calcul´ee grˆace `a la formule d´emontr´ee par Mitani (cf. [14]) :

sL= " 2λgνF(νOνF)nOρng,Fout+nORrTg,β1outYnF+nO −1 F,ev Le nF F LenOO ρ2 g,evCpgW nF+nO−1 nF β nF+nO+1 #1/2 G(nF, nO)1/2exp µ −β 2α ¶ (2.47)

o`u la fonction G est d´efinie par : G(nF, nO) =

Z ∞ 0 ξnF µ ξ + Φ − 1 LeO ¶nO e−ξdξ.

νF et νO sont les coefficients stochiom´etriques respectivement du fuel et de l’oxydant, α =

(Tout− Tev)/Toutet β = αTα1/Tout. nF et nO sont des constantes pas n´ecessairement ´egales `a νF et

νO. On se reportera `a [14] et [16] pour les d´etails.

2.3.3 Obtention d’un syst`eme stationnaire

Pour que le syst`eme ´etudi´e, compos´e d’une zone d’´evaporation et d’une zone de combustion soit stationnaire, il faut que la vitesse en fin de zone d’´evaporation (c’est-`a-dire la vitesse en x = xev)

soit ´egale `a la vitesse de flamme sL calcul´ee `a l’´equation 2.47. Pour ce, d’apr`es l’´equation 2.6, il

faut que la vitesse d’entr´ee des gouttes et du gaz soit telle que : uin=

ρg,evsL

ρg,in

(2.48) C’est ainsi que la vitesse d’entr´ee du fluide est calcul´ee.

16

Rapport de stage de C. Saulnier

SOLUTION ANALYTIQUE

Annexe B

LES of turbulent spray combustion in

aeronautical gas turbines [67]

Proceedings of the ECCOMAS Thematic Conference on Computational Combustion, 2005

St´ephane PASCAUD, Matthieu BOILEAU, B´en´edicte CUENOT CERFACS, Toulouse

Thierry POINSOT IMFT - CNRS, Toulouse

Abstract : Reduction of pollutants emission or altitude re-ignition, strongly influenced by turbulent mixing and fuel spray evaporation, are critical issues for aeronautical gas turbine design. To understand unsteady spray combustion in industrial burners, Large Eddy Simulation (LES) is a unique and powerful tool. Its potential has been widely demonstrated for turbulent gaseous cold and reacting flows. Its exten-sion to two-phase turbulent reacting flows is an obvious research path for the future. In the present work, an Euler-Euler formulation, together with a turbulent sub-grid scale model and a turbulent combustion model, is used to solve the conservation equations in each phase and the exchange source terms for mass, momentum and heat transfer. A stabilised turbulent spray flame in an aeronautical gas turbine is consi-dered for application. Due to complex geometry, an unstructured mesh is used. The partially premixed flame structure revealed by LES is detailed. In particular, the role of evaporation and recirculation zones on the stabilisation mechanism is emphasized. Finally, LES and RANS results are compared.

CONFERENCE´ ECCOMAS

LARGE EDDY SIMULATION

OF TURBULENT SPRAY COMBUSTION

IN AERONAUTICAL GAS TURBINES

S. Pascaud

, M. Boileau

, B. Cuenot

and T. Poinsot

2_{IMFT, Toulouse, France}

Abstract Reduction of pollutants emission or altitude re-ignition, strongly influenced by turbulent mixing and fuel spray evaporation, are critical issues for aeronautical gas turbine design. To understand unsteady spray combustion in industrial burners, Large Eddy Simulation (LES) is a unique and powerful tool. Its potential has been widely demonstrated for turbulent gaseous cold and reacting flows. Its extension to two-phase turbulent reacting flows is an obvious research path for the future. In the present work, an Euler-Euler formulation, together with a turbulent sub-grid scale model and a turbulent combustion model, is used to solve the conservation equations in each phase and the exchange source terms for mass, momentum and heat transfer. A stabilised turbulent spray flame in an aeronautical gas turbine is considered for application. Due to complex geometry, an unstructured mesh is used. The partially premixed flame structure revealed by LES is detailed. In particular, the role of evaporation and recirculation zones on the stabilisation mechanism is emphasized. Finally, LES and RANS results are compared.

Keywords Large Eddy Simulation, spray, evaporation, combustion, complex geometry

CONTEXT

Large Eddy Simulations (LES) are rapidly becoming standard tools to study

combustion in many modern combustion devices [1-6]. However, even though

multiple proofs of the validity of the LES concept have been obtained for

gaseous combustion, LES for two-phase flow combustion remains a much more

difficult topic for which very few recent studies are available [7-10].

Consid-ering that most fuels used for aeronautical applications are liquid, the need

for two-phase combustion LES tools is obvious. The turbulent spray

combus-tion involves several different physical phenomena such as particle dispersion,

vaporisation, mixing and combustion. The dispersion is highly linked to the

characteristics of the spray. Experimental studies on dispersion of solid

parti-cles [11] and vaporised droplets [12] have shown the influence of the Stokes

number on the droplet trajectories. The isolated vaporising droplet model is a

simple but useful description of the vaporising spray, required to understand

the physics [13-17]. Different approaches have been used to analyse

vaporis-ing turbulent sprays : experiments [12, 18, 19], Direct Numerical Simulations

(DNS) [20], LES [21] and Reynolds Averaged Navier Stokes (RANS) [18] or

Conf´erence ECCOMAS

turbulent spray combustion : experiments [22, 23], DNS [24-26], LES [27] and

RANS [28, 29]. These recent studies exhibit the complexity of the multiple

interactions between turbulence, two-phase flows and combustion. The aim of

this paper is to use LES to analyse these unsteady correlations.

Modelling of the dispersed phase raises the question of the choice of the

method used to couple the liquid and the gas phases in a LES formulation. In

the Lagrange approach, the liquid phase is computed using a particle tracking

method in which each droplet (or group of droplets) is computed

individu-ally giving its trajectory, velocity, temperature and diameter [7]. In the Euler

approach, the liquid phase is homogeneized and solved for using a set of

con-servation equations for the liquid volume fraction, the liquid phase velocity and

temperature, and the first/second order moments of the size distribution [30,

31]. The Lagrange framework is used in many applications because phenomena

like droplet break-up, dispersion, interaction with walls and droplet/droplet

in-teraction are easier to model. This choice may be revisited for the computation

of unsteady spray combustion in complex geometry. The first argument comes

from the numerics : LES are high CPU consumers; run on parallel computers.

However, a significant amount of work is still needed to implement efficiently

Lagrange algorithms on parallel computers [10]. Moreover, the cost of the

Lagrange treatment increases rapidly with the number of droplets while the

parallel efficiency decreases. On the other hand, Euler techniques are directly

parallelised with the same algorithms than the gas phase computations. Another

major issue for Lagrangian reacting two-phase LES is the number of droplets

per cell required to provide a correct description of the liquid phase. LES being

less dissipative than RANS, enough Lagrangian droplets must be used at each

time step in each cell to provide a smooth and accurate continuous field of

fuel mass fraction. Because the fuel vapour distribution, directly produced by

the discrete droplet evaporation source terms, controls the propagation of the

front [32, 33], this is crucial for two-phase flame computations. Very limited

experience on this question is available today but it is likely that combustion

requires much more particles than usually used for dispersion or evaporation

studies, leading to uncontrolled CPU costs. Another advantage of Lagrange

methods is that they naturally allow to track multisize droplet clouds. Size

dis-tribution controls the flame regime in many instances and must be taken into

account. However, recent studies have demonstrated that Euler techniques may

also be extended to include multisize liquid sprays [20, 34, 35]. Finally, a more

general question regarding LES of two-phase flow combustion is the difficulty

of specifying inlet conditions for the liquid phase: close to the fuel injectors,

the droplet velocity and size distributions are not known with precision due

to the complexity of primary atomization. The question then arises whether

it is worth computing the dispersion of the fuel droplets with a high-precision

Lagrange method while using very approximate injection conditions. Close to

the fuel injectors, the liquid phase is even not organized as droplets but more

as liquid blobs [36] showing that the Lagrange method cannot even be used

there. In these high-loading zones where no real drop exists, using the Euler

approach may actually be more compatible with the physics of the liquid phase

and the large uncertainties related to the fuel injection conditions.

In this paper, the Euler framework is chosen for LES. The parallel solver

AVBP is described and the computation of a turbulent stable spray flame at

atmospheric pressure in an industrial gas turbine sector is presented.

EQUATIONS

The Euler approach leads to similar conservation equations for both the

carrier phase and the dispersed phase, on which LES filtering is then applied [37,

38]. For the carrier phase, the classical LES set of equations is recovered. It

is coupled with the dispersed phase equations through phase exchange terms.

The Favre averaged (defined as : ˜

φ =

) governing equations for gas and

liquid mass and species conservation, momentum and total energy read :

∂

∂ t

(ρ) +

∂

∂ x

(ρ ˜

u

)

=

Γ

¯

(1)

∂

∂ t

(ρ ˜

Y

) +

∂

∂ x

(ρ ˜

Y

u

˜

)

=

∂

∂ x

( ¯

ρ ¯

D

∂ ˜

Y

∂ x

) −

∂

∂ x

( ¯

ρY

) + ¯˙

ω

+ ¯

Γ

δ

(2)

∂

∂ t

(ρ ˜

u

) +

∂

∂ x

(ρ ˜

u

u

˜

)

=

−

∂

∂ x

¯

p +

∂

∂ x

¯

τ

−

∂

∂ x

( ¯

ρ u

) + ¯

I

(3)

∂

∂ t

(ρ ˜

E) +

∂

∂ x

(ρ ˜

E ˜

u

)

=

−

∂

∂ x

¯

q

+

∂

∂ x

( ¯

τ

u

˜

) −

∂

∂ x

( ¯

ρ E

)

+

ω

¯˙

+ ¯

I

u

˜

+ ¯

Π

(4)

∂

∂ t

(α

ρ

) +

∂

∂ x

(α

ρ

u

˜

)

=

−

Γ

(5)

∂

∂ t

(α

ρ

u

˜

) +

∂

∂ x

(α

ρ

u

˜

u

˜

)

=

−I

(6)

∂

∂ t

(α

ρ

˜h

) +

∂

∂ x

(α

ρ

˜h

u

˜

)

=

−

Π

(7)

∂

∂ t

(n

) +

∂

∂ x

(n

u

˜

)

=

0

(8)

In these equations, ρ (ρ

) is the gas (liquid) mass density, Y

is the mass

fraction of the species k and u

(u

) is the gas (liquid) velocity. E is the total

Conf´erence ECCOMAS

non chemical energy of the carrier phase defined as : E = e

+

u

u

where e

is the sensible energy and h

is the liquid sensible enthalpy. Finally, α

is the

volumic liquid fraction and n is the droplet density.

The stress tensor ¯

τ

is assumed newtonian and the heat diffusion ¯

q

is

determined by the Fourier law. The unresolved turbulent fluxes Y

= g

u

Y

−

˜

V

Y

˜

and E

= g

u

E − ˜

u

E are estimated by a classical gradient diffusion

˜

assumption. The subgrid stresses u

=

u

_g

u

− ˜

u

u

˜

are modeled with the

WALE formulation [39].

The mass transfer

Γ

is expressed by the Spalding model [40] for spherical

droplets :

Γ

= α

6

d

Sh(ρD

)ln(1 + B

)

(9)

where the Sherwood Number Sh is equal to 2 and the Spalding number is

de-fined by : B

= (Y

−Y

)/(1 −Y

) with

denoting the value at interface.

The momentum transfer is defined as I

= F

+ u

Γ

where F

is the

stan-dard drag force of a spherical droplet equal to −α

4d

ρ

|u

− u

|(u

− u

).

The drag coefficient C

is, for Re

≥ 1000 , taken constant at 0.44 and equal to

24

Red

(1 + 0.15 Re

0.687

d

) otherwise, with the droplet Reynolds number defined as

Re

=

|u

− u

|.

The enthalpy exchange is :

Π

=

Λ

+

Φ

(10)

where

Λ

is the contribution of mass transfer and

Φ

is the heat flux through

the interface. The enthalpy transfer linked to evaporation is taken as

Λ

=

h

Γ

where h

is the sensible enthalpy of vaporised fuel. The heat flux is

defined by

Φ

= α

λ Nu

(T

− T ) where Nu is the Nusselt number and T

is

the temperature at the interface.

The reaction rate ˙

ω

and the heat release ˙

ω

are modeled by an Arrhenius

law [41] with coefficients fitted by a genetic algorithm [42] from a reduced

chemistry [43] to the present one-step chemistry : JP10 + 14 O

*

)

10 CO

+

8 H

O using criteria such as flame speed and thickness. The fuel JP10 is

a substitute for kerosene and has the same thermochemical properties. The

flame/turbulence interaction is modeled by the thickened flame model [6].

THE LES CODE

The LES solver AVBP is a finite volume code, computing the equations for

both phases with the same numerical method on the same unstructured mesh.

It is explicit in time and takes into account the variations of molecular weights

and heat capacities with temperature and mixture composition. Boundary

con-ditions used for inlets and outlet are non-reflective NSCBC concon-ditions with

relaxation coefficients [44]. Symmetry conditions are used on the burner sides,

while non-slip adiabatic walls are used elsewhere. The unstructured mesh is

composed of 2.3 million of tetrahedral elements. In this application, the time

step is about

∆

t = 0.2µs. To calculate one turnover time of the swirled flow

τ

' 3.5ms , the LES computation of the turbulent spray flame takes one day

on SGI ORIGIN 3800 on 64 processors.

CONFIGURATION

The configuration is a 1/16

part of an annular combustion chamber. The

geometry takes into account the main swirled inlet, the primary jets, the dilution

jets and the cooling films that preserve the lower and upper walls from the flame.

A sketch of the geometry and a view of the mesh are presented on Figure 1. The

four primary jets are located on upper and lower walls, between the centered

vertical plane and the sides of the burner, leading to highly turbulent impacts.

Ten dilution jets are placed downstream to create a "cold wall" composed of

cooling air, limiting the outlet temperature to preserve the downward turbine

structure.

a.

b.

Figure 1: a. Geometry of the configuration, b. Mesh view (central plane)

INLET CONDITIONS

Air at 525 K is injected by two annular swirlers located upstream of the

computational domain. The inner swirler is located between r∗ = r/r

= 0

and 0.4 , where r

is the maximum radius of the outer swirler and r∗ is

defined as illustrated on Figure 2a. Using a cylindrical referential, the velocity

components of the main swirled inlet (Figure 2b) are normalised by the bulk

velocity U

and noted with the symbol

. The normal and radial components

Conf´erence ECCOMAS

u

and u

are strongly influenced by both swirlers : their value is higher around

the external radius of each swirler. The Reynolds number based on the bulk

velocity U

and the maximum injection radius r

is equal to Re = 15000.

The liquid fuel cone is defined by specifying α

' 10

in the inner zone

and zero elsewhere, and taking the liquid velocity equal to the inlet gaseous

velocity. The initial diameter of the kerosene droplets is 15µm , and their

temperature is 288 K. The Stokes number, based on the droplet relaxation time

τ

=

2

18µ

= 2.0 ms , is equal to St =

τp

τswirl

' 0.6. This means that the droplet

trajectories are quite correlated to the carrier phase motion. This correlation

rises when the diameter of the vaporising droplets decreases.

a.

b.

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

u*

1.0

0.8

0.6

0.4

0.2

0.0

r*

inner zone

Figure 2: a. Definition of r*, b. Radial profiles of u∗_n( ), u∗_r( ◦ ) and u∗_θ ( )

LES RESULTS

Dynamics

The swirled inlet generates a precessing motion and a recirculation zone,

starting close to the fuel injection and stopped by the primary jets. The

transver-sal half length of the recirculation zone is r∗ ' 0.4 − 0.6. An instantaneous

view of the dynamics on the vertical central plane is presented on Figure 3a.

The location of the primary jets, the dilution jets and the fuel injection is

in-dicated. The solid line shows the zone where the axial velocity is opposite to

the main flow direction. The impact of the primary jets on each other strongly

influences the dynamics as illustrated on Figure 3b, where the y-component of

the velocity is plotted on the plane of the primary jets.

CONFERENCE´ ECCOMAS

a.

b.

Figure 3: a. Back flow line, b. Primary jets : y-velocity field

Precessing Vortex Core

In its review on vortex breakdown [45], Lucca-Negro classifies the different

topologies of swirling flows using the swirl number. One of these topologies

at high swirl number is the precessing vortex core (PVC) : due to the swirl,

the axial vortex breaks down at the stagnation point S and a spiral is created

around a recirculation zone as illustrated on Figure 4a. In this application, the

swirl number (based on r

defined on Figure 3a) is equal to :

S =

1

r

uwr

dr

u

r dr

= 0.44

(11)

Using the transverse plane A-B defined on Figure 5a, the backflow line is plotted

on Figure 4b at six successive times marked with a number from 1 to 6 and

separated by 0.5 ms. The precessing motion is then illustrated in the present

configuration with a turnover time equal to τ

' 3.5ms , corresponding to a

frequency of f

' 286 Hz.

Conf´erence ECCOMAS

a.

S

b.

Figure 4: a. Sketch of the PVC ; b. Backflow line at six successive times

Dispersion

Due to their relatively low Stokes number, the droplets motion is controlled

by the carrier phase so that the recirculation zone of the carrier phase and of the

dispersed phase are similar. This can be observed on Figure 5a, where the u = 0

isolines of both phases are superposed. Since the droplets are constrained by

the recirculation zone, they accumulate in a region close to the injector where

the droplet density and the volumic fraction increase above the injection value.

Radial dispersion by the swirl is also visible on Figure 5b.

a.

b.

Figure 5: a. accumulation ; b. radial dispersion

CONFERENCE´ ECCOMAS

Evaporation

Evaporation can be characterised on Figure 6 by the mass and heat transfer

fluxes

Γ

and

Π

(see Equation 10 and Equation 11). The evaporation zone is

located where fuel vapour is created. To describe the resulting fuel vapour

distribution, the mixture fraction and local equivalence ratio are respectively

defined by :

Z

=

sY

−Y

+Y

sY

+Y

(12)

φ

=

Z

1 − Z

1 − Z

Z

(13)

with Y

= 0.233 , Y

= 1 and Z

= 0.066. In the evaporation zone, the

mass transfer is high and leads to high variations of the local equivalence ratio.

This zone is globally very rich as illustrated by the field of equivalence ratio

on Figure 6a. The heat conduction

Φ

controlled by the temperature difference

is relative and heats the droplets (reducing the air temperature as illustrated on

Figure 6b) while the enthalpy transfer

Λ

linked to the mass transfer is positive.

The global heat transfer

Π

takes positive values where the heat released by the

flame front accelerates the evaporation.

a.

b.

Figure 6: a. Equivalence ratio φ and mass transferΓ, b. Heat transferΠ

Flame structure

The chemical reaction takes place where fuel vapour, oxidant and hot gases

are simultaneously present. The evaporation zone brings fuel vapour and the

Conf´erence ECCOMAS

recirculation zone brings hot gases : the flame front, presented on Figure 7a,

is attached to the evaporation zone and stabilised by the recirculation zone. It

is actually composed of two successive fronts separated by the primary jets.

In order to distinguish premixed and diffusion flame fronts, the Takeno index

T =

∇

Y

.

∇

Y

and an indexed reaction rate ˙

ω

= ˙

ω

F|.|∇YO|

are used. The

flame structure is then divided in two parts : ˙

ω

= + ˙

ω

in premixed regime

and ˙

ω

= − ˙

ω

in diffusion regime. Results are shown on Figure 7b. Close to

the dispersion and evaporation zones, the turbulent mixing between reactants

create a rich partially premixed spray flame limited by the primary jets. By

continously creating pure fuel vapour, the evaporation process reinforces the

partially premixed regime. An important point is that the reaction rate is very

low there. The unburned fuel vapour and the cold air of the dilution jets then

create a second flame front in the diffusion regime. This is clear through the

Takeno index but also through the superposition of the reaction rate and the

stoichiometric line.

a.

b.

Figure 7: a. Reaction rate ˙ωF, b. Indexed reaction rate ˙ω_F∗

Stabilisation mechanism

Classical combustion models of twin-fluid atomized jet spray [46, 47] are

used to characterise the main competitive phenomena for flame stabilisation :

1. the air velocity must be low enough to match the turbulent flame velocity :

the dynamics of the carrier phase (and in particular the main recirculation

zone) stabilise the flame front on a stable pocket of hot gases

2. zones where the local mixture fraction is within flammability limits must

exist : combustion occurs between the fuel vapour radially dispersed by