Modélisation de l'écoulement diphasique dans les injecteurs Diesel

is worth citing, because this is an attempt to createa transition model from large scale models to smallscale models.Interface location is not defined by the model• Methods that do not take in account the interfacelocation are the continuum models, and thetwo-fluids models. The continuum models, or mixturemodels, consider a single fluid whose densityis defined by an equation of state. G enerally, thisequation is a simple barotropic one. Some examplesare presented in table 2. The main attractivefeature of these models is this equation of state,but this is also their weakness : this equation needsto represent the whole cavitation physics, dependingon the configuration of the flow. Delannoy andKueny [54] have developped a pocket model on ahydrofoil. Their equation of state is barotropic,and considers three domains : the incompressibleliquid, the incompressible vapour, and the regionof compressible homogeneous mixture. This regionis a sine function of ρ versus P . Chen and Heister[55] have criticized this model by saying thattemporal terms in the equation of state must betaken in account. Pressure field history has to appearin the local density, as shown in the Rayleigh-Plesset equation [56]. The Chen and Heister algorithmdistinguishes three zones : the cavitatingregion, the region where P < P sat (the pressure isthen readjusted to P sat in this zone), and the regionwhere there is neither collapse nor cavitation growing.This model considers also the two pure phasesas incompressible. Schmidt [21] has developped amodel particularly adapted to instationary cavitatingflows in common-rail injectors. This model considersthe liquid as compressible, and is based on theWallis [22] study on HEM 1 concerning sound velocityin two-phase flows. The liquid, the vapour andthe liquid/vapour mixture are considered as compressible.The numerical simulation results fit withthe Chaves experimental data on the exit velocity[7] for a typical axisymmetrical singled-hole nozzle.• The two-fluid models are usually used for largescaledflows, i.e. typically bubbly flows, in order totake in account each phase turbulence and the bubblesdrag. The approach consists in solving a setof Navier-Stokes equations for each phase, and tolink them one to each other using a mass transferterm in the mass conservation equation (see equation7), where N = 2 for a two-phase flow and k isthe phase subscript. For instance, Alajbegovic et al.[57] have made a two-fluid model using a simplifiedRayleigh-Plesset equation [58] as the mass transfer1 Ho m o g en eo u s E q u ilib riu m Mix tu reterm :∂α k ρ k∂t+ α kρ k v k,i∂x i=N∑l=1, l≠kΓ kl (7)Γ 21 = ρ 2 N ′ ′ ′ 4π R 2 ∂R∂t = −Γ 12 , (8)w here R is the av eraged b u b b le rad iu s and N ′ ′ ′ theb u b b le d ensity, w hich is link ed to the v oid fractionα 2 b y :{N ′ ′ ′ N′ ′ ′=′ ′ ′0 if α 2 ≤ 0.52(N 0 − 1)(1 − α 2 ) + 1 if α 2 > 0.5(9)T his heu ristic form u la is su p p osed to tak e in account the b u b b les coalescence as soon as the v oidfraction is great enou gh. In that case, the m od el isnot ad ap ted to high α 2 v alu es, for the “transitionfrom bu bbly to slu g, or even fi lm regim e” [59]. Lam -b ert [60] has also u sed a tw o-fl u id m od el to com p u tethe internal injector fl ow . A p articu lar attentionhas b een focu sed on tu rb u lence m od elling for thetw o p hases. T he liq u id is su p p osed com p ressib leand ob ey s to the T ait eq u ation :ρ l = ρ r ef[1 − C ln( B + PB + P r ef)] −1(10)T he resu lts lead to negativ e p ressu re v alu es (w hichcan b e encou ntered in ind u strial cases [24]), w hosev alu es are great (m ore or less 200 b ar) com p aringto u su al v alu es of the ord er of sev eral b ar. C onsequ ently, the fl ow d y nam ics resu lts are certainly d istorted. F ew com m ercial cod es d eal w ith cav itation.In the actu al F LUE NT v ersion, a tw o-fl u id m od el isp rop osed , and the cav itation can b e m od elled u singa m ass transfer term b ased on Alajb egov ic m ethod(see eq u ation8) [57, 61]. P sa t and the b u b b le d ensityare the p aram eters that can b e m od ifi ed b y theu ser. T he collap se is not tak en into accou nt b y thesolv er, i.e. b u b b les can only grow . F u rtherm ore, theb u b b les are consid ered as sp herical and m u st b e farsm aller than the com p u tational cell size. In fact,this m od el is p articu larly ad ap ted to large scaledfl ow s, w hich is not the case of Diesel injectors.C onsid ering the p articu larities of the ex istingm od els, w e can conclu d e that interface track ing andfront cap tu ring m ethod s are not ap p rop riate for theinjector fl ow confi gu ration. T he com p u tational cellnu m b er w ou ld b e too large for actu al com p u ter capab ilities. T he tw o-fl u id m ethod is not ad ap tedto large v ap ou r entities, i.e. w hen som e cells arefi lled of v ap ou r, w hich is ou r case in ou r confi gu ration.C onseq u ently, the right ap p roach shou ld b eto m od el the fl ow thank s to a continu u m m ethod ,b ased on an ap p rop riate eq u ation of state.