Modélisation de l'écoulement diphasique dans les injecteurs Diesel

CAV2001:sessionB6.005 6Depending on the exit flow configurations, several cases have to be considered.2.4.1 Subsonic outflowIn that case, all λ i ’s are positive, except for i = 1. That means that the L i ’s can be estimatedfrom the interior values (as waves move outward), thanks to the equations 15, whereas the L 1 valueis obtained by considering the chamber pressure (P ch ) as constant. If the outlet pressure is notclose to the P ch prescribed value, incoming wave will enter the domain in order to bring the outletpressure value back to P ch .L 1 can be simply expressed by :L 1 = κ(P − P ch ), (16)where a mechanical analogy with the spring is obvious. The main point now is to determine κ inorder to get the best behaviour as possible at the exit. It can be noted that, by setting κ = 0, weget the so-called “perfectly non-reflecting” conditions.2.4.2 Sup ersonic outflowAs we have seen in section 2.2, sound speed is strongly dependent on void fraction. As cavitationcan reach the injector exit, local sound speed can decrease to very low values, leading to localsupersonic flow. In that case, all L i ’s are estimated from the interior of the domain as no wave canenter from the exit. The set of equations 15 is then used to compute L i values. This statement isvery important : when cavitation reaches the injector exit, no relaxation over the exit pressure P chis made, and cavitation can normally leave the domain, without any numerical collapse.2.4.3 Subsonic inflowThe topology of the flow in Diesel injector can lead to “hydraulic flip” (see Y ule et al. 1998,Tamaki et al. 1998, Soteriou et al. 1995), as recirculation zones reach the orifice outlet. In thatcase, the flow enters the domain and the λ i ’s are negative except λ 4 . So L 4 can be calculated fromthe interior values (see equation 15), and the other wave amplitudes are given by the followingequations :L 1 = κ(P − P ch ),(17)L 2 = L 3 = 0.2.4.4 Sup ersonic inflowThis case hardly happens but to be consistent one has to model this configuration too. Forsupersonic inflow, all propagation velocities are negative, so all L i values need to be modeled fromoutside of the domain.L 1 = κ(P − P ch ),L 2 = L 3 = 0,(18)L 4 = κ(P − P ch ).3 Validation of the cavitation model3.1 Collapse of a symmetric bubbleIn order to validate the numerical scheme and the barotropic equation of state, one has tosimulate a well-known test case : the collapse of a symmetric bubble in an infinite domain. Thestudy of the bubble dynamics has been performed by Rayleigh and Plesset, considering that :– the bubble shape remains spherical,