{'fraction_non_alphanumeric': 0.02525085455948837, 'fraction_numerical': 0.008931524975190208, 'mean_word_length': 4.939751146037983, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Large Eddy Simulation of diesel injector opening with a two phase cavitation model

In the current paper, indicative results of the flow simulation during the opening phase of a Diesel injector are presented. In order to capture the complex flow field and cavitation structures forming in the injector, Large Eddy Simulation has been employed, whereas compressibility of the liquid was included. For taking into account cavitation effects, a two phase homogenous mixture model was employed. The mass transfer rate of the mixture model was adjusted to limit as much as possible the occurrence of negative pressures. During the simulation, pressure peaks have been found in areas of vapour collapse, with magnitude beyond 4000bar, which is higher that the yield stress of common materials. The locations of such pressure peaks corresponds well with the actual erosion location as found from X ray scans.


Introduction
Diesel injection systems play a fundamental role in internal combustion engines since they affect the formation of the fuel spray, atomization and combustion, the formed emissions and the engine efficiency. The jet velocities formed are of the order of 500m/s, with upstream pressures around 2000bar. Current trends show injection pressures to even rise to 3000bar, in order to meet the future EU legislations in emissions. However, higher pressure levels causes very high velocities through the tight passages in the Diesel injector and strong accelerations in sharp direction changes (corners, fillets etc.), which lead to local static pressure dropping below the saturation pressure and causing cavitation. Furthermore, cavitation may lead to cavitation erosion damage and serious degradation of the injector performance, even catastrophic injector failure, which could damage the engine, if the injector tip breaks off. There have been many efforts for the prediction of the Diesel injector erosion, see e.g. the work of Koukouvinis et al. [1]. The aim of the current work is to simulate the flow inside a Diesel injector in a more fundamental level, including compressibility effects of the liquid phase and also using a Large Eddy Simulation for describing turbulence. The reason for employing compressibility effects is the large density difference in the injector, which may reach 10% [2], not to mention the high liquid velocities that can reach a Mach number of 0.5. Furthermore, resorting to Large Eddy Simulation techniques is because RANS/URANS are inadequate for capturing the complicate vortex patterns which affect cavitation formation [3], while even modified RANS turbulence models are situational [4]. To the authors knowledge, a similar work does not exist in literature; Sezal et al. [5] and Salvador et al. [6] numerical work on injectors includes compressibility effects, but not the needle motion of an actual injection cycle.

Numerical background
Numerical simulations presented in this work are based on a the solution of the Navier Stokes equations, using a commercial pressure-based solver, Fluent [7]. The equations solved consist of the continuity and momentum equations, while the energy equation has been omitted since heat effects were ignored. However, since Diesel properties vary significantly with the pressure levels in the injection systems, both liquid phase viscosity and density are assumed variable, as functions of pressure. For density, the Tait equation of state was used: where ρ sat,L = 747.65kg/m 3 is the density at saturation pressure p sat =1.2bar and temperature of 395K. The temperature was estimated through simplified 1D analysis for the pressure levels in the injector [8] and is an estimated average during the injection event. The bulk modulus is B=110MPa and n=7.15. The Tait equation of state is a compromise in the sense that is known to predict accurately the behaviour of liquids, whereas more accurate data are not available in the pressure ranges involved. For Diesel liquid kinematic viscosity, v, a relation provided by N. Kolev [2] is used, applied for the same temperature level as above: (2) For simulation of cavitation effects, an additional transport equation is solved for tracking the vapour phase, of the form: where a is the vapour fraction, ρ v is the vapour density, u is the velocity field and R e , R c are the mass transfer rates for condensation (c) and evaporation (e), prescribed by the Zwart-Gerber-Belamri model [9]. The vapour properties used, are density ρ V =6.5kg/m3, the dynamic viscosity µ v =7.5µPa.s. The two phase model, assumes mechanical equilibrium between the two phases, i.e. both liquid and vapour phase share the same pressure and velocity fields. The mass transfer terms have been tuned in order to limit the existence of negative pressures inside the computational domain as much as possible; after the tuning the minimum pressure inside the injector is approximately -20bar. The aim of the two phase model employed is to replicate the growth and, more importantly, Rayleigh collapse of vapour structures which leads to high pressures during cavitation collapse and to cavitation erosion. In order to capture the complicated turbulent structures inside the sac volume and orifice, the Wall Adapted Local Eddy-viscosity (WALE) LES model was employed [10], since it has much better performance in wall-bounded flows, as the one developing inside the injector.

Case set-up
The Diesel injector tip geometry is shown in fig. 1a; it has 5 tapered orifice holes. The injector tip inlet and outlet pressure boundary conditions were calculated using 1D CFD tools. Note also that at the end of the orifice of the injector an additional hemispherical volume was added ( fig. 1b), in order to move the influence of the outlet boundary further away from the orifice, especially considering that cavitation structures may reach or even exit the orifice, as will be shown later. The configuration resembles the injection test benches where fuel is squirted into a collector filled with liquid. The needle motion is assumed to be in the axial direction only, so any eccentricity effects were omitted. In reality eccentricity effects are significant during the early opening and late closing phases, but introduction of such effects will impose the simulation of the complete injector geometry, which will be much more complicated. In the current simulation, only 1/5 th of the domain was solved and periodic boundary conditions have been employed at the sides of the domain. The computational domain was split in a set of moving, deforming and stationary zones, as shown in fig. 1b. The computational mesh used is block-structured, with the exception of a zone in the sac prior to the orifice entrance, which is unstructured tetrahedral (see fig. 1c). Inflation layers were introduced in the area of interest, such as sac and orifice to capture the boundary layers. Mesh motion was introduced by a smoothing algorithm at low lifts and the mesh layer addition at higher lifts. The mesh size was approximately 1 million cells at the beginning of the simulation; the needle motion introduces additional cell layers inside the volume denoted with blue colour in fig. 1b, so the actual cell count increases over time. A bounded central scheme (hybrid between central and second order upwind) was used for momentum, while second order upwind for density and QUICK for volume fraction. Time advancement was performed with an implicit, second order, backward differentiation with a time step of 5ns, in order to be able to capture the complicated turbulent patterns. The implicit integration avoids the time step restriction due to compressibility effects, which would further limit the time step to even lower values.

Simulation results
In fig. 2a the complicated cavitation patterns inside the sac volume and the injector orifice are shown. Also in fig. 2b the evolution of the maximum pressure on the wall is shown. From the simulation results, cavitation starts to develop at the edges of the jet formed at low lifts between the needle and needle seat, due to shear layer instability. Later on, cavitation extends inside the sac volume, due to the vortex formation. At the same time, cavitation collapses on the needle produce strong shock waves, with pressure levels exceeding 4000bar and even peaking at 10000bar. When the needle opening is high enough, e.g. at ~39µm pressure starts to rise inside the sac volume and cavitation structures collapse, causing sporadic pressure peaks on the sac wall. Eventually, cavitation is limited only inside the orifice, at the upper orifice surface. Note that cavitation may extend outside the injector orifice.

Conclusion
This paper outlines the methodology and results from Large Eddy Simulation inside a Diesel injector, including liquid compressibility effects. The turbulence model used is capable of capturing the complicated vortex patterns inside the injector sac and orifice, which in turn affect the cavitation formation location and collapse. Cavitation collapses produce strong shockwaves, with pressure levels