Baumgarten

move radial outwards with velocity vr2. Altogether, this results in a larger spray divergence at the cavitation side. The model is thus capable of producing asymmetric primary sprays according to the nozzle hole flow. The high break-up energy per unit mass (high collapse energy, low density) results in small droplets and a large spray divergence near the nozzle. Consequently, zone 2 forms the outer spray region, which appears optically dense but does not contain much mass.

Break-up of the liquid zone: Outside the nozzle, zone 1 forms the extremely dense core in which energy is dissipated due to chaotic collision, break-up, and coalescence processes. The chaotic nature of these dissipation processes is modeled by using an efficiency Κ1, which is sampled randomly for each ligament between 0 and 1 and which reduces the effective break-up energy of zone 1 to E1,eff = Κ1E1. As in the model of Nishimura and Assanis [91], it is assumed that all breakup energy is present as turbulent kinetic energy the velocity fluctuations uturb = (2E1,eff /3m1) of which cause a disruptive turbulence force Fturb. Mass is split off the liquid cylinder of zone 1 until the surface tension force Fς is equal to Fturb:

The remaining cylinder with diameter

is transformed into a spherical drop, and its turbulent kinetic energy is used to determine its spray angle (radial velocity). The disintegration of the split mass into droplets with a velocity component normal to the main spray direction is modeled in the same way as for zone 2, but the direction Μ of the child parcel in the x-y- plane is sampled from a uniform distribution between 0° and 360°. Altogether this results in a symmetric full-cone spray (angle Ι1, Fig. 4.12). It forms the dense inner spray region with high momentum in the axial direction and is responsible for the spray characteristics in the far field of the nozzle. Because of the small breakup energy per unit mass, the average droplet diameter is larger and the spray angle is smaller than in zone 2.

Cavitation bubble dynamics: It is assumed that the bubbles leaving the nozzle experience a sudden rise in pressure from the vapor pressure pvap to the pressure pφ of the gas inside the chamber and that the bubbles collapse inside zone 2. In order to calculate the time-dependent development of the bubble radius R(t) during the collapse process, the equation of Herring [50] and Trilling [141] (see Prosperetti and Lezzi [108]),

Using this equation, the actual values k(tcoll) and Eturb available for break-up are estimated.

Using their primary break-up model in combination with the Kelvin-Helmholtz model for secondary break-up, the authors have shown that experimentally observed asymmetric spray structures due to asymmetric nozzle hole flows as well as penetration, spray angle, and droplet sizes could be well predicted. Figs. 4.15 and 4.16 show the results for an asymmetric nozzle hole flow resulting in an asymmetric distribution of cavitation and liquid zone inside the primary ligament as shown in Fig. 4.13. Diesel fuel (rail pressure: 65 MPa) is injected in compressed air (5 MPa, 25°C).

In Fig. 4.15, the results of a statistical analysis (frequency distribution) of the relevant primary break-up parameters during the disintegration of the primary ligaments are shown. As expected, zone 2 has the highest break-up energy per unit mass because of the collapse of cavitation bubbles and its low mass, Fig. 4.15a. This results in larger primary spray angles and smaller droplets compared to zone 1 (Figs. 4.15b and 4.15c). The frequency distribution of the formation of new droplets at different circumferential positions Μ in the x-y-plane, see also Fig. 4.13, is, as implied by the model, proportional to the thickness of the cavitation zone in case of zone 2, and equally distributed in case of zone 1, Fig. 4.15d. The combination of the diagrams gives the following picture of the 3D primary spray: zone 1 forms a dense full-cone spray consisting of big droplets (spray center). Zone 2 forms the outer spray region, whose asymmetry is directly dependent on the asymmetric distribution of the cavitation zone inside the nozzle holes.

b

a

Fig. 4.15. Statistical analysis of relevant parameters during primary break-up [12]

4.1.6 Sheet Atomization Model for Hollow-Cone Sprays

In direct injection spark ignition (DISI) engines, nozzles producing a hollow-cone spray are usually used in order to achieve maximum dispersion of the liquid phase at moderate injection pressures of about 5 to 10 MPa. Compared to the boundary conditions in the case of diesel engines, the backpressures and temperatures are small and the use of full-cone sprays would result in poor mixture formation and increased wall impingement. Hollow-cone sprays are typically characterized by small droplet diameters, effective fuel-air mixing, reduced penetration, and thus high atomization efficiencies. Fig. 4.17 shows two nozzle concepts, an inwardly opening pressure-swirl atomizer and an outwardly opening nozzle. In case the of a swirl-atomizer, the fuel passes through tangentially arranged swirl ports and gets a rotational motion inside the swirl chamber, Fig. 4.18. The centrifugal motion of the liquid forms a hollow air core.

Fig. 4.17. a Inwardly opening pressure-swirl atomizer, b outwardly opening nozzle

Fig. 4.18. Schematic illustration of the tangentially arranged swirl ports

Fig. 4.19. Break-up of the liquid sheet into ligaments and droplets

First, a liquid film with initial thickness h, spray angle Τ , axial velocity u, and total velocity Uinj is formed, which penetrates into the combustion chamber. The liquid sheet becomes thinner because of the conservation of mass as it departs from the nozzle. Further on, Kelvin-Helmholtz instabilities are assumed to develop and grow on the film surface resulting in a first break-up and in the formation of ring-shaped ligaments. These ligaments finally break up into droplets.

Schmidt et al. [125] have shown that when the film exits the injector as a free liquid sheet, the tangential velocity (which is present in the case of a swirl atomizer) is completely transformed into radial velocity, and the trajectory approaches a straight line. The angle Τ from the centerline is determined by the ratio of radial and axial velocity. Hence, at the point of film break-up and drop formation, the swirl velocity of the drops can be neglected. This is a very convenient approach because the swirl velocity at the nozzle orifice is usually unknown. Nevertheless, because the swirl velocity is responsible for the formation of the spray half-cone angle Τ, the spray cone angle must be known from experimental investigations in this case. Alternatively, the detailed liquid film movement model of Gavaises et al. [39] could be used at this point in order numerically to predict the spray angle, the reduction of the film thickness, and the decay of the swirl velocity during the film disintegration process.

The disintegration of the liquid film is modeled in direct analogy to the KelvinHelmholtz model for diesel sprays, see Sect. 4.2.4. The model, which is presented in detail in Senecal et al. [129], assumes that a two-dimensional, viscous and incompressible liquid sheet with thickness 2h moves with velocity U through a quiescent, inviscid, and incompressible gas. A spectrum of infinitesimal disturbances

is imposed on the sheet surface producing pressure and velocity fluctuations in both the liquid and the gas. The amplitudes of these disturbances grow due to the liquid-gas interaction. In Eq. 4.73, Κ0 is the initial wave amplitude, k = 2Σ/Ο is the wave number, and Ζ = Ζr + iΖi is the complex wave growth rate. It is assumed that the most unstable disturbance with growth rate Ζr = : causes sheet break-up. Hence, a dispersion relation Ζ = Ζ(k) is derived, from which the most unstable wave can be deduced. In ref. [45], it has been shown that two modes of film dis-

where the value of ln(Κb /Κ0) is taken to be 12 as suggested by Dombrowski and Hoper [29]. It must be noted that the quantity U in Eq. 4.75 is the relative velocity between liquid and gas while Uinj in Eq. 4.77 is the absolute velocity of the sheet. However, it is assumed in the numerical implementation that the relative velocity is equal to the absolute velocity because the liquid is usually injected into quiescent gas.

The diameter of the primary ligaments is obtained from a mass balance. It is assumed that every tear in the sheet forms a primary ligament. The resulting diameter is given by

Ks is the wave number of the fastest growing surface wave. Because the sheet thins as it departs from the nozzle, the position of first the break-up determines the ligament diameter. The sheet half-thickness hb at the point of break-up is

h0 ū0.5Γ0 cos(Τ).

The subsequent disintegration of the ligament into drops is calculated using another stability analysis, which is based on an analogy to Weber’s result for the growth of waves on cylindrical, viscous liquid columns. The wave number for the fastest growing wave on the ligament is

As proposed in Dombrowski and Johns [30], it is assumed that break-up occurs if the amplitude of the wave is equal to the radius of the ligament, and that one drop is formed per wavelength. A mass balance gives the drop diameter:

The starting conditions of the drops formed at the point of break-up are used as input for the CFD calculation of spray formation. The parcels are injected into a hollow-cone with a mean spray angle Τ from the spray axis. The exact trajectory of each discrete parcel is determined by randomly distributing the angle over a range Τ ρ Τ, where Τ | 10 deg. is the dispersion angle. In the numerical implementation, it is assumed that the sheet does not interact with the gas phase. The sheet is represented by dormant parcels, which are injected with velocity Uinj (Eq. 4.70), and do not experience any drag nor undergo any break-up, evaporation, or