Baumgarten

where p is a random number uniformly distributed in the interval [0, 1].

The most important limitation of the model of Naber and Reitz [90] is that the phenomenon of droplet shattering (splash), which occurs at high impact energy and is important in effecting wall spray dispersion and vaporization, is ignored. This is especially important if the distance between nozzle and wall becomes small. Furthermore, the formation of a liquid film adhering to the wall and its effect on the impingement behavior of drops are not included. Under normal engine conditions, the wall temperatures are below those used in the experiments of Wachters and Westerling [144], and, especially under cold-starting conditions, the transition criterion and the determination of momentum loss during impact are not applicable any more.

Bai and Gosman [11] have developed a more detailed impingement model considering also the splash regime. In the case of a dry wall, the stick and spread regimes are combined and called the adhesion regime, and the stick regime is neglected in the case of a wetted wall because of its typically very low impact energy.

For a dry wall, the transition criterion between adhesion and splash is given as

where the coefficient A is depends on the surface roughness rs, Table 4.6. In the case of an already wetted wall the transition Weber numbers are:

In Eq. 4.273 it is assumed that a liquid film can be approximated by a very rough dry wall. Next, the post-impingement characteristics have to be modeled. In the adhesion regime, the incoming droplets are assumed to coalesce to form a local liquid film. In the rebound regime, the droplet bounces off the wall and does not break up, but loses a small part of its kinetic energy by deforming the liquid film. Bai and Gosman [11] use a relationship developed by Matsumoto and Saito [85] for a solid particle bouncing on a solid wall in order to determine the rebound velocity components,

droplet classes are derived from an energy balance. It is assumed that the splashing energy Ek,s

which consists of kinetic and surface energy, is the difference between the kinetic

energy Ek of the incoming parcel and the critical kinetic energy Ek,crit, below which no splashing occurs. Ek,crit is believed to be lost in deformation, dissipation, and in

producing liquid film energy. Ek,crit is calculated from the critical Weber number:

From experimental data on the size-velocity correlation of the new droplets after splashing, the relation

is deduced. This relation expresses the observation, that the larger the size of the new droplets, the smaller the magnitude of their velocity. Finally, application of the tangential momentum conservation law yields

The wall friction coefficient cf is assumed to be in the range of 0.6 to 0.8, and Τ1 and Τ2 are the ejection angles of the two new parcels. Τ1 is randomly chosen inside an assumed ejection cone (approximately 10°< Τ1 < 160°), and Τ2 is determined from Eq. 4.284. Together with

the above set of equations can be solved in order to calculate the normal and tangential velocity components of the two new parcels.

The model of Bai and Gosman [11] has been successfully validated against experimental data. A typical spray pattern (side view, half spray) simulated with this model and showing all the characteristic features of a wall spray is given in Fig. 4.66. The computational parcels are shown as black points.

Fig. 4.67 shows comparisons between predicted and measured adhered fuel ratio on the wall, wall spray radius, and wall spray height. Comparisons of local droplet diameters are not given. The experiments were performed by Saito et al. [123], and the boundary conditions are given in Table 4.7.

188 4 Modeling Spray and Mixture Formation

Fig. 4.66. Simulation of an impinging spray [11], t = 2 ms after SOI, atmospheric conditions

Table 4.7. Boundary conditions for the experiments of Saito et al. [123]

Measurement and simulation agree well. The cusp in the curves of Fig. 4.67a corresponds to the end of injection. After the end of injection, the adhered fuel ratio, which is defined as total mass deposited on the wall divided by the total mass injected at that time, increases, because the deposited mass continues to increase for a short time while the injected mass remains constant. The predictions of wall spray radius (Fig. 4.67b) and wall spray height (Fig. 4.67c) capture the trend that an increase of injection pressure results in larger values of wall spray radius and height. The droplets in the wall spray get larger tangential velocities due to the increased gas velocities along the wall, and larger normal velocities because of a more intense splashing.

Fig. 4.67. a Measured and predicted adhered fuel ratio, b wall spray radius, c wall spray height, [11]

Other detailed wall-impingement models have been developed by O’Rourke and Amsden [95], Lee and Ryou [72], and Stanton and Rutland [132]. Differences between these models and that of Bai and Gosman [11] appear mainly in the splash regime.

In the model of O’Rourke and Amsden [95], the film momentum equation is used in combination with experimental results from Mundo et al. [89] in order to derive the spread-splash transition criterion

where

Furthermore, the spread-splash transition criterion is determined by using an empirical correlation based on the experimental results of Mundo et al. [89], and only one instead of two new parcels is created in the splash regime. The size and number of the splashed new droplets are determined from the conservation of mass and from experimental data.

4.8.3 Wall Film Modeling

A detailed modeling of wall film evolution includes the simulation of all effects increasing or decreasing the liquid mass on the surface of a wall cell like deposition of droplets, evaporation and film movement, see Fig. 4.68. Film movement can be due to the gravity force or due to exchange of momentum between gas and film or between incident droplets and film. Different classes of wall film models exist. One method is to set up and solve the governing equations for wall films. The other method is to simplify the treatment and to use empirical correlations for the relevant effects.

An early model for the simulation of wall film evolution and heat transfer from the wall to the film or to the impinging drops was developed by Eckhause and Reitz [31]. This model includes the impingement model of Naber and Reitz [90], Sect. 4.8.2. The model distinguishes between a flooded (a wall film is present) and a non-flooded case (no wall film), see Fig. 4.69. According to the model of Naber and Reitz [90], only droplets with Wein > 80 (slide) contribute to the formation of a liquid film. The film thickness Γ is calculated separately for each surface cell by taking the total mass of fuel on the surface of a wall cell and dividing by the fuel

density Υl and the wall cell area Acell. The transition criterion between the nonflooded and the flooded case is given as Γ = 2 µm. The total heat transferred to a film or a drop is

Fig. 4.68. The major physical phenomena governing film flow [133]

For the case of slide, Eckhause and Reitz [31] distinguish between the flooded and the non-flooded regime. In the non-flooded regime the contact area is calculated according to Eq. 4.295 during the time step of first contact, and then the area is A = (Σ4) din2 during the time the droplet slides along the wall. Again, Nu = 2.0 is used and the corresponding surface temperature is the wall temperature. In the flooded regime the droplet becomes part of the wall film. The film temperature is the mass average temperature of all droplets belonging to the film, and the contact

area A is equal to the surface area Acell of the respective wall cell. The gas-phase heat transfer is calculated based on the modified law of the wall (Reitz [116]),

while the liquid-phase heat transfer is based on the heat transfer correlation Nu = 3.32·Pr1/3 for the boundary layer flow.

The temperature change of a film element or a drop is given by

where cv,l is its specific heat capacity and m is the liquid mass of the fuel on a wall cell (flooded case) or the one of a single drop (non-flooded case and rebound).

A much more detailed wall film model, which simulates thin fuel film flow on solid surfaces of arbitrary configuration, is presented in Stanton and Rutland [132, 133]. The major mechanical and physical processes like mass, momentum, and pressure contributions to the film due to spray impingement and splashing effects as well as the effects of shear forces, piston acceleration, and gravity are included. The following assumptions are used in the formulation of the model: First, the liquid film is treated as a thin film and the momentum and continuity equations are integrated across the film thickness x&3, see Fig. 4.70. Thus, a two dimensional film is regarded that flows over a three dimensional surface. Fig. 4.70 shows the local coordinate system used for the description of a wall film cell.

Fig. 4.70. Wall film cell and local coordinate system of the film model