Injection pressure pinj 200bar Injection duration ∆ti 0.4ms Injected massminj 26.6mg Liquid temperature TC7H16 20◦C
Table 2.1: Ambient and injection settings of the reference case.
4.3 · 105nodes and 2.4 · 106 elements (half-cylindrical box). Figure 2.3 presents an arbitrary cut through the mesh.
The convergence criteria for the presented calculations are set to5.0·10−5 for the RMS residual target and to1%for the conservation target, cf. Section 1.3. Further simulation and modelling parameters are summarised in Table 2.2.
Eulerian timestepDT 5.0·10−6s Discretisation of convective terms high resolution
Turbulence model SST
Number of parcel integration time
steps per element 10
Drag model Schiller-Naumann
Breakup model none
Table 2.2: Parameter settings and convergence criteria.
Figure 2.2: Section plane through the middle of the cylindrical geometry used in simulations.
Figure 2.3: Arbitrary cut through the mesh with edge length ≤ 1mm of the tetrahedra.
2.2.2 Spray initialisation
To define the primary spray the following quantities have to be set:
• the injection area and positions where Lagrangian parcels are initiated.
The actual opening area of the injector is not resolved in the computational mesh and a simplified injection region in form of a small ring is addressed. It is situated near the nozzle exit in the first cell layers to avoid incorrect backward flow along the nozzle wall. The ring width is assumed constant, i.e. opening and closing phase are neglected. The starting positions of the parcels are distributed randomly on the injection ring with a virtual width of0.03mm.
• The mass flow rate of the injected liquid.
Mass conservation gives
˙ minj =
Z
ρ·vnozzle exitdAinj. (2.1)
As mentioned the opening area Ainj at the injector tip is not resolved. Further-more, the spatial profile of the velocity across this area is unknown. Therefore, a simplified mass flow rate is used based on the needle lift function. A measurement of that is provided for an injection pulse width of∆ti = 1.0ms, see Figure 2.4(a).
(a) Measured and scaled needle lift function for
∆ti= 1ms.
(b) Setting in the simulation.
Figure 2.4: Injected mass flow rate.
• The number of Lagrangian parcels.
A number of40 000parcels is chosen to represent the total injected mass ofminj ≈ 26.6mg. The number injected per timestep is directly related to the mass flow rate with fewer parcels injected during the opening and closing phase of the injector.
• The injection velocity vector, which is determined by its amount vinj and the known cone angle of 90◦.
vinj is time-dependent and shows a spatial profile across the injector opening area.
Yet, measurements near the nozzle exit are very difficult due to the high spray density and vinj is not known for the considered injector and injection conditions.
The final function, see Figure 2.5(a), neglects the spatial profile and is deduced as follows:
– A nozzle flow calculation (courtesy Dr. Wolfgang Kern of BMW) is used to define the ramping functions during the opening and closing phase, see Figure 2.5(b). The progression of the maximal velocity values is considered here because these (and not the means) determine the penetration of the spray front.
– The results of the nozzle flow calculation refer to a slightly different nozzle and a pulse width of∆ti = 1.0ms. Cavitation is not included in the calcula-tion and only the lower part of the injector is considered where the pressure drop occuring in the upper part is not accounted for. Hence, the calculated stationary value of vstat ≈232m/s is rather uncertain.
A simple Bernoulli approximation, 0.5·vstat2 + 1bar
ρ = 200bar
ρ , (2.2)
(a) Setting in the simulation. (b) Results from a nozzle flow calculation. The progression of the maximal velocity values is ap-proximated to define the ramping functions during the opening and closing phase of the injector.
Figure 2.5: Injection velocity.
Figure 2.6: Comparison of different stationary velocitiesvstat at t= 0.25ms for the measured drop size distribution with D32 = 18µm. vstat = 210m/s refers to the progression of the mean velocity in Figure 2.5(b), vstat = 232m/s to that of the maximal velocity andvstat = 241m/s is the value deduced from the Bernoulli equation and finally used in the setup.
is also overestimated there: it has been observed that the ejected liquid sheet only breaks up in distinct drops in a distance of several millimeters after the nozzle exit due to turbulent and aerodynamic forces, see Figure 2.7. In an Euler-Lagrangian approach this cannot be modelled and the initialised individual drops experience larger drag.
Figure 2.7: Breakup of the injected liquid sheet several millimeters after the nozzle exit.
• The drop size distribution.
Its initialisation presents a well-known problem in spray simulations. Primary breakup cannot be modelled in a satisfactory way yet. For the considered injector and injection conditions, measurements of the drop sizes are only available in a distance of 40mm to the nozzle exit: At the outer border of the spray cone, a distribution with a Sauter mean diameter of D32 ≈ 15µm is obtained, see Fig-ure 2.8. Measuring even closer to the spray cone, the diameter increases and the distribution functions show a value ofD32≈18µm. They seem to stabilise there, yet, measurements have not been performed crossing the spray cone completely and drop sizes might still be larger in the core.
Using the measured distributions directly for spray initialisation at the nozzle exit without any further breakup model applied, it is not possible to reproduce the spray penetration of the transmitted-light images with the measured diam-eter distribution of D32 ≈ 18µm for the chosen velocity value vstat ≈ 241m/s.
Therefore, the distribution is scaled with a factor of1.2resulting inD32≈21µm.
Even larger drops, which also lead to good results just before wall interaction, see Figure 2.9(b), are not justified because the penetration of the main mass flow is then too fast in the beginning, cf. Figure 2.9(a).
In Figure 2.10 the numerical predictions of spray propagation resulting with the finally chosen setup are compared with the transmitted-light images. The overall agreement is rather good - the more so as an impartial evaluation of the transmitted-light images is also difficult.
Figure 2.8: Drop distributions considered for initialisation.