Comparison to Roisman’s/Horvat’s data and interpretation

The fact that single drop models fail supports the importance of drop/drop interactions for dense sprays. Kuhnke’s and especially Roisman’s/Horvat’s model should provide better results. Both models are not deterministic as Elsässer’s single drop model but depict secondary distributions using random numbers. Consequently, individual single outcomes can vary significantly and a comparison with measured mean values, analo-gous to Figure 5.4, is rather restricted. Therefore, a more meaningful comparison of the distributions is postponed to the simulation of the experiment in Section 6.2.

Yet, for Roisman’s/Horvat’s model not only the final model correlations, which could be applied to the respective conditions, are available but also the underlying data, which

with −→e_γ being normal to the target surface. For convenience, these quantities are re-named to U_b, D_b and U_a, D_a where the subscripts b and a denote “before” and “after”

impact, i.e. primary and secondary drops. Derived quantities are calculated using the means, e.g. Re_b = ^ρ·Ub_µ^·Db.

For the comparison in this section, the current experimental data are exceptionally evaluated in the same way as Roisman’s/Horvat’s data, which is important for a rea-sonable analysis as described in Section 4.5. U_a and D_a are determined for secondary drops assigned to the outside of the spray cone. All measurements are included in the comparison due to the insignificant influences of the wall temperature and the mean primary Reynolds number.

(a)Diameter ratio. (b) Ratio of wall-normal velocity components.

Figure 5.5: Comparison of Roisman’s/Horvat’s data with the measured data.

Figure 5.5 shows the secondary to primary ratios of diameters and wall-normal velocity components versus the mean impinging Reynolds number for Roisman’s/Horvat’s data (Re_b >500) in comparison to the new data obtained in this work:

• The behaviour of the diameter ratio is different although the values are of the same order.

• Especially the normal velocity values³ of Roisman’s/Horvat’s data scatter over a large range in contrast to the new values which only display a negligible parameter influence.

• The data sets refer to different ranges of the Reynolds numbers with higher values for the new measurements. A gap around Reb = 2000contains no data points.

3Note that these are not used directly for modelling in Roisman’s/Horvat’s model. Instead a relation between the volume flux ratioΓ_V and the energy flux ratioΓ_Etot is considered, see Section 1.4.2.

It is deduced that the impact phenomena are different for both data sets. In Rois-man’s/Horvat’s model, corona splash is stated: Finger-like jets are created at unstable crown rims and break up into secondary drops. Their diameters are assumed to scale in the same way as the finger radius, which finally gives D_a/D_b ∝ Re^−0.5_b , see Sec-tion 1.4.2. This presumes an undisturbed crown formaSec-tion for an individual spray drop, i.e. a small spray density. In contrast to the relatively sparse full cone spray considered by Roisman/Horvat, the disturbance of crown formation by neighbouring drops cannot be neglected for the dense hollow cone spray addressed in this work. This can be proven by a rough estimation which shows that the necessary time and space are not available:

• Using the non-dimensional time τ = t·v_prim/D_prim, reference [82] describes the maximum time of crown propagation as

τ_max = 5.44·Z with Z = Weprim

Re^0.5_prim = D^0.5_prim·v_prim^1.5 ·ρ^0.5·µ^0.5

σ . (5.2)

Inserting the liquid properties of isooctane from Table 3.1 and approximating the mean spray impact properties of the experiment as v_prim ≈ 35m/s and D_prim ≈ 50µm, one gets τ_max ≈220 and t_max≈0.31ms respectively.

Note that a lot of parameters, like the wall film thickness, are known to influence the crown formation, cf. [39], [43]. They are neglected at this point.

• Equation 5.2 is based on a mass and momentum balance in the axial crown di-rection and some results from [95] and [10]. These references also give estimates for the crown expansion in form of its radius R_crown:

– In [95] the growth of crowns formed on the normal impact of an ethanol drop chain on a thinly wetted wall is described. R_crown(τ)/D_prim =K ·(τ −τ₀)ⁿ is obtained with the non-dimensional time τ = t·2·π·f. f denotes the drop frequency and τ₀ a constant, equivalent to a virtual time origin. The exponent is set to n = 0.5.

– In [10] this behaviour is confirmed for the impact of single water drops.

Using τ = t·v_prim/D_prim, n ≈ 0.4 is found. This exponent is shown to be independent of the film thickness and of the impact velocity in contrast toK which is found to depend on the impacting Weber number (We_prim <1000).

However, the fits cannot be used to calculate the maximal radius R_crown,max be-cause the values ofK andτ₀ are not known in the current case. Therefore, an in-tuitive and possibly rather small estimate is chosen withR_crown,max = 1.5·D_prim/2.

The area occupied by the crown is then given asR²_crown,max·π.

• The injected mass flux m˙inj = 19.4g/s in the experiments is known from the injector characteristics. For a distance between target and injector of d_inj.point ≈ 37mm and a main impact Θ-area of ∆Θ_prim ≈15^◦ on the target, see Figure 4.21 in Section 4.3.2.1, the impinging mass flux density can be estimated as qm,prim ≈ 21kg/(s·m²). Assuming spherical drops of uniform drop sizeD_prim ≈50µm, this value leads to a number flux densityq_Nr,prim ≈4.6·10¹¹/(s·m²).

• Finally, the number of drops which impact during crown formation can be assessed as N_impact = q_Nr,prim·t_max·R²_crown,max·π ≈0.6. This means, that approximately

Due to this assessment ideal corona splash is ruled out as main source of secondary atomisation in case of dense spray impact. Yet, spray density is not the only important parameter where Roisman’s/Horvat’s and the new data sets differ: As can be seen in Figure 5.5, they also cover considerably different values of the Reynolds number that describes the ratio between inertial and viscous forces.

For Roisman’s/Horvat’s spray, viscous forces are important which becomes obvious in the scaling D_a/D_b ∼ Re^−0.5_b . An increase in the Reynolds number is equivalent to a decrease of viscosity, i.e. to an increase in the tendency to break up into smaller drops.

For the data gathered in this work, however, the viscous forces do no longer play a role and inertial forces strongly dominate, which is expressed in negligible influence of Re_b. Inertial forces also outweigh other forces, e.g. surface tension, which can be seen if other characteristic numbers, e.g. the Capillary number (which shows a size of ≈1) or the Weber number, are considered, see Section 1.4.1. The overall weighting of forces is changed and inertial forces clearly represent the decisive forces in the spray impact. As a consequence, large parameter influences are no longer to be expected in this regime.

This explains well the results observed in the present experiment.

Summarised, spray impacts considered by Roisman/Horvat and those regarded in this work differ due to the mutual influence of a different spray density and of a changed force weighting. Due to the former, undisturbed corona splash cannot represent the impact phenomena in the measurements of this work and due to the latter, inertial forces dominate strongly and cause negligible parameter influences.

Unfortunately, no data could be found in literature to close the gap around Re_b ≈2000 in Figure 5.5. Further experiments are hence necessary to catch the transition from low to high Reynolds numbers as well as from sparse to dense sprays and to finally formulate a global model.