Multimodal distributions of agricultural-like sprays: A statistical analysis of drop population from a pressure-atomized spray

This paper focuses on the statistical analysis of a droplet population produced by a pressure-atomized jet spray, laying in the second-wind-induced regime, far behind the nozzle. The droplet size and axial velocity derived from droplet tracking velocimetry measurements are shown to follow bimodal distributions and their modeling is tackled in the framework of turbulence and of combustion applications, respectively. In addition, the existence of subsets of droplets showing specific behaviors is brought to light from the analysis of the experimental droplet-size–velocity joint probability distribution function (PDF). Such subsets can be precisely defined using the properties of the size and axial-velocity distributions. Finally, the trend of the joint PDF is depicted due to a quadratic relationship which is derived in the context of combustion and shown to work here as well, far behind the nozzle.

Figure 1

Schematic of the experimental setup used by Felis et al. [8], with $\vec{g}$ the gravity field, $d$ the droplet diameter, and $(u,v)$ the droplet axial and transversal velocities, respectively, along $x$ and $y$.

Figure 2

Distributions of (a) $d/{\langle d\rangle }_{\mathcal{V}}$, (b) $u/{\langle u\rangle }_{\mathcal{V}}$, and (c) $v/{\langle u\rangle }_{\mathcal{V}}$ for experimental data provided by Felis-Carrasco [10]. (a) A logarithmic scale and (b), (c) a semilogarithmic scale are used.

Figure 3

Statistical moments of (a) $d/{\langle d\rangle }_{\mathcal{V}}$, (b) $u/{\langle u\rangle }_{\mathcal{V}}$, and (c) $v/{\langle u\rangle }_{\mathcal{V}}$ for experimental data provided by Felis-Carrasco [10]. The blue lines represent $\langle ·\rangle$ (+), ${\langle ·\rangle }_{\mathcal{V}}$ ($\times$), and $\sigma$ ($*$). The red lines represent $S$ ($\square$) and $\kappa$ ($\circ$).

Figure 4

Fit of the marginal distribution of $d/{\langle d\rangle }_{\mathcal{V}}$ at the axial position $x/d_n=800$ by the distributions from (a) Kooij et al. [26] and (b) Novikov and Dommermuth [15].

Figure 5

Fit of the marginal distributions of $d/{\langle d\rangle }_{\mathcal{V}}$ at the axial position (a) $x/d_n=400$ and (b) $x/d_n=600$ by the droplet-size distribution from Novikov and Dommermuth [15].

Figure 6

Distributions of $u/{\langle u\rangle }_{\mathcal{V}}$ for $x/d_n=800$ at the radial positions (a) $y/d_n=0$, (b) $y/d_n=-8$, (c) $y/d_n=-20$, and (d) $y/d_n=-32$.

Figure 7

(a) Fit of the distribution of $u/{\langle u\rangle }_{\mathcal{V}}^{y/d_n=20}$ at the radial position $y/d_n=20$ by a log-normal distribution (23b) and (b) fit of the distribution ${\mathcal{P}}_{u/{\langle u\rangle }_{\mathcal{V}}}$ over all the radial positions by the model $f\left(u\right)$. The dotted lines represent the experimental data and the solid line represents the models. Both graphs use a semilogarithmic scale.

Figure 8

Droplet-size–velocity joint PDF at (a) $x/d_n=400$ and (b) $x/d_n=800$.

Figure 9

Size-velocity joint PDF for $x/d_n=600$ and droplet subsets discriminated with the characteristics of the size and axial-velocity distributions.

Figure 10

(a) Comparison of two reference velocities for fitting the centerline of the size-velocity joint PDF at $x/d_n=600$ by Eq. (27) where the fit parameter $K$ is set to $3.15{\langle u\rangle }_{\mathcal{V}}^6\times {10}^{-3}$. (b) Comparison of different values of $K$ for fitting the centerline of the size-velocity joint PDF for $x/d_n=600$ by Eq. (27). The arrow indicates increasing values of the fitting parameter $K$.

Figure 11

Comparison of the experimental size PDF and the PDF reconstructed from the size given by Eq. (27) with (a) $K=3.15{\langle u\rangle }_{\mathcal{V}}^6\times {10}^{-3}$ and (b) $K=7.00{\langle u\rangle }_{\mathcal{V}}^6\times {10}^{-3}$. The pluses indicate the relative difference between the two PDFs. The dotted and dashed lines represent relative differences of 0.3 and 0.1, respectively.