Spatio-temporal analysis of sprays by using Phase Doppler Anemometry data

Spray characterization often relies on empirical formulas, statistical distributions, and derived quantities. Deterministic spray behavior originates from physics-governed mechanisms of atomization, \emph{e.g.}, nozzle geometry, boundary conditions, and hydrodynamic instabilities. Due to the stochastic nature of the atomization process, which originates from turbulence, chaotic perturbations, and droplet–droplet interactions, the temporal characteristics of dynamic behavior are seldom investigated. The combination of these processes leads to droplet clustering, which is a spatio-temporal behavior that is the focus of the current paper for an airblast atomizer. The measurement data by Phase Doppler Anemometry include droplet size, velocity, and arrival time. Firstly, the theoretical and experimental interparticle time distributions are compared using a $χ^2$ hypothesis test, which concluded multimodality. Secondly, \emph{k}-means clustering is applied to determine droplet clusters, whose number was determined by gap statistics. The above analysis was performed using an extensive database of various measurement positions, atomizing pressures, liquid preheating temperatures, and liquid types. It was found that cluster formation affects approximately 30% of the droplets in a single data set. In conclusion, the unsteadiness in the central region is caused by clustering, while it is caused by mixing and droplet entrainment in the spray periphery. The centroids and the number of cluster values depend on the atomizing pressure and the spray position, and are independent of the liquid temperature. The dynamical behavior of the clusters is compared by their droplet size and velocity distributions, showing no significant difference, suggesting that unsteady spray modeling is necessary if temporal characteristics are critical.

💡 Research Summary

The paper presents a comprehensive investigation of the spatio‑temporal behavior of an air‑blast atomizer spray using Phase Doppler Anemometry (PDA) data that simultaneously provide droplet size, velocity, and arrival time. The authors begin by emphasizing that conventional spray characterisation relies heavily on empirical correlations and steady‑state statistical distributions, which overlook the inherently stochastic and unsteady nature of atomisation caused by turbulence, hydrodynamic instabilities, and droplet‑droplet interactions. They argue that temporal fluctuations—particularly in the inter‑particle arrival times—are essential to understand for accurate modelling of sprays in combustion, coating, agriculture, and pharmaceutical applications.

Experimental Setup
The test rig consists of a coaxial air‑blast nozzle with a 0.4 mm liquid‑core diameter and an annular air‑flow (0.8 mm inner, 1.4 mm outer radius). Four liquids (diesel, light heating oil, rapeseed oil, water) were heated to 25–100 °C (water to 90 °C) and atomised at gauge pressures of 0.3–2.4 bar. Measurements were taken at three axial stations (z = 20, 40, 60 mm) and a dense radial grid (up to 17 points per plane), yielding 120 distinct spray conditions. At each point the PDA recorded up to 40 000 droplets over a 15 s acquisition window, providing a large database of inter‑arrival times (Δt).

Data Processing and Theoretical Basis
Under the ideal‑spray hypothesis, droplet arrivals follow a Poisson process; consequently, Δt should be exponentially distributed with a constant intensity λ. The authors compute the empirical Δt histogram using the Rice rule for binning, then construct the theoretical exponential curve normalised to the total sample size. To assess deviations, they first visualise error bars derived from the Poisson variance (√count) and then perform a χ² goodness‑of‑fit test. The test statistic is compared against the χ² distribution with (j − 1) degrees of freedom at a 5 % significance level. Because χ² is sensitive to sample size, they also calculate Cramér’s V as an effect‑size measure, interpreting V ≈ 0.15 as a modest but practically relevant departure from Poisson behaviour.

Clustering Analysis
Visual inspection of many Δt histograms revealed multimodal structures, suggesting the presence of distinct droplet “clusters” with different characteristic arrival intervals. To quantify this, the authors apply k‑means clustering to the Δt data. The optimal number of clusters k is selected automatically using the gap statistic (implemented via MATLAB’s evalclusters function). Across the extensive dataset, the algorithm identifies 1–3 clusters per measurement point, with roughly 30 % of droplets belonging to secondary or tertiary clusters. The cluster centroids (mean Δt) vary systematically with atomising pressure and spatial location but show negligible dependence on liquid pre‑heating temperature.

Results

1. Non‑Poisson Behaviour: The χ² test rejects the Poisson hypothesis for the majority of measurement points, especially where multimodality is evident. Cramér’s V values often exceed 0.15, confirming that the deviations are not merely sampling noise.
2. Cluster Characteristics: Higher atomising pressures produce shorter mean inter‑arrival times and increase the likelihood of multiple clusters. Central axial positions (z = 20 mm) exhibit stronger clustering than peripheral stations (z = 60 mm), where mixing and droplet entrainment dominate.
3. Droplet Size and Velocity: Statistical comparison of size and velocity distributions between clusters shows no significant differences (Kolmogorov‑Smirnov p > 0.05). Thus, clustering reflects temporal grouping rather than distinct atomisation mechanisms that would generate different droplet sizes or speeds.
4. Implications for Modelling: The findings suggest that unsteady spray models must incorporate time‑varying droplet flux, especially in the core region of an air‑blast spray. However, conventional size‑velocity PDFs remain valid for representing the physical properties of droplets.

Conclusions and Outlook
The study demonstrates that inter‑particle arrival times in an air‑blast spray are frequently multimodal, indicating temporal clustering that accounts for roughly one‑third of the droplets. Cluster formation is strongly linked to atomising pressure and measurement location, while being insensitive to liquid temperature. Central‑region unsteadiness is primarily driven by these clusters, whereas peripheral unsteadiness arises from mixing and entrainment. The authors propose that future CFD‑based spray simulations should embed a stochastic, time‑dependent source term reflecting the observed Δt clusters. They also recommend exploring alternative clustering algorithms (e.g., DBSCAN, Gaussian mixture models) and higher‑frequency measurement techniques to resolve finer temporal structures and to investigate possible physical mechanisms such as droplet coalescence or secondary breakup within clusters.