Analysis and classification of droplet characteristics from atomizers using multifractal analysis

Atomizers find applications in diverse fields such as agriculture, pharmaceutics and combustion. Among the most commonly found atomizer classes of designs are pressure swirl, airblast and ultrasonic atomizers. However, it has thus far not been possible to identify the class of an atomizer from spray characteristics. We perform multifractal detrended fluctuation analysis on the droplet inter-arrival times, diameters and axial velocities of pressure swirl, airblast and ultrasonic nebulizer sprays to quantify the differences in complexity in the respective signals. We show that the width of the multifractal spectrum of the signals of droplet diameters and the inter-arrival times, measured at the edge of the spray are robust atomizer identifiers. Further, we show the presence of correlations among the droplet diameters which are otherwise considered as random or derived from a log-normal distribution. This study can be further generalized to classify fluid mechanical systems or biological sprays using an appropriately chosen single point measurement in the flow field.


Results
We acquired the droplet characteristics of the sprays from the PSA the ABA, and the ultrasonic nebulizer at various axial and radial locations using a Phase Doppler Interferometer (PDI). The data for the PSA was taken from Dhivyaraja et al. 34 , and the data from the ABA was taken from Rayapati et al. 35 The droplet characteristics include the diameters, axial velocities and the inter-arrival times.
Traditionally, the sprays are typically characterized by their mean droplet diameters (D ), as shown in Fig. 1. The droplet diameter distribution of the sprays produced from PSA and ABA are similar and hence cannot be used to differentiate between the sprays produced from these atomizers. The distribution of D fails to capture the difference in the atomization mechanisms of the ABA and the PSA (Fig. 1). However, distribution of D of the ultrasonic nebulizer spray is quite different from the PSA and the ABA.
Multifractal analysis of spray characteristics. Quantification of complexity using the width of multifractal spectrum (W). W quantifies the complexity and the multifractality embedded in a signal 36 . The width indicates the range of scaling exponents required to describe the signal. A signal with a higher width is described to be more complex and multifractal than the one with a smaller width.
Width of multifractal spectrum of the inter-arrival times (W τ ). We observe that W τ of the PSA does not change with axial distance and hence they are plotted together (Fig. 2). In the case of PSA, the measurements were performed at the axial locations of 11 mm, 22 mm, 33 mm, and 44 mm. On the other hand, W τ of the airblast spray  Width of the multifractal spectrum of the inter-arrival times of the pressure swirl, the airblast and the ultrasonic nebulizer sprays. The width of the multifractal spectrum (W τ ) of the inter-arrival times of the spray are shown for axial distances of 9.5 mm, 12.5 mm and 25 mm for the ABA, at 11 mm, 22 mm, 33 mm and 44 mm for the PSA and at 5 mm for the ultrasonic nebulizer. W τ of the spray formed from the ABA at an axial location of 25 mm is much higher in general than the spray produced from the PSA at all axial locations. The green arrow in the figure depicts a significant difference in the widths of the spectra corresponding to the PSA and the ABA when measured at the spray central axis towards the far field of the atomizer. The maroon arrow depicts a significant difference between the spectra corresponding to the ultrasonic nebulizer from both the ABA and the PSA when measured near the edge of the spray. A significant variation in W τ with axial distance from the atomizer exit is seen for ABA. The mean error bars of different atomizers is shown at radial locations corresponding to the right edge of the sprays. (2019) 9:16218 | https://doi.org/10.1038/s41598-019-52596-6 www.nature.com/scientificreports www.nature.com/scientificreports/ increases as we move farther from the injector exit and hence, they are grouped together accordingly. In the case of ABA, a strong counter-flow toroidal recirculation zone (CTRZ) leads to the presence of long-range correlations in the inter-arrival times, thereby increasing W τ near the spray central axis. For the ultrasonic nebulizer, measurement is performed at an axial distance of 5 mm. W τ of the ultrasonic nebulizer is maximum at the edge of the spray and minimum near the spray central axis unlike the PSA and the ABA.
There is a significant difference between W τ of the nebulizer spray and those of the PSA and the ABA near the edge of the spray (indicated by maroon arrow). Hence W τ can distinguish the ultrasonic nebulizer near the edge of the spray. When measurement is performed close to the spray central axis, W τ can differentiate between the PSA and the ABA (shown by the green arrow).
Fundamentally, the PSA, the ABA and the nebulizer are different in the nature of momentum exchange between the droplets and the surrounding gas phase. The droplets formed in an ABA spray are typically moving slower than the surrounding gas phase. In other words, the ABA spray is best described as containing droplets embedded in a turbulent swirling gas jet. The gas jet, as a result, has several embedded coherent structures which are responsible for clustering and velocity field inhomogeneity. The turbulent swirling gas jet which drives the droplets enables a high correlation. Even in the case of the ultrasonic nebulizer, the incoming air carries the droplets upwards. We can expect high complexity, thereby high W τ near the edge since that is the inlet of the air. In contrast, the gas phase velocity field in a PSA spray is established due to entrainment initiated by the moving droplets. The high inertia droplets are not as affected by the surrounding air as much as in the case of the ABA or the ultrasonic nebulizer. As a result, the inter-arrival times signal of PSA is likely to be less complex. Kantelhardt 14 proposed two possible sources of multifractality in a signal. The first one is due to the broad probability distribution of the values in the signal. The second is the presence of long-range correlations for the different magnitudes of fluctuations in the signal. These two sources of multifractality are differentiated by computing the width of the multifractal spectrum for the randomly shuffled signal and comparing with the original one. If the randomly shuffled data retains multifractality, it is because of the presence of a broad probability density function whereas, the reduction in the width of the spectrum for the randomly shuffled data indicates the presence of long-range correlations.
In order to isolate the source of multifractality due to the long-range correlations among different orders of fluctuations, we compare W τ with the randomly shuffled inter-arrival times (Fig. 3). We observe that W τ of the randomly shuffled inter-arrival times of the airblast spray measured at an axial location of 25 mm (indicated in the legend as Shuffled ABA 2) show a significant drop from the original signal (indicated as ABA 2). This reduction in W τ indicates the presence of long-range correlations among the large and small scale fluctuations in the inter-arrival times. The reduction in W τ is also seen in the pressure swirl spray at the spray central axis. Nonetheless, the reduction in W τ is not as significant at other axial and radial locations of the spray from the ABA (ABA1 and Shuffled ABA1), the PSA (PSA and Shuffled PSA) and the ultrasonic nebulizer (Nebulizer and Shuffled nebulizer). The resultant multifractality even after random shuffling of the data is because of the broad probability distribution of the inter-arrival times.
Width of multifractal spectrum of the droplet diameter signal (W D ). Further, we estimate W D of the droplet diameters (Fig. 4). W D does not vary much with the axial location and hence they are grouped together. W D is higher for the droplet diameter series of the spray obtained from the ABA and the ultrasonic nebulizer than those www.nature.com/scientificreports www.nature.com/scientificreports/ obtained from the PSA. The difference in W D is significant near the spray edge. Thus, W D is a robust classifier of the PSA when measurement is performed at the edge of the spray.
W D for the shuffled data shown in Fig. 5 depends only on the width of the droplet diameter pdf. The diameter distributions for all the atomizers are compared in Supplementary Fig. S3. In comparison, W D for the actual data is a function of the droplet diameter as well as any temporal correlation structure embedded in the data. One can observe that W D for the ABA and the ultrasonic nebulizer (Fig. 5a,c) are markedly different, indicating the presence of long-range correlations among different orders of fluctuations in addition to the broad droplet diameter pdf. However, W D is not so different for the PSA (Fig. 5b). The long-range correlations might be due to the presence of vortices in the sprays from the ABA which centrifuges like-sized droplets towards the edge of the spray, thereby causing the correlations. In contrast, the lack of strong vortical interactions precludes such significant long-range correlations in the spray from a PSA (Fig. 5b). The strong interaction between the air and the droplets near the edge of the spray in ultrasonic nebulizer cause high W D . In contrast to the traditional view that the droplet diameters are random 37,38 , we report the existence of correlations in the airblast spray using W D and the Hurst exponent of the diameters (H D ) (Figs 4 and 7).
In spite of the random shuffling of the diameter time series, the multifractal spectrum does not collapse into a monofractal one (W D ∼ 0 for a monofractal spectrum and around 0.06 for a random noise signal depending on the realization). We conjecture that the droplet inertia is a reason for the presence of multifractality. For validation, we performed multifractal analysis on the droplet diameter series of an aerosol plume (typically produced by a medical nebulizer) where droplet inertia is extremely low. We compare D and u of the aerosol plume with those of pressure swirl, airblast and the ultrasonic nebulizer sprays near the spray central axis. The spray central axis is where D is minimum for the airblast and the pressure swirl sprays (Fig. 1).
For the aerosol plume with an air supply pressure of 50 psi, , which implies that the data is uncorrelated (Refer Fig. 7) and = . ± . W 0 1 0 04 D (close to random noise). Thus, we hypothesize that the droplet inertia is the primary cause of multifractality and the presence of correlations in the droplet diameters.
We also examined the variation of the width of the multifractal spectrum of the droplet axial velocities. However, W u does not show a clear distinction between the spray characteristics of the ABA, the ultrasonic nebulizer and the PSA (not shown here). From Figs 2 and 4, we observe that W τ and W D can be are robust measures to distinguish between the spray characteristics of the PSA, the ABA and the ultrasonic nebulizer, when measurement is made at the edge of the spray at any axial location from the atomizers. In contrast, this difference is not visible in the mean droplet diameters (Fig. 1). Hence, the width of the multifractal spectrum is a stronger measure to validate the empirical models developed for predicting the ABA, the nebulizer and the PSA performance.
Quantification of the persistence of the sprays using the Hurst exponent (H). Hurst exponent is a measure of the amount of correlation in the signal. H is indicative of the persistence of a signal. A signal is persistent if a large value is more likely to be followed by a large value and a signal is anti-persistent if a large value is more likely to be followed by a small value 39 . For a persistent signal, H lies between 0.5 and 1 whereas, for an anti-persistent signal, H lies between 0 and 0.5. H is 0.5 for an uncorrelated signal such as white noise.
The Hurst exponent of multifractal spectrum of the inter-arrival times (H τ ). The persistence in the inter-arrival times encodes information about the clustering of the droplets. The hollow markers in Fig. 6 indicate the cases where no conclusion can be reached as far as persistence is concerned as the inter-arrival times at those radial We observe from Fig. 6 that near the edge of the spray, the inter-arrival times from the ABA are persistent (long-term positive correlations). Based on the initial fluctuations, if the τ between two droplets is small, then the next droplet arrives faster than expected, enabling clustering near the edge of the spray obtained from the ABA. On the other hand, the inter-arrival times of the spray from the PSA are uncorrelated towards the edge of the spray. Alternatively, near the central axis (the region in between the two vertical lines), the τ-signal is persistent for the pressure swirl spray and anti-persistent for the airblast spray. The τ-signal is almost always persistent for the ultrasonic nebulizer at all the radial locations. Thus, for the spray produced from the PSA, clustering can happen near the spray central axis, for the ABA clustering can happen towards the edge of the spray and for the nebulizer, clustering can happen anywhere.
Surprisingly, near the central axis of the airblast spray, the inter-droplet time signal is anti-persistent (a conclusion that can be drawn with 95% confidence). This indicates that the droplet clustering is not observed in ABA near the spray central axis. However, a mechanism responsible for the presence of this anti-persistent nature is yet to be determined. This is consistent with the experimental study on droplet clustering reported by Henien et al. 40 .  www.nature.com/scientificreports www.nature.com/scientificreports/ Thus H τ contains information related to the location of droplet clustering in a spray, thereby reducing the need for further analysis using imaging or performing analysis on the histogram of the inter-arrival times. H τ also shows a distinction in the spray characteristics of PSA, ultrasonic nebulizer and ABA; however, the difference is not as significant as the width of the multifractal spectrum.

The Hurst exponent of the multifractal spectrum of the droplet diameter signal (H D ).
We observe that near the edge of the spray, the droplet diameter series of the spray obtained from the PSA are mostly uncorrelated or slightly persistent (Fig. 7). Contrarily, those obtained from the ABA and the ultrasonic nebulizer are persistent at almost all radial locations (Fig. 7). In all the sprays, droplet diameter series are persistent near the spray central axis indicating the presence of long-range correlations.
We also studied the Hurst exponents of the axial velocities (H u ) of both the sprays. Not surprisingly, the sprays from both ABA, PSA and ultrasonic nebulizer have persistent axial velocities. Classically, this is a well-known characteristic of two-phase turbulent flows 41 . From this study, we observe that H u cannot be used to distinguish between the sprays from the atomizers.
We observe that all the multifractal measures are scattered in all the figures (Figs 2-7). This is because, we grouped data points corresponding to different axial locations of an atomizer as our objective is to classify the atomizers. The radial locations corresponding to the edge of the spray are different for different axial locations. The other reasons for scatter include turbulence and inherent fluctuations in the experiment.
In addition to the comparison of the multifractal measures across different atomizers, we also compared the variation of multifractal measures for the same atomizer at different Reynolds (Re) and Weber (We) numbers. We considered the spray characteristics from the PSA at various radial locations at a Re of 1030 ± 5 for two We, 121.5 and 206. Alternatively, we considered the spray characteristics from the PSA at various radial locations at a We of 210 ± 5 for two Re, 1034 and 1364. We observed that the variation of multifractal measures, such as H τ and W D , with radial location, is same irrespective of the flow conditions ( Supplementary Figs 1 and 2 shown in Supplementary material). Hence, the multifractal measures do not vary with We and Re for the same atomizer. We can conjecture that the measures are dependent on the underlying atomization process and hence can be used to distinguish two atomizers.

Discussion
In this study, we investigated the droplet characteristics of the sprays obtained from the pressure swirl, the airblast atomizers and the ultrasonic nebulizer using MFDFA for the first time. We showed that the Hurst exponent and the width of the multifractal spectrum of the signals of the droplet diameters and the inter-arrival times unveil more information about the complexity in the signals than the standard measures such as mean droplet diameter and mean axial velocity.
We propose that the long range correlations resulting in high widths of multifractal spectrum in the ABA and the ultrasonic nebulizer are due to the strong interaction between the air flow and the droplets. W τ is an efficient discriminant of the ultrasonic nebulizer from the PSA and the ABA when the measurement is performed near the edge of the spray irrespective of the axial location.
We also note the presence of different long-range correlations in the droplet diameters in the ABA and the ultrasonic nebulizer mainly near the edge of the spray. We propose that these correlations are a result of the presence of vortices in the flow field of the airblast sprays which eject the like-sized particles towards the edge of the sprays. In addition to these, we conjecture that the multifractality in the droplet diameter series is due to the droplet inertia. We report that W D can distinguish the PSA from the other two atomizers, provided the measurements are made at the edge of the spray. Thus, using W τ and W D , one can distinguish between the three atomizers when the measurement is performed near the edge of the spray. We related H τ to the droplet clustering at various radial locations and we report that it can be used as a measure to identify cluster locations instead of the spatial imaging of the droplets or other statistics. Further, H D and W D show the presence of correlation in the droplet diameters in the sprays and hence the droplet diameters cannot be considered as random while modeling the atomization process.
The measures, the Hurst exponent and the width of multifractal spectrum should be used to validate the models in addition to the traditional measures such as droplet size distribution, axial velocity distribution. We showed that MFDFA of the measurements can be used for classifying different classes of atomizers, in addition to improving understanding of atomization process. Thus our analysis shows that the inverse problem of identifying the class of flows from the sparse data is possible using MFDFA. This study can be further extended to classify fluid mechanic systems with similar hardware using a single point measurement in the flow field such as classifying sprays produced during biological process such as breathing, coughing etc. This classification can also aid in developing efficient defense strategies such as classification of submarines based on measurement of velocity from its wake.

Methods
Multifractal detrended fluctuation analysis. The brief description of MFDFA is given below.
, , and N is the length of the signal, a mean subtracted cumulative signal (Y) is generated as The new signal is then partitioned into N p non-overlapping windows of equal size p. The local trends are detrended using a linear fit y i . To account for the scaling of fluctuations at multiple scales in the signal, a structure function F ( ) p q of order q and span p is defined as The slope of the linear regime in the logarithmic plot of F p q versus various span sizes, p gives the generalized Hurst exponent H q of order q. The Hurst exponent corresponding to the correlation of the signal is the generalized Hurst exponent of order 2 (H 2 or H).
These generalized Hurst exponents can be represented using a spectrum of singularity exponents f(α) using a Legendre transformation where α is a singularity exponent. The Legrendre transform to obtain the singularity spectrum is given as 42 The multifractal spectrum is represented as the plot between f(α) and α. The width of the multifractal spectrum is given by α α = − W max m in . The Hurst exponent provides information related to the long term correlations and memory of a signal. The width and skewness of the multifractal spectrum determine the complexity and multifractality of a signal. For instance, for a monofractal signal, the multifractal spectrum collapses to a single point.
We perform MFDFA on the spray characteristics, the droplet diameters, the inter-arrival times and the axial velocities. The data is divided into segments of 7000 data points each. The multifractal measures such as the width of the multifractal spectrum and the Hurst exponent are computed for each window. To compute the multifractal spectrum for these signals, all possible scales ranging from 16 to 1024 are considered since the data appears to be very noisy. The mean of the values computed for each window is plotted and the standard deviation of these measures gives the error bar shown in the figures representing a 95% confidence interval. We plotted the mean Hurst exponent and the width of the spectrum at each location and the error bar is given by the standard deviation of these measures.
Experimental method. The PSA, the ABA and the ultrasonic nebulizer are shown in Fig. 8. In the PSA, water (indicated by blue arrows) enters the atomizer inlet and fills the annulus. The top view of the swirl chamber is shown in Fig. 8a. The liquid enters into the swirl chamber (indicated in pink color) through four tangential inlet slots (the water flow is indicated with blue arrows) and converts the kinetic energy of the liquid into swirl energy. A thin conical swirling liquid sheet exits the final orifice with an air-core formation in the atomizer center. The liquid sheet further disintegrates into drops and forms a hollow cone spray. A pre-filming airblast atomizer is shown in Fig. 8b. The pink color indicates the primary and secondary swirler region of the atomizer. The air and water flows are indicated with black arrows and blue arrows. Initially, the slow-moving thin liquid film is formed in the pre-filmer. Then the primary and the secondary air with high swirl velocity interact with the thin liquid film and forms a spray with fine droplets. The ultrasonic nebulizer is shown in Fig. 8c. The air and water flows www.nature.com/scientificreports www.nature.com/scientificreports/ are indicated with black and blue arrows, respectively. In the ultrasonic nebulizer, the piezoelectric transducer (shown in red) immersed inside the water, produces the ultrasonic actuation. The acoustic waves generated due to the ultrasonic actuation, travel towards the free surface of the water, and form an aerosol (shown as blue droplets). The air flow is then used to transport the aerosol.
The spray characteristics such as droplet diameter, axial velocity and its inter-droplet arrival times are simultaneously measured using Atrium ® Phase Doppler Interferometer (PDI). PDI is a point based measurement technique in which two laser beams of same wavelength, but shifted in frequency, intersect in space. The intersecting region forms the measurement volume. The detector receives a Doppler signal from a drop passing through the measurement volume. The system can measure drop sizes within a range of 0.5 to 1500 μm with an accuracy of 0.5 µm and the velocity values within a range of −100 to 300 m/s with an accuracy of ±0.2%. The measurement volume is traversed along different radial locations in the mid-plane of spray at different axial locations. At each measurement locations, more than 10,000 samples were captured. The sampling location along the radial direction is equally binned such that the last bin has a minimum sampling rate of at least 10 Hz. For the spray from PSA measurements are performed at the axial locations of 11 mm, 22 mm, 33 mm and 44 mm from the atomizer exit. Similarly for the air blast spray, the measurements are performed at the axial locations are at 9.5 mm, 12.5 mm and 25 mm from the atomizer exit. Finally, for the nebulizer, the measuments are performed at an axial location of 5 mm from the atomizer exit.
Detailed descriptions of the measurement techniques, experimental setups of the pressure swirl and the airblast atomizers are given in Dhivyaraja et al. 34 and Rayapati et al. 35 , respectively.