4.1 Macroscale results

Figure 4. (a) Plot of the flow rate $Q$ (l min $^{-1}$ ) as a function of dimensional supply pressure $\unicode[STIX]{x0394}p$ (bar) for all the atomizers. (b) Plot of dimensionless flow rate $Re$ as a function of dimensionless supply pressure $\overline{\unicode[STIX]{x0394}p}$ . It can be seen that $Re\sim \overline{\unicode[STIX]{x0394}p}^{\unicode[STIX]{x1D6FC}}$ with $R^{2}>0.99$ . Here, $\unicode[STIX]{x1D6FC}=0.46\pm 0.009$ with $99.7\,\%$ confidence. For reference $Re\sim \sqrt{\overline{\unicode[STIX]{x0394}p}}$ is also plotted. The value $\unicode[STIX]{x1D6FC}=0.46$ is a better estimate than $1/2$ for these data over $Re$ varying by a factor of $15$ .

Figure 4(a) shows the variation of volume flow rate ( $Q$ , l min $^{-1}$ ) for all the atomizers at varying supply pressure conditions ( $\unicode[STIX]{x0394}p$ , bar). Firstly, it can be observed that the atomizer flow rate increases as the supply pressure increases. In addition, as the atomizer scale increases from $1$ x to $10$ x, the flow rate at a constant supply pressure also increases. Figure 4(b) shows the data from figure 4(a) plotted in non-dimensional terms as $Re$ versus $\overline{\unicode[STIX]{x0394}p}$ , by defining a dimensionless pressure $\overline{\unicode[STIX]{x0394}p}=\unicode[STIX]{x0394}p\,D_{o}^{2}/\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}^{2}$ and $Re$ is calculated from equation (2.5).

Experiments were carried out on different atomizers operating such that the same dimensionless supply pressure conditions were achieved. As expected, they yield the same dimensionless flow rate ( $Re$ ) for all atomizer scales. In addition, the data over dimensionless supply pressures varying over three orders of magnitude collapse onto one power-law curve for all the atomizers. It is no surprise that the Weber number ( $We$ ) plays no role in this scaling process; data corresponding to multiple $We$ at a constant $Re$ collapse to one point. A Froude number, $Fr=u_{s}/\sqrt{gD_{o}}$ , where $g$ is acceleration due to gravity, is calculated for the range of sprays considered. It was observed that $Fr$ ranged from 9 to 307, indicating that the effect of gravitational force on the spray is negligible.

A best fit of the data was generated using the maximum likelihood estimation technique (Coles Reference Coles2001). As can be seen from figure 4(b), the flow rate data collapse onto one curve for different values of $\overline{\unicode[STIX]{x0394}p}$ , given by $Re\sim \overline{\unicode[STIX]{x0394}p}^{\unicode[STIX]{x1D6FC}}$ , where $\unicode[STIX]{x1D6FC}=0.46\pm 0.009$ . The error bounds on the power-law exponent are $\pm 0.009$ with $99.7\,\%$ confidence, clearly suggesting that $0.5$ is outside the confidence bounds. Most standard textbooks report that the flow rate of the atomizer varies as the square root of the supply pressure (Lefebvre Reference Lefebvre1989; Bayvel & Orzechowski Reference Bayvel and Orzechowski1993). We would like to suggest that $0.46$ is a more accurate exponent to determine the actual flow rate variation in pressure swirl atomizers. The percentage error between the two exponents – $0.5$ and $0.46$ – could be significant in many instances. For example, as we attempt to approach the limits of low or high flow rates, such as in the current case, this difference in the exponents could play a major role. Finally, the exponent from figure 4(b) was derived based on measurements over three orders of magnitude variation in the dimensionless pressure. Based on the above discussion, we would also like to suggest that a more accurate definition of a dimensionless flow number ( $FN$ ) for pressure swirl atomizers is $FN=Re/\overline{\unicode[STIX]{x0394}p}^{0.46}$ . It was observed that $FN=0.067\pm 0.016$ for our design and is independent of the atomizer scale. The pressure drop in the inlet ports was independently calculated using the friction factor approach. From these calculations, it was found that at least 50 % of the pressure drop (typically 80 %) occurred in the swirl chamber, with the remaining pressure drop occurring in the inlet ports. This again confirms that the flow rate is governed more by the air core dynamics than by the viscous losses in the inlet port.

The physical reason for the deviation of the exponent from $0.5$ lies in the fact that the ‘square-root law’ was originally derived for flow through an orifice (Lefebvre Reference Lefebvre1989) and is appropriate in instances where the cross-sectional area available to the flow is constant. For example, diesel injectors have been shown to follow such scaling (Payri et al. Reference Payri, Bermudez, Payri and Salvador2004). In the case of pressure swirl atomizers, the effective cross-sectional area available to the flow is a function of the flow rate. This is due to the presence of an air core at the atomizer exit (Lefebvre Reference Lefebvre1989) and therefore it should not be expected that the flow rate scales as the square root of the supply pressure. Several studies have analysed the effect of $Re$ on the air core relative diameter. For example, Halder, Dash & Som (Reference Halder, Dash and Som2002), Som (Reference Som2012) and Amini (Reference Amini2016) have observed that the air core relative diameter increases slightly as $Re$ increases. Amini (Reference Amini2016) has reported that $d_{a}/D_{o}\sim Re^{0.1}$ . Owing to this, the effective cross-sectional area available to the liquid is reduced. As a result, the flow rate varies with a power-law exponent less than $0.5$ .

Figure 5. Instantaneous photographs of the spray from different atomizers, all operated at $Re\approx 1035$ : (a) $We=206$ from the $3$ x, (b) $We=122$ from the $5$ x and (c) $We=64$ using the $10$ x atomizer. Each image shows a scaled area which is $13.5D_{o}\times 27D_{o}$ . The scale bar shown in all images corresponds to 8 mm. The scaled breakup length (in $D_{o}$ units) in all cases is relatively independent of $We$ at this value of $Re$ .

We next discuss the near-nozzle flow structure in each of the atomizers obtained using high-speed imaging at 10 000 f.p.s. Figure 5(a–c) shows photographs of the near-nozzle breakup structure for the $3$ x, $5$ x and $10$ x sprays all showing a region $z/D_{o}\times r/D_{o}=13.5\times 27$ . Here $r$ and $z$ are the radial and axial coordinates as shown in figure 3(a). In other words, the field of view in all these images was scaled with $D_{o}$ . All such images were obtained at the constant spray condition, $Re\sim 1035$ , but varying $We$ as ( $206,122,64$ ). Firstly, it can be observed that the spray cone angles as well as the breakup lengths for these three sprays are nearly the same and independent of $We$ . Secondly, it can be observed that the wavelength of the observed waves in figure 5(c) for the $10$ x spray is finer than that observed in figure 5(a,b) for the $5$ x and $3$ x sprays. In conclusion, it can be seen from these images that $We$ (again) plays only a minor role in determining the dynamical structure of the liquid sheet. We will revisit these three sprays again when we discuss the unsteady characteristics analysed using POD in § 4.4.

4.2 Microscale results

Drop size and velocity spectra were measured on all sprays using PDI for the different atomizers and at varying $Re$ and $We$ . These measurements were made at various equally spaced radial locations in the midplane of the spray. All measurements were made at a fixed dimensionless downstream location ( $\overline{z}=z/D_{o}=19$ ). A schematic showing the measurement locations is shown in figure 3(a). We will begin with a discussion of the liquid-flux-based and number-flux-based global Sauter mean diameter calculated for the different operating conditions and listed in table 1. These values were calculated following the approach proposed by Tratnig & Brenn (Reference Tratnig and Brenn2010). A regression analysis of these data showed that the qualitative variation of $D_{32}$ with both $Re$ and $We$ follows the previous literature.

As mentioned before, we are interested in investigating if the spatial structure of mean drop size and velocity distributions in a spray exhibits dynamical similarity. In addition, we are interested in identifying the parameter space where viscous effects inside the injector affect the drop size spectrum in the spray. For this purpose, we define a similarity variable

The above choice of $\unicode[STIX]{x1D701}$ is motivated by the similarity variable for a round jet $r/z$ , but has been modified for the fact that spray cone angle can vary unlike in a round turbulent jet. Here, $\unicode[STIX]{x1D703}_{m}$ is the half cone angle of the spray; and $z\tan \unicode[STIX]{x1D703}_{m}$ is the radial distance to the edge of the spray, which implies that $|\unicode[STIX]{x1D701}|=1$ denotes the spray edge. In effect, $\unicode[STIX]{x1D701}$ represents the fractional radial distance to the edge of the spray at a given $z$ location allowing both positive and negative values to denote the two spray edges visible to the camera.

Figure 6. (a) Plot of dimensionless Sauter mean diameter ( $D_{32}/t_{v}$ ) versus $\unicode[STIX]{x1D701}$ with the viscous liquid sheet thickness ( $t_{v}$ ) chosen as the length scale. (b) Plot of dimensionless Sauter mean diameter ( $D_{32}/\unicode[STIX]{x1D6FF}_{o}$ ) versus $\unicode[STIX]{x1D701}$ with the exit boundary layer thickness ( $\unicode[STIX]{x1D6FF}_{o}$ ) chosen as the length scale. The legend represents the $(Re,We)$ corresponding to each dataset, and the same legend applies to both panels. All data were obtained at $\overline{z}=19$ . In (a), the profiles for various flow conditions nearly collapse onto a single curve, except for the data corresponding to $(Re,We)$   $=$  (364,76). This shows that the inviscid length scale is not appropriate for $Re\leqslant 1000$ . In (b), all profiles for various flow conditions nearly collapse onto a single curve for $|\unicode[STIX]{x1D701}|<1/2$ . It appears that $\unicode[STIX]{x1D6FF}_{o}$ is a suitable length scale for  $Re$ .

Figure 6(a) presents the Sauter mean diameter non-dimensionalized using the viscous length scale ( $t_{v}$ ). The corresponding dimensional data are presented in appendix C for completeness. Figure 6(a) presents a plot of $\overline{D_{32}}=D_{32}/t_{v}$ versus $\unicode[STIX]{x1D701}$ for all operating conditions considered. It can be seen that the scaled $\overline{D_{32}}$ data generally collapse onto a single curve over the range of atomizer length scales, $Re$ and $We$ considered. This indicates that $\overline{D_{32}}$ exhibits self-similar behaviour. Upon closer examination, one can observe that, in the region defined by $|\unicode[STIX]{x1D701}|\leqslant 0.5$ , the data collapse is nearly perfect, except for the data corresponding to the $1$ x atomizer (which happens to be for a low $Re=364$ ).

Figure 6(b) is similar to figure 6(a) except that the boundary layer thickness at the nozzle exit ( $\unicode[STIX]{x1D6FF}_{o}$ ), as calculated from the Bloor–Ingham theoretical boundary layer analysis, is chosen as the length scale. As can be seen from figure 6(b), the data for the low- $Re$ spray show a much better collapse along with the other high- $Re$ data. From figure 6(a,b), one can conclude that $\unicode[STIX]{x1D6FF}_{o}$ is a better choice of length scale if the obligation is to choose a scale that will apply over a wider range of  $Re$ .

In figure 6(a,b), the data do not collapse for $|\unicode[STIX]{x1D701}|>0.5$ , since this region is approaching the edge of the spray. In this region, there is a dependence of the scaled $\overline{D_{32}}$ with $Re$ . In other words, a central core region of the spray can be identified where the scaled $\overline{D_{32}}$ shows self-similar behaviour following the predictions of Kostoglou (Reference Kostoglou2003) and Rayapati et al. (Reference Rayapati, Bhamidipati, Peddieson and Panchagnula2010). From this observation, one can write a self-similar radial Sauter mean diameter profile for pressure swirl sprays as $D_{32}(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D6FF}_{o}h(\unicode[STIX]{x1D701})$ . Here, $h(\unicode[STIX]{x1D701})$ is a function that defines the self-similar variation of $\overline{D_{32}}$ as shown in figure 6(b). The number flux and the volume flux information is observed in the region $|\unicode[STIX]{x1D701}|\leqslant 0.5$ , which is the central region of the spray consisting of a very high number flux of small drops effectively resulting in a low volume flux. This is the region where similarity arguments hold and not in the outer region $|\unicode[STIX]{x1D701}|>0.5$ .

The self-similarity exhibited by the $D_{32}/\unicode[STIX]{x1D6FF}_{o}$ profiles in the $\unicode[STIX]{x1D701}$ coordinate is novel, interesting and deserves discussion. In many ways, a pressure swirl spray appears to be analogous to an annular gas jet, where an inner mixing zone contains a toroidal vortex which is responsible for mass being transported towards the centreline (Ko & Chan Reference Ko and Chan1978). Since pressure swirl sprays are generally hollow-cone, the inward radial droplet transport into the core region is fluid dynamically induced. Therefore, it is reasonable to expect self-similar mean droplet size profiles in this region.

Figure 7. Plot of number flux and volume flux versus radial location for $(Re,We)=(1029,122)$  (a) and $(1057,64)$  (b). The PDI measurements are captured from the centre of the spray to the edge. The number flux will be a maximum in the spray centre where the volume flux is a minimum. This indicates that the core region of the spray consists of a large number of small drops. PDI measurements are performed by traversing up to where the number flux drops to a value less than $100~\text{s}^{-1}$ .

The scaling in figure 6 is most appropriate only for $|\unicode[STIX]{x1D701}|<1/2$ . Outside this region, it is likely that ballistic motion of large drops will be responsible for breakdown of similarity. This conclusion is also supported by comparing the number flux to volume flux presented in figure 7 for two different operating conditions. Figure 7 shows a plot of number flux ( ${\dot{n}}$ ) and volume flux ( $\dot{q}^{\prime \prime }$ ) versus $r$ for two representative $Re$ and $We$ cases. As can be inferred from figure 7, the core region of the spray where the similarity is most clearly observed consists of large numbers of small droplets. Since the small drops in the spray are likely to show ‘diffusive’ behaviour, it is likely that similarity is observed in this part of the spray. The discussion pertaining to figure 6 suggests that similarity is only restricted to $|\unicode[STIX]{x1D701}|<1/2$ . When droplet ballistic motion dominates spray transport, the scaling in (4.1) is likely to break down.

We will next discuss drop size distributions and their characteristics. As is well known, PDI yields joint drop size–velocity probability density function (p.d.f.) information at one sampling location. Therefore, when one performs several such pointwise measurements in a spray, it is not clear which of the points is representative of the nozzle performance. In contrast, it would be more useful to characterize a global size–velocity p.d.f. as though one were sampling all drops at an entire cross-sectional axial location. Such a p.d.f. would be insensitive to external factors and, more importantly, would be a direct measure of the nozzle performance. In this study, we use the approach suggested by Tratnig & Brenn (Reference Tratnig and Brenn2010) to calculate such a global p.d.f. Assuming symmetry, a given radial location of the probe can be construed to represent half the annular slice of the spray as shown in figure 3(a). Following Tratnig & Brenn (Reference Tratnig and Brenn2010), we calculate a dimensionless global joint particle size–velocity number p.d.f., $g(\overline{x},\overline{u})$ , at a given axial location. Here, $\overline{x}$ is a dimensionless particle size and $\overline{u}$ is a dimensionless axial velocity component given by $\overline{u}=u/u_{s}$ ; $g$ is computed based on the number flux density of drops ${\dot{n}}(\overline{x},\overline{u};r)$ . In other words, ${\dot{n}}(\overline{x},\overline{u};r)\,\text{d}\overline{x}\,\text{d}\overline{u}\,\text{d}t$ is the number of drops of dimensionless size in the range $(\overline{x},\overline{x}+\text{d}\overline{x})$ and dimensionless axial velocity magnitude in the range $(\overline{u},\overline{u}+\text{d}\overline{u})$ at a radial location $r$ in a time interval ( $\text{d}t$ ). Mathematically, $g(\overline{x},\overline{u})$ can be computed as

The global p.d.f. presented in equation (4.2) is a number-flux-based reconstruction from pointwise PDI data. An analogous volume-flux-based reconstruction can also be invoked. These p.d.f.s describe the joint drop size and velocity spectrum at a given axial location. This p.d.f. would change from one given axial location to another only if significant drop coalescence or secondary atomization were present. In other words, mere rearrangement of drops between radial or azimuthal locations in the spray would not alter $g(\overline{x},\overline{u})$ . The limits of numerical integration were chosen such that at least $99.99\,\%$ of the drops were included in the p.d.f. computation. In addition, as can be seen from figure 7, ${\dot{n}}$ (as well as $\dot{q}^{\prime \prime }$ ) decreases exponentially as a function of radial location. Since ${\dot{n}}\sim \text{e}^{-r}$ , the integral in (4.2) which is weighted by ${\dot{n}}r$ converges rapidly. From the measurement process, we have ensured that the global p.d.f.s contain at least 99.99 % of the droplets in the spray both by number and by volume. Finally, the outermost radial location for each $(Re,We)$ pair was chosen such that at least 99.99 % of the drops were sampled in each case.

The marginal p.d.f.s of the joint size–velocity p.d.f. in each of the dimensionless coordinates (size, $\overline{x}$ , and axial velocity, $\overline{u}$ ) are given by ${\mathcal{X}}(\overline{x})$ and ${\mathcal{U}}(\overline{u})$ , respectively. These p.d.f.s can be obtained from (4.2) as

In addition to the global and marginal p.d.f.s, we would also like to investigate if a size–velocity correlation exists. Towards this end, we define three velocity-conditioned size p.d.f.s. The drops are classified into three quantiles based on their velocity, denoted herein as slow- ( $S$ ), medium- ( $M$ ) and fast-moving ( $F$ ) drops. Each quantile contained an equal number of drops. Let $\overline{u}_{s}$ be the upper limit of the slow-moving drop quantile and let $\overline{u}_{m}$ be the lower limit of the fast-moving drop quantile. The global p.d.f. of dimensionless drop size conditioned on velocity is calculated for each of the three groups as

Figure 8. Plot of the scaled global p.d.f. of (a) drop size ( $x$ ) scaled by viscous length scale $(t_{v})$ based on (2.6), (b) drop size ( $x$ ) scaled by boundary layer thickness $(\unicode[STIX]{x1D6FF}_{o})$ calculated analytically from Bloor–Ingham analysis for each $(Re,We)$ and (c) drop size ( $x$ ) scaled by $D_{32}$ calculated for each p.d.f. The legend, which is the same for all panels, shows $(Re,We)$ for each dataset. The scaled drop size p.d.f. based on the boundary layer thickness as well as $D_{32}$ show a universal behaviour especially for $x/\unicode[STIX]{x1D6FF}_{o}$ and $x/D_{32}>1$ .

In order to compare drop size p.d.f.s from different atomizers, we define an appropriately non-dimensionalized $\overline{x}$ . Figure 8(a) is a plot of the scaled global drop size p.d.f., ${\mathcal{X}}(\overline{x})$ , versus $\overline{x}$ with $t_{v}$ (the viscous liquid sheet thickness) chosen as the length scale for non-dimensionalization ( $\overline{x}=x/t_{v}$ ). The scaled drop size distribution for the $1$ x spray indicates coarser atomization (in relation to $t_{v}$ ), even though the absolute p.d.f. (in $\unicode[STIX]{x03BC}\text{m}$ ) indicated a finely atomized spray (see appendix C for details). As can be observed from figure 8(a), the p.d.f.s for $Re>1000$ show good collapse and the data for low $Re$ show improvement especially for $x/t_{v}>0.5$ . Again for $x/t_{v}<0.2$ , the p.d.f.s show $Re$ dependence. This indicates that viscous effects cannot be ignored at low  $Re$ .

Figure 8(b) is again a plot of the same p.d.f., ${\mathcal{X}}(\overline{x})$ versus $\overline{x}$ , but with a different length scale for non-dimensionalization. Here, $\unicode[STIX]{x1D6FF}_{o}$ is calculated analytically from the Bloor–Ingham analysis and chosen as the length scale. Consequently, $\overline{x}=x/\unicode[STIX]{x1D6FF}_{o}$ . As can be observed from this figure, the p.d.f.s for all $Re$ , including the low- $Re$ spray, collapse onto a single curve for $x/\unicode[STIX]{x1D6FF}_{o}>1$ . For $x/\unicode[STIX]{x1D6FF}_{o}<1$ , the p.d.f.s show $Re$ dependence, where smaller drops are increasingly probable for larger $Re$ . This implies that drops in the spray smaller than the boundary layer thickness ( $\unicode[STIX]{x1D6FF}_{o}$ ) are formed from a different physical process, such as secondary breakup, and hence do not collapse with $\unicode[STIX]{x1D6FF}_{o}$ as the length scale. Therefore, we would like to suggest that models of drop size distribution should take this difference in physics into account by invoking two piecewise models, one each for $x/\unicode[STIX]{x1D6FF}_{o}>1$ and $x/\unicode[STIX]{x1D6FF}_{o}<1$ . To the best of our knowledge, this is the first report where experimentally measured mean drop size as well as drop size p.d.f.s have both been shown to be dynamically similar and to exhibit universal scaling. Figure 8(c) is a plot of the p.d.f. of $x/D_{32}$ . Interestingly, again, one can observe behaviour similar to that in figure 8(b).

It may be recalled that ${\mathcal{X}}(\overline{x})$ is the global drop size p.d.f. at an axial location $z/D_{o}=19$ . If secondary atomization and coalescence are insignificant after this axial location, this drop size p.d.f. is likely to remain unchanged as a function of axial distance. Therefore, the universality of the collapse shown in figure 8(b,c) is even more profound in that a distribution constructed from these experimental data represents a universal representation of the drop size p.d.f. in a pressure swirl spray, which is insensitive to both $Re$ and axial location. We refrain from suggesting a particular mathematical form of this distribution, as that discussion will distract the reader from our primary emphasis of showing universality. In conclusion, we would like to argue that $\unicode[STIX]{x1D6FF}_{o}$ , the boundary layer thickness at the nozzle exit, plays a key role in determining the liquid sheet thickness and therefore the global drop size spectrum in a spray. It is gratifying to see that p.d.f.s of $x/D_{32}$ also show self-similarity, but p.d.f.s of $x/\unicode[STIX]{x1D6FF}_{o}$ (with $\unicode[STIX]{x1D6FF}_{o}$ calculated analytically) showing similarity offers a route to predict the p.d.f. from first principles.

The length scale choice, as has been discussed before, can be either a sheet thickness or a boundary layer thickness. The sheet thickness, in turn, can be calculated purely based on the atomizer’s geometrical parameters which does not include the effect of viscosity or can be calculated after including the effect of viscosity. For low $Re$ values, the sheet thickness is influenced by viscosity. In other words, the sheet thickness is greater due to the viscous boundary layer and hence results in lower mean axial velocity of the liquid sheet. In addition, at low $Re$ , the effect of viscosity delays the breakup process. It has been argued by Lefebvre & Suyari (Reference Lefebvre and Suyari1986) that for $Re>1000$ (in our nomenclature), the experimental sheet thickness is likely to be close to the inviscid sheet thickness. The critical value of $Re$ , identified from our experiments, agrees with this prediction of Lefebvre & Suyari (Reference Lefebvre and Suyari1986). We find that the probability of finding smaller drops appears to be sensitive to $Re$ and $We$ . In other words, $\unicode[STIX]{x1D6FF}_{0}$ appears to provide a distinguishing scale that can, in fact, be used to define ‘large’ and ‘small’ drops from the point of view of universality of the p.d.f.

Figure 9. (a) Plot of scaled mean velocity, $\overline{u}=u/u_{v}$ , versus $\unicode[STIX]{x1D701}$ . The data show universal scaling especially for $|\unicode[STIX]{x1D701}|<1/2$ . (b) Plot of the global scaled drop velocity p.d.f. The drops are divided into three quartiles denoted by $S$ , $M$ and $F$ indicating slow-, medium- and fast-moving drops. (c) Plot of scaled volume flux $\dot{q}^{\prime \prime }/u_{v}$ versus $\unicode[STIX]{x1D701}$ . Inset in (c) shows the same data plotted in log scale. The legend shows $(Re,We)$ for each dataset, and is the same for all panels. The p.d.f. in (b) exhibits an asymmetry towards slower-moving drops. Also, the data collapse around a universal p.d.f., especially for $\overline{u}<1$ .

We will next discuss the behaviour of the drop velocity spectra. Figure 9(a) is a plot of the dimensionless mean axial velocity $\overline{u}$ versus $\unicode[STIX]{x1D701}$ . Here, $\overline{u}=u/u_{v}$ . This choice stems from the expectation that the droplet axial velocity $u$ scales with $u_{v}=Q/\unicode[STIX]{x03C0}D_{o}t_{v}$ , which is a velocity scale associated with the viscous liquid sheet effusing from the nozzle. Figure 9(a) contains data from several atomizers and for multiple values of $Re$ . As can be observed from figure 9(a), the data appear to show dynamical similarity. In addition, it can be observed that the scaled axial velocity shows a central core ( $|\unicode[STIX]{x1D701}|<1/2$ ) that is moving slower than the drops near the edge. From the above, we can conclude that $\overline{u}$ is generally a function of $\unicode[STIX]{x1D701}$ . As can be seen in figure 9(a), the droplet mean velocity is maximum near the edge of the spray ( $|\unicode[STIX]{x1D701}|=1$ ) and minimum near $\unicode[STIX]{x1D701}=0$ , indicating hollow cone behaviour. On a related note, effervescent sprays as well as many other sprays have been shown to exhibit self-similar mean velocity profiles (Panchagnula & Sojka Reference Panchagnula and Sojka1999; De Vega, Rodríguez & Lecuoa Reference De Vega, Rodríguez and Lecuoa2000).

Velocity profile similarity has been discussed in the context of single-phase jets, which (in that case) implies both flux and advection velocity similarity. However, in the case of sprays, dispersed phase flux is different from drop velocity owing to spatially varying volume fraction. In order to demonstrate flux scaling, figure 9(c) is presented. Figure 9(c) is a plot of the dimensionless mean volume flux $\dot{q}^{\prime \prime }/u_{v}$ versus $\unicode[STIX]{x1D701}$ . Here $\dot{q}^{\prime \prime }$ ( $\text{cm}^{3}~\text{cm}^{-2}~\text{s}^{-1}$ ) is the mean liquid volume flux. As can be seen from this figure, the data show reasonable similarity scaling especially for $|\unicode[STIX]{x1D701}|<1/2$ given that PDI flux measurements are known to be noisy. The scaling is not as clean as in a gas jet. In addition, the data do not scale for $|\unicode[STIX]{x1D701}|>1/2$ owing to the fact that this region could involve large drops exhibiting ballistic motion. The scaling is most appropriate when studying the inner core region and liquid flux distribution is a better parameter to demonstrate similarity. The dimensional mean volume flux versus radial location information is added in appendix C.

The continuous phase motion in the central core region is in fact induced by the droplet motion at $|\unicode[STIX]{x1D701}|=1$ . In that sense, the central core region of a pressure swirl spray is conceptually similar to a wake. Most of the droplets at source are injected into a region near $|\unicode[STIX]{x1D701}|=1$ . The droplets are then transported into the core region ( $|\unicode[STIX]{x1D701}|<1/2$ ) by the continuous phase. The variation of the mean droplet size in this central core region is therefore governed by an ‘advection–diffusion’ process, where the particle diffusivity has a size dependence (Kostoglou Reference Kostoglou2003). In the absence of particle inertia, it is expected that such a random walk process could be responsible for droplet transport. This could be a reason for the self-similarity exhibited by $\overline{D_{32}}$ and drop size p.d.f. discussed in the context of figure 6.

Figure 9(b) presents a plot of the scaled axial velocity p.d.f., ${\mathcal{U}}(\overline{u})$ , for the different cases considered. As can be seen from this figure, all the data (including the data from the $1$ x spray) collapse onto a single curve for $\overline{u}<1$ . Secondly, the velocity p.d.f. curves exhibit significant non-Gaussian behaviour in the form of skewness towards slower velocities. The various datasets appear to show differences for $\overline{u}>1$ . This implies that drops moving slower than the parent liquid sheet exhibit a scaling independent of $Re$ and $We$ . However, for drops whose axial velocity is greater than the liquid sheet velocity, i.e. for $u/u_{v}>1$ , remnant effects of both $Re$ and $We$ persist in the p.d.f. This, again, implies that the physical mechanism that created drops which move faster than the parent liquid sheet (in spite of drag forces which attempt to slow these drops) is different. This implies, again, that two piecewise models of the p.d.f. are required – one for $u/u_{v}<1$ and second for $u/u_{v}>1$ . The differences in the drop size p.d.f. curves in figure 8(a) for $x/t_{v}>1$ are related to the differences in the velocity p.d.f. curves in figure 9(b), since large drops are typically moving faster. The fact that both drop size and velocity marginal p.d.f.s shown in figures 8(b) and 9(b) collapse onto a universal curve points to a conclusion that dynamical similarity in sprays can be extended to model even microscale characteristics, if appropriate scales are chosen.

Figure 10. Plots of scaled global drop size p.d.f. conditioned on velocity. The legend shows $(Re,We)$ for each dataset, and is the same for all panels. Panels (a), (b) and (c) show plots with boundary layer thickness $\unicode[STIX]{x1D6FF}_{o}$ chosen as the length scale and represent slow-, medium- and fast-moving drops, respectively. The p.d.f. for the faster[moving drops in (c) exhibits universal behaviour independent of the values of $(Re,We)$ .

As mentioned before, $g(\overline{x},\overline{u})$ is a joint size–velocity p.d.f. that contains information about the size–velocity correlation. In order to analyse the magnitude of this correlation, we plot a drop size p.d.f. conditioned on velocity, which was earlier defined in equation (4.4). Figure 10(a–c) shows plots of ${\mathcal{X}}_{s}(\overline{x})$ , ${\mathcal{X}}_{m}(\overline{x})$ and ${\mathcal{X}}_{f}(\overline{x})$ , respectively, with $\unicode[STIX]{x1D6FF}_{o}$ chosen as the length scale for non-dimensionalization ( $\overline{x}=x/\unicode[STIX]{x1D6FF}_{o}$ ).

As can be seen from these figures, the data for even the low- $Re$ , $1$ x spray also collapse onto the other data for high $Re$ , especially for $x/\unicode[STIX]{x1D6FF}_{o}>1$ . In other words, drops larger than the boundary layer thickness exhibit similarity with increase in $Re$ . However, minor differences are observable for all the p.d.f.s, especially for $x/\unicode[STIX]{x1D6FF}_{o}<1$ and in the case of slow-moving drops, ${\mathcal{X}}_{s}(\overline{x})$ shown in figure 10(a). On the other hand, the collapse is nearly perfect for the fast-moving drops, ${\mathcal{X}}_{f}(\overline{x})$ in figure 10(c). For $x/\unicode[STIX]{x1D6FF}_{o}<1$ , i.e. drops smaller than the boundary layer thickness, the physical mechanism causing their formation, and hence the scaling with $Re$ , could be different. This adds another hitherto unknown dimension to the literature on dynamical similarity in sprays, in that conditional p.d.f.s of drop size also exhibit universal behaviour over a wide range of $Re$ , if the drop size is scaled by $\unicode[STIX]{x1D6FF}_{o}$ and velocity by $u_{s}$ . The universal collapse is most (least) evident from the conditional p.d.f. of the fastest (slowest) quartile of drops in figure 10(a,c). This further corroborates the residual $Re$ and $We$ effect observed for the smaller drops (which are also the slowest) in figure 8(c).

We have calculated the distribution of Stokes number $(Stk)$ in the spray for each condition as an alternative means of classifying drops based on their response time. It was observed that $Stk\ll 1$ in all cases. The reason for this fact lies in the physics that the air flow in these sprays is induced by diffusion of liquid momentum to a less dense phase and therefore the two phases are more or less in equilibrium. In contrast, air-blast sprays are fundamentally different. The near-nozzle flow structures such as breakup length, liquid film thickness and sheet frequency have been investigated by Déjean, Berthoumieu & Gajan (Reference Déjean, Berthoumieu and Gajan2016a ,Reference Déjean, Berthoumieu and Gajan b ) in the context of planar air-blast sprays. In contrast to the present study, local droplet Stokes number could play a significant role in such air-blast sprays where the drops and air are not necessarily in local equilibrium.

4.3 Mechanisms of droplet generation

Figure 11. Instantaneous image from an $Re=1364$ and $We=214$ spray. Inset images show magnified views of regions (a) and (b). Shear-induced breakup can be observed in region (a) while Kelvin–Helmholtz–Rayleigh–Taylor (KH–RT) instability-induced breakup can be observed in region (b).

We have shown that the scaled probability distribution of large drops is insensitive to $Re$ and $We$ , while that of the small drops is. Several different drop formation mechanisms have been proposed in the literature; the reader is referred to a recent review article by Villermaux (Reference Villermaux2007) for a comprehensive discussion. We will herein briefly present qualitative experimental evidence that at least two mechanisms of breakup – one that results in large drops and the other that results in small drops – occur simultaneously in our experiments.

Figure 11 depicts an instantaneous image obtained from a spray at $Re=1364$ and $We=214$ . Importantly, two regions are magnified and shown as insets. The inset (a) shows ligaments being formed from the surface of an intact liquid sheet, which will likely result in small drops. These drops are typically smaller than the boundary layer thickness. The mechanism associated with the formation of these drops has been widely discussed in the literature as being one originating from shear (see Lasheras et al. Reference Lasheras, Villermaux and Hopfinger1998). In contrast, the inset (b) shows drops forming from the disintegration of a flapping edge of the liquid sheet. This Kelvin–Helmholtz–Rayleigh–Taylor (KH–RT) based mechanism creates large drops, typically much bigger than the boundary layer thickness. $Re$ and $We$ are dimensionless numbers based on the bulk flow properties and known to govern the liquid sheet flapping dynamics and the probability of finding large drops. In contrast, the smaller drops form from shear-induced K–H instability at the liquid free surface. These two mechanisms are known to exhibit different scaling behaviour (Lasheras et al. Reference Lasheras, Villermaux and Hopfinger1998) and will therefore result in different scaled probabilities of finding large and small drops.

4.4 Dynamical modes of the liquid sheet

We have shown scaling of flow rate, mean drop size, mean axial velocity as well as drop size and velocity p.d.f.s with the appropriate scale choices. These quantities are time-averaged and hence can be construed as steady-state measures of the spray. In contrast, high-speed shadowgraph images captured in the experiments contain information about the unsteadiness. In order to analyse the scaling behaviour of the dynamical aspects of a spray, we treat a video of the spray as a time series of spray structures represented in individual images. POD is used to identify the dominant spatial modes in the spray as well as temporal frequencies.

The POD technique is widely used to identify the dominant dynamical structures in the flow field (Sampath & Chakravarthy Reference Sampath and Chakravarthy2014; Rajamanickam & Basu Reference Rajamanickam and Basu2017). A comprehensive overview of the technique is available in Chatterjee (Reference Chatterjee2000). For the purpose of this study, a variation of POD for image analysis introduced by Sirovich (Reference Sirovich1987) was employed. In this method, the high-speed shadowgraph video of a spray is converted into a set of images. Each instantaneous $m\times n$ pixels image contains information about the spatial variation of the drops in the flow field as grey scale intensity and is assembled into a column vector. The column vectors corresponding to successive time steps are assembled into a matrix $\unicode[STIX]{x1D63C}$ which is of size $mn\times N$ , where $N$ is the number of time instants sampled. In the POD method, one is interested in identifying the singular values of this matrix $\unicode[STIX]{x1D63C}$ , as well as its spatial modes or basis functions from the space–time series data captured in the high-speed shadowgraph imaging experiments. The singular value decomposition (SVD) (Alter, Brown & Botstein Reference Alter, Brown and Botstein2000) of matrix $\unicode[STIX]{x1D63C}$ has the form

where $\unicode[STIX]{x1D650}$ is an $mn\times N$ matrix containing the spatial mode information and $\unicode[STIX]{x1D651}$ is an $N\times N$ matrix containing the temporal mode information corresponding to each spatial mode. The energy content of each mode is calculated from the singular values which are the diagonal elements of $\unicode[STIX]{x1D72E}$ . A Fourier transform of the temporal mode $\unicode[STIX]{x1D651}$ is performed to identify the dominant frequency of a particular spatial mode. We are interested in understanding the two most dominant spatial modes in each spray at the same dimensionless flow conditions. The first mode (mode  $0$ ) is usually the time-averaged steady state and the second mode (mode  $1$ ) corresponds to the most energetic unsteady mode. A power spectral density (PSD) plot is then constructed to identify the dominant temporal frequency in the image sequence.

Figure 12. Contour plots of the mode shapes obtained from POD for $Re\approx 1035$ but varying values of $We$ . The steady mode (mode 0) is shown for: (a)  $We=135$ obtained using $3$ x, (b)  $We=79$ obtained using $5$ x and (c)  $We=42$ obtained using $10$ x. The first unsteady mode (mode 1) is shown for: (d)  $We=135$ obtained using $3$ x, (e)  $We=79$ obtained using $5$ x and (f)  $We=42$ obtained using $10$ x. Each image has a scaled window size of $27D_{o}\times 13.5D_{o}$ . The wavelength associated with the antisymmetric flapping motion of the liquid sheet is shown in (f). The $\unicode[STIX]{x1D706}/D_{o}$ value was observed to be approximately the same ( ${\approx}6$ ) in (d–f). This value is close to the theoretical predictions from linear stability analysis of Xianguo & Tankin (Reference Xianguo and Tankin1991).

Figure 12 shows contour plots of the dominant spatial modes in the $3$ x, $5$ x and $10$ x spray at $Re\approx 1000$ . The values of $We$ for each case are indicated in the caption. Each image shows the same dimensionless spatial region of $27D_{o}\times 13.5D_{o}$ in width and height. Figure 12(a–c) depicts the mode 0 in the three sprays, which is the time-averaged steady mode. In all three cases, this was the most dominant spatial structure in the spray as identified from the POD analysis. It can be seen that these contour plots are nearly identical, implying that the breakup length scales as $D_{o}$ and is independent of $We$ at this value of  $Re$ .

Figure 12(d–f) depicts the second most energy-containing unsteady mode (referred to as mode 1). As can be seen from these images, the flapping motion of the conical liquid sheet exiting the spray atomizer is well represented. The wavelength of the propagating wave can be estimated from the distance along the sheet between the centres of red and blue contour regions. It can be seen that this wavelength is approximately the same in all the three images in figure 12(d–f). Since each image represents the same dimensionless field of view, it can be concluded that the dimensionless wavelength ( $\unicode[STIX]{x1D706}/D_{0}$ ) associated with the flapping motion is approximately the same. In addition, the wavelength in figure 12(e) is within $20\,\%$ of the theoretical prediction of Xianguo & Tankin (Reference Xianguo and Tankin1991) from a linear stability analysis of a planar liquid sheet. We have also measured the wavelength for a single atomizer at several $Re$ conditions for $Re>1000$ . From these measurements, we have seen that the dimensionless wavelength appears to be relatively independent of $Re$ for all these cases, again implying that inviscid theory is sufficient. The flapping motion of the liquid sheet shows nearly similar spatial characteristics in terms of the shape of the most energetic unsteady mode.

The phase difference between the left and right edges of the conical liquid sheet in figure 12(d–f) is also a point to note. Let us consider figure 12(f), for an example. A horizontal line through the image at approximately half the height in the image would pass through a red region on the left-hand edge and a blue region on the right-hand edge of the conical liquid sheet. Since these POD modes are derived from instantaneous snapshots, the red and blue correspond to the presence and absence of the liquid sheet at that point in the image, respectively. Therefore, the liquid sheet appears to be oscillating such that the left edge and the right edge of the spray are out of phase. This implies that the most dominant unsteady mode is antisymmetric, as has been pointed out by Shen & Li (Reference Shen and Li1996).

Figure 13. (a) Plot of dimensional dominant liquid sheet flapping frequency ( $\unicode[STIX]{x1D6FA}$ ) of the most energetic unsteady mode (mode 1) as a function of the dimensional flow rate ( $Q$ ) for the $1$ x,  $3$ x,  $5$ x and $10$ x atomizers. The inset in (a) shows the PSD plot for one case. The dominant flapping frequency is identified from this PSD plot. It is observed that above a critical value of $Re>1000$ , the liquid sheet exhibits a second peak in the PSD plot (see inset in b). The frequencies corresponding to the second peak are shown using open symbols. (b) Plot of Strouhal number versus Reynolds number calculated from the data in (a). The grey region indicates the accuracy of measurement of $St$ ; $St$ is nearly constant and independent of $Re$ for $Re>1000$ .

We are next interested in the most dominant frequency associated with the sheet flapping motion. Information pertaining to the time variation of the modal amplitude is contained in the $\unicode[STIX]{x1D651}^{\text{T}}$ matrix in (4.5). The PSD is then calculated from $V$ . The frequency corresponding to a maximum power in the PSD ( $\unicode[STIX]{x1D6FA}$ ) is identified for each case. See the inset of figure 13(a) for a plot of the PSD for one case, also depicting the peak frequency ( $\unicode[STIX]{x1D6FA}$ ). In order to quantify the error in the peak frequency, we have performed the POD analysis on multiple subsets of the images. The repeatability error in the peak frequency from at least 10 different subset realizations of the experimental data was less than $5\,\%$ .

Figure 13(a) shows a plot of the most dominant dimensional flapping frequency ( $\unicode[STIX]{x1D6FA}$ ) as a function of the dimensional flow rate $Q$ for the $1$ x, $3$ x, $5$ x and $10$ x atomizers. Firstly, it can be observed that, for a given atomizer, $\unicode[STIX]{x1D6FA}$ increases with increasing $Q$ . In addition, as the scale of the atomizer decreases, the dimensional flapping frequency increases sharply. It is particularly noteworthy that the dimensional flapping frequencies in the $1$ x case are between $3$ and 10 kHz. We would like to reiterate that the high-speed camera was operated at 50 000 f.p.s. for this case, in order to resolve motion at this frequency.

Figure 13(b) is a plot of the data in figure 13(a) in a dimensionless framework. In figure 13(b) we plot $St$ versus $Re$ , where $St=\unicode[STIX]{x1D6FA}t_{s}/u_{s}$ . As can be observed from this figure, $St$ initially increases with $Re$ . At low $Re$ , it appears that viscous effects cause the dimensionless flapping frequency to be less than that obtained under high- $Re$ conditions. At high- $Re$ conditions, $St\sim 2\times 10^{-3}$ and is approximately constant. This order of magnitude of $St$ has also been observed in experiments on air-blast sprays by Wahono et al. (Reference Wahono, Honnery, Soria and Ghojel2008).

For $Re>1500$ , an interesting feature is observed in the flapping dynamics. For these cases, a second non-harmonic higher-frequency peak in the PSD was observed (see inset in figure 13 b). While a second sharp peak is observed at $\unicode[STIX]{x1D6FA}=1067~\text{Hz}$ , it is still of a lower energy than the dominant peak at $\unicode[STIX]{x1D6FA}=343~\text{Hz}$ . The $St$ corresponding to these higher frequencies is also indicated in figure 13(b) as open symbols. Except for a case $Re\approx 2000$ , the $St$ corresponding to the lower frequency always has a greater energy. In both cases of $St$ (based on either high or low frequency) $St$ appears to be independent of $Re$ , for $Re>1500$ . Finally, the difference in the magnitudes of the energies between the two peak frequencies is observed to decrease with increasing $Re$ . This implies that a critical value of $Re$ could exist where a crossover may occur to the higher- $St$ mode. Similar dynamics has been observed for vortex shedding behind a cylinder by Bearman (Reference Bearman1969) where he showed that the higher value of $St$ is approximately $2.5$ times the lower $St$ value. As Bearman (Reference Bearman1969) suggests, it is likely that beyond a critical value of $Re$ , the vortex shedding dynamics transitions to a ‘supercritical’ regime. The present results appear to suggest a similar transition in the case of a flapping liquid sheet.