2.1. Description of the high-pressure atomization facility

A specialized custom-built pressurized coaxial jet facility is shown in figure 1, with the capability to change the ambient air pressure surrounding the coaxial jet from 1 to 5 times the standard atmospheric pressure. This is achieved by enclosing the coaxial nozzle in a 190 litre stainless steel tank, and feeding the coaxial gas jet with highly pressurized air. Increasing ambient air pressure results in higher air density $\rho _g$, which we report as $\rho _g/\rho _0$, where $\rho _0 = 1.2\ {\rm kg}\ {\rm m}^{-3}$ is the ambient gas density at standard conditions. The value of $\rho _g$ can be approximated (neglecting temperature changes as a slow isothermal process) by the ideal gas law $\rho _g = P_g m_g / R T$, where $P_g$ is the air pressure in units of Pa, $m_g = 28.97\,{\rm g}\,{\rm mol}^{-1}$ is the molar mass of air, $R = 8.3145\,{\rm m}^{3}\,{\rm Pa}\,{\rm K}^{-1}\,{\rm mol}^{-1}$ is the universal gas constant, and $T$ is the air temperature in kelvins (the term ‘air’ refers to the gas phase for the rest of the paper).

Figure 1. Schematic representation of the pressurized coaxial jet facility: (a) front view, and (b) top view, including the angles between the four viewing windows. Panel (c) is an inset of the movable nozzle, showing the inner and outer diameters of the liquid inlet, $d_l$ and $D_l$, as well as the inner diameter of the air inlet, $d_g$.

The two-fluid coaxial atomizer that produces the spray is shown in figure 1(c), which has been described extensively in Machicoane et al. (Reference Machicoane, Bothell, Li, Morgan, Heindel, Kastengren and Aliseda2019, Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020) and Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022). Briefly, the liquid and gas jets flow continuously during operation, with liquid jet diameter $d_l=2$ mm and annular gas jet diameter $d_g=10$ mm. Liquid and air flow rates are regulated to be steady in time by proportional integral derivative (PID) controls in the flow loop, with user inputs mediated through a National Instruments LabView program. An additional PID circuit is used to regulate the ambient air pressure, which is required as varying atomizing air flow rates will increase the ambient air pressure inside the facility if the air injected is not allowed to exit at the correct rate. Glass imaging windows are built in to enable visible light imaging and laser Doppler velocimetry (LDV) measurements. Due to the fixed position of the glass windows, the nozzle is placed on a two-axis traverse system to enable measurements at different locations inside the spray. More details on the facility, including the construction process, are given in Burtnett et al. (Reference Burtnett, Morgan, Dahlstrom, Aliseda and Heindel2021).

2.2. Parameter range studied

The experimental parameters are listed in table 1. We employ purified water for the liquid phase, with the liquid flow rates (and Reynolds number) kept constant throughout the full parameter range for which experiments were conducted. The co-flowing air is controlled within a range of atomizing air flow rates characterized by the momentum ratio $M = \rho _g U_g^{2}/ \rho _l U_l^{2}$, which is the ratio of the gas dynamic pressure to that of the liquid. Previous experiments described in Lasheras, Villermaux & Hopfinger (Reference Lasheras, Villermaux and Hopfinger1998) and Lasheras & Hopfinger (Reference Lasheras and Hopfinger2000) found that primarily, the momentum ratio dictates the atomization behaviour of the liquid jet, as confirmed for this set-up under atmospheric conditions (Machicoane et al. Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020). The momentum ratio dictates the different liquid breakup modes. When the co-flowing gas velocity is low (low $M$, between 1 and 10), the liquid column undergoes mainly a bag-like breakup process, whereas in fast co-flowing air (high $M$, greater than 10), the liquid column is ‘stripped’ at its surface and forms ligaments, which then break up via a Rayleigh–Taylor instability (Aliseda et al. Reference Aliseda, Hopfinger, Lasheras, Kremer, Berchielli and Connolly2008). Swirl, or angular momentum in the air jet, as described for atomization control experiments in previous studies, e.g. Machicoane et al. (Reference Machicoane, Bothell, Li, Morgan, Heindel, Kastengren and Aliseda2019, Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020), is not studied here.

Table 1. Main physical parameters characterizing the experiments.

Another important parameter is the liquid–gas density ratio $\rho _g/\rho _l$, which varies as the ambient air pressure is increased. Specifically, as the air density $\rho _g/\rho _0$ increases from 1 to 5, $\rho _g/\rho _l$ decreases from 820 to 160; i.e. at the highest air pressure, the water is only approximately 100 times denser than the ambient (and co-flowing) air. Finally, the liquid Weber number $We = \rho _g (U_g-U_l)^{2}d_l/\sigma$, which also influences the liquid breakup process, ranges from 30 to 400. In summary, two independent parameters, the momentum ratio $M$ and the gas density ratio $\rho _g/\rho _0$, result in a total of 20 different experimental conditions in which the near-field dynamics of the coaxial jet atomization, as well as the mid-field droplet size and velocity profiles, are studied. Intrinsic liquid and gas properties, such as the dynamic viscosity and air–water surface tension $\sigma$, are dependent mainly on temperature and are largely unaffected by pressure (Lo et al. Reference Lo, Carroll and Stiel1966; Vargaftik et al. Reference Vargaftik, Volkov and Voljak1983). The isothermal set-up ensures that temperature does not change significantly between experiments, and thus does not affect these properties. All results represent the effect of the change in air density only.

3.1. High-speed shadowgraphy imaging

The near-field of the spray is the region where primary breakup of the liquid jet occurs, and we define it in this study as $0 < x/d_g < 9$, in the axial direction starting at the nozzle. In the near-field, the small time scales associated with the liquid atomization process require the use of high-speed imaging to observe the flow with sufficient temporal resolution. We obtain high-speed videos of the jet using a shadowgraphy method (figure 2), with a light source and an imaging system placed opposite each other, with the spray situated in between, such that the liquid column appears as a dark region against the light source. The illumination system comprises a DC LED source (Veritas Constellation 120E) with a diffuser (Westcott Scrim Jim) to ensure a uniform light background, and the imaging system is a high-speed camera (Phantom v12) paired with a Nikon 180 mm f/4 lens and a $2\times$ teleconverter, placed 1 m away from the nozzle. This results in a field of view $30\,{\rm mm} \times 30\,{\rm mm}$, and spatial resolution $37\,\mathrm {\mu }{\rm m}\,{\rm px}^{-1}$ (magnification approximately 0.6). The image acquisition frame rate is 10 000 frames per second, resulting in total acquisition time 1.5 s, with $3\,\mathrm {\mu }{\rm s}$ exposure time for each frame.

Figure 2. Schematic of the high-speed shadowgraphy set-up.

After the images are obtained, further processing is required in order to eliminate biases from, for example, uneven lighting effects, as illustrated in figure 3. These procedures are similar to those described in Machicoane et al. (Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020). First, a background subtraction procedure is performed to remove the vignetting effect of the optics. Next, a normalization is applied on each frame, dividing the brightness values throughout the field by the value of the brightest pixel, such that the values of the dimmest and the brightest pixels are 0 and 1, respectively. A binarizing procedure follows, where regions having intensity values ${>}0.5$ are considered as the liquid phase being present, and are assigned binary value 1, and values below ${>}0.5$ are assigned value 0. We note that this imaging and thresholding method informs only the presence or absence of liquid along the line of sight, and not the amount of liquid present throughout the spray. Finally, to correct for reflections of the light source on the liquid core, a filling procedure is used to fill in gaps in the binarized images, where regions with intensities below the threshold but with perimeters above the threshold intensity are adjusted such that these regions are also considered as being occupied by the liquid phase.

Figure 3. (a) Original image, (b) image after background subtraction, (c) image after normalization and binarization, and (d) final image after filling in gaps due to lighting conditions.

We obtain high-speed shadowgraphy videos of the near-field spray for four different momentum ratios $M$, and five different air density ratios $\rho _g/\rho _0$. Figure 4 shows representative images of the spray at the various conditions. Qualitatively, one can already infer that the momentum ratio $M$ dominates the near-field breakup dynamics. An increase of $M$ from 5 to 25 (which corresponds to the liquid Weber number $We$ increasing from 37 to 189 at $\rho _g/\rho _0 = 1$, and from 35 to 186 at $\rho _g/\rho _0 = 5$) induces the transition from a bag-breakup regime to a ligament-breakup regime (Chigier & Reitz Reference Chigier and Reitz1996). Further increases in the momentum ratio result in the liquid column becoming shorter, similar to observations in Machicoane et al. (Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020) and Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022). In contrast, the effect of increasing the air density ratio $\rho _g/\rho _0$ is not apparent immediately in the instantaneous images. Statistical analysis of the videos is required to elucidate the effects of increased ambient air pressure on a spray with similar momentum ratios, as shown in the next subsection.

Figure 4. Instantaneous images of the spray at different momentum ratios $M$, increasing from (a,e,i) to (d,h,l), and different air density ratios $\rho _p/\rho _g$, increasing from (a–d) to (i–l).

3.2. Core length statistics

For each image obtained, we extract the intact liquid core length, defined as the axial length of a continuous region where liquid is present, as shown in figure 5(a). Doing this over the full acquisition sequence results in a time-resolved liquid core length signal $L_B(t)$. With total acquisition time 1.5 s, $L_B(t)$ converges to an average value $\langle {L_B} \rangle$ (figure 5c) such that the mean and r.m.s liquid core lengths can be compared for different momentum ratios and ambient pressures. Error bars associated with the data points represent statistical uncertainties, calculated based on 95 % confidence intervals and using the random uncertainty of the time-resolved signal (Bendat & Piersol Reference Bendat and Piersol2011).

Figure 5. (a) Instantaneous image of the spray showing the liquid intact core length $L_B$ for case $M = 5$, $\rho _g/\rho _0 = 1$. (b) Plot showing a time series of the evolution of $L_B$ in time, for the same case $M = 5$, $\rho _g/\rho _0 = 1$. (c) Convergence plot of the liquid intact core length. (d) Autocorrelation example of the liquid core length.

As shown in figure 6(a), the average liquid core length trends mainly with momentum ratio, with a reciprocal relationship also observed in Machicoane et al. (Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020). Increasing the air density results in a subtle but measurable increase in the liquid core length, for all momentum ratio cases. The r.m.s. fluctuations of the liquid core length, $L'_B$ are shown in figure 6(b). The r.m.s. fluctuations decrease mainly with increasing momentum ratio, however, the trend of the r.m.s. fluctuations with increasing air density is less pronounced. Considering that higher momentum ratios ($M > 25$) result in more complete atomization, the r.m.s. fluctuations appear to decrease with higher air density, at these high momentum ratios. For the lower momentum ratios ($M \leq 25$), however, the r.m.s. fluctuation change with air density is either positive or flat (no correlation). In any case, the momentum ratio still dictates the mean and r.m.s values of the liquid intact core length.

Figure 6. Liquid core length characteristics as functions of air density ratio $\rho _g/\rho _0$. (a) Mean liquid core length $\langle L_B \rangle /d_g$. (b) R.m.s. fluctuation ${L'_B}/{d_g}$. (c) Characteristic time scale derived from the autocorrelation $\tau _{L_B}$. Symbols represent different momentum ratios as indicated in the legend in (c). Dashed lines represent a linear fit of the data points for each momentum ratio. Where not visible, error bars associated with the data points are smaller than the data symbol itself.

We also leverage the time-resolved measurement to compute the autocorrelation time of $L_B(t)$, or the time required for fluctuations in the liquid core length to become uncorrelated, which informs a characteristic time scale of the liquid core length. The autocorrelation is defined as

(3.1)\begin{equation} C_{L_B}(\Delta t) = \frac{\langle L'_B(t)\,L'_B(t+\Delta t) \rangle}{\langle L'_B(t)^{2} \rangle} , \end{equation}

where $L'_B = L_B-\langle L_B \rangle$ is the fluctuation of the intact liquid core length about the mean. The point where the value of $C_{L_B}$ decays to $1/{\rm e}$ (as illustrated in figure 5d) is taken as the characteristic correlation time of the liquid core, $\tau _{L_B}$. Note that in the autocorrelation plot, $C_{L_B}$ also crosses the zero point, showing an anti-correlation where a long core length is usually followed by a short core length after 2 ms (and vice versa).

In figure 6(c), we compare the characteristic time scale of the core length, $\tau _{L_B}$, for different momentum ratios and air densities. Similar to previous results, the characteristic time scale appears to be determined mainly by the momentum ratio, with a negative effect of increasing air density. This is also true only for the higher momentum ratios ($M \geq 25$); for the lowest momentum ratios where the atomization regime is different ($M=5$), there is no noticeable correlation of $\tau _{L_B}$ with increasing air density.

3.3. Spreading angle and virtual origin

From the binarized images shown in figure 3(d), an ensemble average can also be conducted for the full image sequences. The resulting liquid probability field (figure 7a) has values ranging from 0 to 1, corresponding to regions where liquid is never present throughout the experiment, to regions where liquid is always present, respectively. Following Machicoane et al. (Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020), we use the time-averaged liquid probability field to derive the spray spreading angle, using the notion that for axial distances of 1–1.5 times the liquid bulk length, i.e. $L_B < x < 1.5L_B$, an axial slice of the field yields a Gaussian profile (figure 7b), and it being true for the different axial distances implies self-similarity. Leveraging this, the standard deviation (s.d.) of the Gaussian is calculated for each axial distance, and the location of $r/d_g = \pm 2\,{\rm s.d.}$ is plotted to define the spray edge. This is followed by fitting a line through the points representing the spray edge for $L_B < x < 1.5L_B$ on both sides. The angle that the two lines subtends is the spreading angle $\theta$, whereas the axial location where the two lines meet is the virtual origin $x_0$.

Figure 7. (a) Probability of liquid presence, P(x,r), averaged over the entire image set (from the binarized shadowgraph images). The location of the spray edge (delineated by red dashes at angle $\theta$, with the virtual origin $x_0$) is defined as $r/d_g = \pm 2\,{\rm s.d.}$. The axial locations of $\langle L_B \rangle$ and $\langle 1.5L_B \rangle$ are marked in white dashes. (b) Profiles of axial slices showing the Gaussian distribution of the liquid presence probabilities.

Figure 8(a) shows, for the different momentum ratios, that generally, the spray spreading angle decreases with increasing air density. The effect is more pronounced in the higher momentum ratio cases, where more complete atomization results in a more unambiguous decrease in the spreading angle corresponding to a narrowing of the jet. Similar trends can be inferred from figure 8(b), which shows that the axial value of the virtual origin decreases with increasing air density (negative values show that the location of the spray virtual origin is above the imaging field, i.e. inside the nozzle). As with the liquid core length, the values of the spray spreading angle and the virtual origin are dominated mainly by the momentum ratio (Machicoane et al. Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020). The spreading angle is a time-averaged manifestation of the Kelvin–Helmholtz instability, which will be shown to have a characteristic wavelength and a corresponding growth rate that is modulated with increasing air density. This is reflected in the overall spray characteristics as a narrowing of the jet and an upstream shift of the jet virtual origin.

Figure 8. Spray characteristics, as functions of air density ratio $\rho _g/\rho _0$, for (a) the spreading angle $\theta$, and (b) the virtual origin $x_0$. Symbols represent different momentum ratios, as indicated in the legend in (a). When not visible, the error bars have sizes similar to or smaller than the data point symbol.

3.4. Frequency analysis

We also leverage the measurements’ time resolution to compute the characteristic frequencies associated with the spray. In Machicoane et al. (Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020), a power spectral density (PSD) analysis is performed on the time-resolved liquid intact core length $L_B(t)$ signal, showing characteristic frequencies of the liquid bulk length that change with the swirl ratio $SR$. In our case, since the effect of pressure on the liquid core length is more subtle, we opted to perform the PSD analysis in a spatially resolved manner, i.e. directly on the time-resolved binarized liquid presence field instead.

For each radial location $r$, an axial average of the binarized liquid presence field (for example, shown in figure 3(d)) is taken at axial distances $0.75 < x/\langle L_B \rangle < 1.25$, where $\langle L_B \rangle$ is the mean liquid bulk length for each case. This is repeated for each time step, resulting in an axial-averaged, time-resolved liquid presence signal $P(t)$ on which the PSD analysis is performed. The results of the PSD analysis are plotted in figure 9, with the radial location $r$ represented in the $x$-axis, the frequencies in the $y$-axis, and the PSD magnitudes in the colour axis. The dominant frequencies in the PSD colour contours are the frequencies for which the liquid phase appears and disappears over time due to breakup. This spatially resolved colour contours of the PSD for each radial location allows a clearer visualization of the characteristic frequencies of the liquid column breakup.

Figure 9. Colour contours for the PSD of the liquid presence field, with the $y$-axis indicating frequencies and $x$-axis indicating radial distances. The air density ratio is $\rho _g/\rho _0 = 1$, and momentum ratio $M$ increases from (a) to (d).

At a low momentum ratio $M = 5$, the magnitudes of the PSD in the colour contours show a peak value at around 150 Hz, which is sustained across the radial span of the jet, as shown in figure 9(a). This value is concurrent with the PSD analysis of the liquid core length signal in Machicoane et al. (Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020), as well as a similar spatially resolved PSD analysis in Delon et al. (Reference Delon, Cartellier and Matas2018). An increase in the momentum ratio (figures 9a–d) results in the spray exhibiting a higher characteristic frequency in its liquid column breakup. This can be interpreted as an effect of the narrowing of the spray spreading angle at higher momentum ratios – when the spray is narrower, more liquid breakup events occur over the same time period in a more confined spatial region, as the liquid flow rate is unchanged. This results in an observed increase in the frequencies characterizing the liquid presence in the spray. Figure 9 also exhibits a shift in the spatially resolved PSD plots to peak only at the edges of the liquid column, indicating a transition from the bag-breakup regime in $M = 5$ (corresponding to $We = 37$), where the bag-breakup process occurs across the radial span of the jet, to the more energetic ligament-breakup regime from $M = 25$ onwards (corresponding to $We > 180$), where liquid breakup occurs on ligaments surrounding the main jet. The findings for this transition align well with previous studies by Chigier & Reitz (Reference Chigier and Reitz1996), where the bag–ligament breakup transition occurs at $We \sim 100$.

Figure 10 shows the results of the PSD analysis performed on the same momentum ratios but for a higher air density ratio ($\rho _g/\rho _0 = 5$). Compared to figure 9, we observe similar spatial patterns in the PSD colour contours, but higher characteristic frequencies in the liquid breakup with higher air density. This indicates that the increase in air density produces an effect similar to that from increasing the momentum ratio, i.e. reducing the spray spreading angle and producing more energetic liquid breakup. However, the similar spatial patterns also indicate that the breakup regimes (bag-breakup transitioning to ligament-formation) are similar between low and high densities, and are well defined by $M$ and $We$. This observation, combined with the previous observations of elongation of the mean intact liquid core length (figure 6a) and the narrowing of the spray (figure 8a), suggests that at high air densities, the momentum transfer from the gas phase to the liquid phase is enhanced, even when the momentum ratio is maintained.

Figure 10. Colour contours for the PSD of the liquid presence field, with the $y$-axis indicating frequencies and $x$-axis indicating radial distances. The air density ratio is $\rho _g/\rho _0 = 5$, and momentum ratio $M$ increases from (a) to (d).

4.1. Phase Doppler interferometry and laser Doppler velocimetry

The mid-field of the spray is defined as the region with axial distances $x/d_g \geq 9$. Physically, this represents the region where the liquid jet has broken up into droplets that are advected downstream and may undergo secondary breakup if they are unstable, i.e. their aerodynamic Weber number is larger than a critical value, usually $We_{crit} > 4$ (Michaelides, Crowe & Schwarzkopf Reference Michaelides, Crowe and Schwarzkopf2016; Chen et al. Reference Chen2017). Quantities of interest in this region include the droplet velocities and diameters. Coalescence of droplets in the mid-field could also be considered, as was done by Lasheras et al. (Reference Lasheras, Villermaux and Hopfinger1998).

In this study, we conduct point measurements of the droplet diameters and velocities using phase Doppler interferometry (PDI) and LDV. This is done with a TSI LDV/PDI system (FSA4000 Signal Processor, PDM1000 Photo Detector Module) using forward scattering in the first refraction mode, where the transmitter and receiver are aligned at angle $30^{\circ }$ as shown in figure 11. Droplet velocities are measured in the axial and radial directions, and droplet diameters are derived from intensity values of the scattered light signal (Albrecht et al. Reference Albrecht, Damaschke, Borys and Tropea2013). These point measurements are repeated along the radial direction to obtain the droplet diameter and velocity profiles across the spray. As mentioned previously, the viewing windows on the pressurized vessel are fixed, so the nozzle is traversed across the LDV/PDI probe volume to collect measurements across the different radial locations of the spray.

Figure 11. LDV/PDI system set-up to study the spray mid-field under high ambient pressures/densities. (a) Front view of the set-up, including the axial distance from the nozzle and the radial span of the sampling locations. (b) Top view of the set-up, including the orientation of the transmitter and receiver modules.

4.2. Droplet size distributions

PDI results are reported in the form of radial profiles of mean diameter in figures 12 and 13. Figure 12 shows the arithmetic mean diameter $d_{10}$, whereas figure 13 shows the Sauter mean diameter $d_{32}$, which can be interpreted as a mean diameter weighted by the volume to surface area ratio. The generalized form of the various diameters reported can be summarized as

(4.1)\begin{equation} d_{m,n} = \left( \frac{\sum d_d^m}{\sum d_d^n} \right) ^{1/(m-n)} , \end{equation}

where $d_d$ represents the absolute diameter of the individual droplet measured at the sampling point. In each figure, the results are plotted in panels (a–c) for the three momentum ratios $M = 25, 39, 56$, and the symbols representing five different air density ratios, $\rho _g/\rho _0 = 1\unicode{x2013}5$.

Figure 12. Radial profiles of the arithmetic mean droplet diameter $d_{10}$ for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend in (c).

Figure 13. Radial profiles of the Sauter mean droplet diameter $d_{32}$ for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend shown in (c).

Trends in the radial profile of the droplet diameters, for both the arithmetic mean (figure 12) and the Sauter mean (figure 13), are dominated by the momentum ratio $M$. As $M$ increases from left to right in figures 12 and 13, the shapes of the droplet diameter profiles change from concave to flat. This is an effect of the spray behaviour transitioning past a critical value of the momentum ratio, $M_C \sim 50$ (Lasheras & Hopfinger Reference Lasheras and Hopfinger2000). Physically, Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022) described a mechanism where the outside shear layer between the atomizing and quiescent air has eddies with associated time scales that increase downstream. At low momentum ratios, the overall droplet time scales are similar to those of the eddies, and droplets interact most with eddies with similar time scales, thus larger droplets reside preferentially towards the outskirts of the spray. However, at momentum ratios beyond the critical value $M_C$, the droplet time scales are smaller than those of the eddies. Only the smallest droplets are transported by the eddies, whereas larger droplets reside within the inner regions of the spray. This is shown in figure 14, where the probability distribution functions (p.d.f.s) of the droplet diameters at the centre are plotted, showing an increase in the probability of larger droplets as the momentum ratio $M$ increases. These p.d.f.s also show that as the air density increases, the probability of small droplets decreases and the probability of large droplets increases, hence increasing the mean droplet diameters as seen earlier, in figures 12 and 13.

Figure 14. Probability distribution functions (p.d.f.s) of the arithmetic mean droplet diameter $d_{10}$ sampled at the centre of the jet, for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend in (c).

For the same momentum ratio, a clear trend can be observed in figures 12 and 13, where the mean droplet diameters become larger as the air density increases. The increase in mean droplet diameters is largely monotonic, and retains similar profile shapes for different air densities, indicating that the critical momentum ratio $M_C$ is independent of the air density. It is also observed that as the sampling locations move away from the spray centre, the mean diameter generally has a maximum at a certain radial distance and then drops towards the edges. This is understood to be the effect of confinement due to the facility, where the mean droplet diameters at the far radial distances are skewed by small recirculating droplets such that the statistical average diameters at the edges always appear to be smaller than the spray characteristics would predict.

4.3. Droplet and gas phases velocity profiles

In this experiment, the parameter space is defined by the momentum ratio $M$ as well as the air density ratio $\rho _g/\rho _0$. Since the liquid flow rate is kept constant throughout the experiment, an increase in $\rho _g/\rho _0$ results in a decrease in the gas velocity so that the momentum ratio is preserved. Thus the mean droplet axial velocity at the spray centreline, $U_0$, is shown to decrease with increased air density for the same momentum ratio $M$, as shown in figure 15(a). Nevertheless, regardless of the momentum ratio or air density, the droplet axial velocity profiles all collapse when normalized by their respective half-width $r_{1/2} = r$ ($U_x=0.5U_0$) in the abscissa, and also by their respective centreline velocities $U_0$ in the ordinate, as shown in figure 15(b). We note that for all cases, the droplet axial velocities are reduced to half the centreline velocity $U_0$ at a radial position $r=d_g$, meaning that $r_{1/2} \approx d_g$ for all cases, as shown by the dashed lines in figure 15(b). These profiles are also shown to match the jet self-similar solution $f(r/r_{1/2}) = (1+47S(r/r_{1/2})^{2})^{-2}$ (Pope Reference Pope2000), where $S = 0.094$ is the spreading rate derived by Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022) for a two-fluid coaxial jet. Overall, at axial distance $x = 9d_g$ downstream of the nozzle, the droplet axial velocity profiles can be considered fully developed.

Figure 15. (a) Mean droplet axial velocity at the spray centre, $U_0$, as a function of air density ratio $\rho _g/\rho _0$. (b) Radial profiles of the mean droplet axial velocity, $U_x$, normalized by their respective centreline velocities $U_0$, including all $M$ and $\rho _g/\rho _0$ data, and compared with the self-similarity solution for a free jet. Symbols indicate different momentum ratios, as indicated by the legends.

While the droplet axial velocities exhibit a canonical profile that is similar across different momentum ratios and air densities, the radial velocity features interesting dynamics, as shown in figure 16. Across different momentum ratios and air densities, the droplet radial velocity profiles $U_r$ show a maximum mean radial velocity approximately 5 % of the axial centreline velocity, with the maximum value being slightly larger for higher air density cases. This suggests that high air densities act to enhance momentum transfer from the gas to the liquid near the spray centreline, in agreement with prior near-field observations. The opposite trend is shown for far radial distances, where higher air densities yield lower radial velocities for droplets in far radial positions. This can be attributed to the droplets experiencing greater drag when travelling radially through denser air at higher air densities, hence reducing the mean radial velocities of the droplets as they cross the region of zero radial velocity in the gas jet.

Figure 16. Radial profiles of the mean droplet radial velocity $U_r$ normalized by their respective centreline velocities $U_0$, for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend in (c).

Simultaneous measurements of the droplet sizes and axial velocities enabled by the LDV/PDI system also allow for characterization of the axial gas velocity $U_{air}$ for each radial location in the jet, taking the mean axial velocity of all droplets with diameters below $2\,\mathrm {\mu }{\rm m}$ at that radial location. By taking the resulting mean axial gas velocity and subtracting it from the mean axial velocity of all droplets (i.e. $U_x$), we obtain a mean slip velocity $U_{slip} = U_x-U_{air}$. Repeating this procedure for different radial locations yields the slip velocity profiles shown in figure 17. As with the radial velocity profiles, the slip velocity profiles are similar across different $M$ and $\rho _g/\rho _0$ conditions. Near the centre of the spray, the negative value of $U_{slip}$ indicates that the droplet velocities are slower than those of the gas phase. At larger radial distances beyond $r = 0.5d_g$, slip velocities are positive, indicating that droplets travel faster than the gas phase, with maximum slip velocities observed in the region $r \sim 2d_g$ for all cases. This can be interpreted as high-momentum droplets retaining their velocity through the slow-moving gas phase, leading to high slip. We note that the slip velocity profiles are normalized by the droplet velocity at the centre, $U_0$, for each case. The droplet slip velocity values are higher for low $P$, as shown in figure 16(a). This has implications for the estimation of key dimensionless parameters such as the droplet Reynolds numbers $\mbox{Re}_{p}$ and the droplet Weber numbers $We_p$, which we will explore in the next subsection.

Figure 17. Radial profiles of the mean slip velocity $U_{slip} = U_x - U_{air}$, normalized by their respective centreline droplet velocities $U_0$, for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend shown in (c).

4.4. Role of dimensionless numbers in the droplet size increase with higher air densities

In this subsection, we present droplet statistics in terms of droplet Reynolds and Weber numbers to analyse potential mechanisms that explain the increase in droplet sizes with ambient pressures (hence air densities), in terms of secondary breakup and transport within the spray. Several important caveats are highlighted here: while the diameter and velocity of each droplet are measured, the gas velocity is known only in an averaged sense (over a period of time where many small droplets cross the LDV/PDI probe volume). Therefore, the corresponding $\mbox{Re}_p$ is also known only in a time-averaged sense. Furthermore, we limit our analysis to droplets from the centre of the spray ($r = 0$) to minimize the effect of not including radial velocities in this analysis (due to the lower data rate in the measurements of this velocity component). With all these considerations, a droplet Reynolds number $\mbox{Re}_p = \rho _g d_d U_{slip}/ \mu _g$ can be defined for each droplet, and the p.d.f.s for $\mbox{Re}_p$ are shown in figure 18 for all cases across different $M$ and $\rho _g/\rho _0$ conditions.

Figure 18. P.d.f.s of the droplet Reynolds number $\mbox{Re}_p$ for droplets sampled at $x = 9d_g$, $r = 0$, for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend shown in (c).

The first observation is that the distribution of $\mbox{Re}_p$ appears to have similar ranges across different $M$ and $\rho _g/\rho _0$, with a maximum $\mbox{Re}_p$ of $\sim 500$. Low $\mbox{Re}_p$ droplets are three to four orders of magnitude more probable than high $\mbox{Re}_p$ droplets. The probability of high $\mbox{Re}_p$ droplets increases with increasing air density, and this trend is more apparent in the high momentum ratio cases (figures 18b,c). This is despite the lower values of the droplet centreline velocity $U_0$ (and their slip velocities $U_{slip}$) for droplets in the high ambient air pressure environment. Overall, the majority of droplets have low slip velocity and $\mbox{Re}_p$, with only a small percentage of droplets reaching high values of $\mbox{Re}_p$. However, the fraction of droplets with high $\mbox{Re}_p$ increases as the ambient air density increases.

A similar analysis is done with the droplet Weber number $We_p = \rho _g d_d U_{slip}^{2} / \sigma$, another non-dimensional parameter that influences the droplet breakup process. Figure 19 shows the p.d.f.s of the droplet Weber number, for all cases across different $M$ and $\rho _g/\rho _0$ conditions. Notably, despite the wide range of $\mbox{Re}_p$ exhibited, the $We_p$ values are small and often less than 3, corresponding to stable droplet sizes in which no secondary breakup occurs (Michaelides et al. Reference Michaelides, Crowe and Schwarzkopf2016; Chen et al. Reference Chen2017). Comparing the statistics between the droplet Reynolds and Weber numbers, it is now apparent that the latter is a more important parameter with regard to secondary breakup, as shown e.g. by the collapse of the p.d.f.s for the different ambient air pressures, especially for the higher momentum ratios as shown in figures 19(b,c). The fact that $We_p$ is the dominant parameter that controls droplet fragmentation and the final droplet size also explains our prior observation of larger mean droplet diameters with higher ambient pressures (figures 12 and 13). In general, droplets undergo fragmentation until their sizes are stable (i.e. $We_p = \rho _g d_d U_{slip}^{2} / \sigma \leq 4$). In a high-pressure environment, the maximum stable droplet size $d_d^{*}$ is now larger due to the droplet slip velocities $U_{slip}$ being smaller (when the momentum ratio $M$ is constant), and is reflected in the average values of the arithmetic mean and Sauter droplet diameters. In summary, for the same momentum ratio, higher ambient air pressures (i.e. densities) result in larger droplets in the spray.

Figure 19. P.d.f.s of the droplet Weber number $We_p$ for droplets sampled at $x = 9d_g$, $r = 0$, for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend shown in (c).

4.5. Liquid recovery rates and momentum transfer between phases

All prior results and analysis shown are essentially droplet characteristics and their dependency on $M$, $\rho _g/\rho _0$ and $r$. In this subsection, we aim to provide an integral description of the spray, including the net momentum transfer between the gas and liquid phases. To this end, several quantities have to be known, including the total liquid volume flow rate $Q_l$ (from which the percentage recovery rate can be inferred) and the total momentum of the liquid phase $\dot {M}_{l,out}$ (from which the momentum gain in the liquid phase can be computed).

First, an estimation of the probe area is needed. Figure 11 shows a laser collimating lens with focal length $f_c = 750\,{\rm mm}$ used on the transmitter, and an imaging lens with $f_i = 250\,{\rm mm}$ on the receiver. With this set-up, the probe cross-sectional area (where droplets are detected as they flow through) can be approximated by a rectangle defined as $A = lw$, where $l=s/(\sin (\theta )(f_i/f_c))$ is the probe volume length, as seen from the receiver, truncated by a spatial filter in the receiver ($s = 150\,\mathrm {\mu }{\rm m}$) and offset at angle $\theta = 30^{\circ }$. The beam width $w$ is dependent on the diameter of the droplet in the probe volume at each measurement. For each droplet, the product of its residence time (or gate time) in the probe volume and its axial velocity is a path length $l$ that is dependent on the droplet diameter. Then the diameter-dependent beam width $w$ can be estimated by averaging the droplet path lengths conditioned by sizes (e.g. $50 < d_d< 75\,\mathrm {\mu }{\rm m}$); in our analysis, we group the droplets into 10 equally spaced bins. Finally, the probe area $A_i = d w_i$, dependent on droplet diameter, can be computed. Implementing this procedure corrects a bias that arises from the Gaussian nature of the laser beam causing smaller droplets to be less detectable at the edges of the probe volume (as they scatter less light) than large droplets (Albrecht et al. Reference Albrecht, Damaschke, Borys and Tropea2013).

Following Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022), the volume flux of liquid calculated for each diameter bin $i$, containing a total number $N_i$ of droplets, is given by

(4.2)\begin{equation} \dot{G}(d_i) = \frac{1}{T_s A_i} \sum_{j=1}^{N_i} \frac{d_{j,i}^{3}}{6}, \end{equation}

where subscript $i$ refers to the diameter bins, and $j$ to the total number of droplets in the $i$th diameter bin. Here, $T_s$ is the total sample time, and $A_i$ is the probe cross-section for droplets corresponding to the $i$th diameter bin. Integrating over all $D$ droplet size bins ($D = 10$ in our analysis) yields $\dot {g}(r)$, the actual volume flux of liquid passing through each location sampled in the spray:

(4.3)\begin{equation} \dot{g}(r) = \sum_{i=1}^{D} \dot{G}(d_i). \end{equation}

The radial profiles of $\dot {g}(r)$ are plotted in figure 20 for all cases across different $M$ and $\rho _g/\rho _0$. For the same momentum ratio $M$, the profiles of $\dot {g}(r)$ appear similar across different air density ratios $\rho _g/\rho _0$, and $\dot {g}(r)$ increases with $M$, indicating that more liquid passes through the probe location per unit time.

Figure 20. Radial profiles of the liquid volume flux $\dot {g}$ at the sampling location for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend shown in (c).

Similar to $\dot {g}(r)$, the momentum flux of liquid passing through the measuring point, $\dot {m}_l(r)$, can be computed by

(4.4)\begin{gather} \dot{\mathcal{M}}_l(d_i) = \frac{1}{T_s A_i} \sum_{j=1}^{N_i} \frac{1}{6} \rho_l U_{j,i} d_{j,i}^{3}, \end{gather}

(4.5)\begin{gather} \dot{m}_l(r) = \sum_{i=1}^{D} \dot{\mathcal{M}}_l(d_i). \end{gather}

Profiles of $\dot {m}_l(r)$ are plotted in figure 21 for all $M$ and $\rho _g/\rho _0$. Across all cases, the profiles generally feature lower momentum flux with higher air densities.

Figure 21. Radial profiles of the liquid momentum flux $\dot {m}_l$ at the sampling location for the three momentum ratios: (a) $M = 25$, (b) $M = 39$, and (c) $M = 56$. Symbols indicate different air density ratios corresponding to the legend shown in (c).

Integrating the profiles in figure 20 yields the total liquid volume flow rate $Q_l = \int _{0}^{R} 2 {\rm \pi}r\,\dot {g}(r) \,{\rm d}r$. This in turn can be used as a measure of the recovery rate by dividing $Q_l$ by the input volume flow rate $Q_{l,in} = 1650\,{\rm mm}^{3}\,{\rm s}^{-1}$, resulting in recovery values in the range from 28 % to 53 % of the liquid injected, as shown in figure 22(a); i.e. not every droplet passing through the PDI probe volume is captured. Similarly, the total momentum in the liquid phase can also be integrated from the radial profiles in figure 21 as $\dot {M}_{l,out} = \int _{0}^{R} 2 {\rm \pi}r\,\dot {m}_l(r) \,{\rm d}r$. To compute the net momentum gained by the liquid phase in the spray, one divides the previous result by the liquid momentum injected, $\dot {M}_{l,in} = \rho _l U_l^{2} A_l$. Figure 22(b) shows that for all cases, there is a net momentum gain in the liquid phase that is larger for higher momentum ratios $M$. This momentum transfer from the gas to the liquid is larger in lower air densities, i.e. for lower $\rho _g/\rho _0$. This observation holds even when the momentum gain is adjusted to account for the liquid phase recovery rate, as shown in figure 22(c), where the overall momentum gain in the liquid phase of a spray atomizer is seen to diminish in an environment with high ambient air pressure (i.e. high air density).

Figure 22. (a) Liquid recovery from the LDV/PDI measurements. (b) Momentum gain in the liquid phase. (c) Momentum gain after adjusting for the liquid recovery. Symbols indicate different momentum ratios, and the $x$-axis shows different air densities.