4.1. Mean axial velocity ( $u_x$ )

The behaviour of the axial velocity of the turbulent annular gas jet is shown in figure 3. Figure 3 $(a)$ presents the radial profiles of the mean axial velocity ( $u_x$ ). Lines and markers represent the hotwire and PDPA data, respectively. The symmetry of the annular gas jet in the absence and presence of spray is demonstrated here at different axial locations. Apart from the obvious decay and spread of the jet with increasing downstream distance, the significant spread of the swirling jet compared with its non-swirling counterpart is evident. In addition, the increase in the difference between the mean velocity of the gas jet in the absence and the presence of the spray can be noted as the gas jet evolves downstream. Modarress et al. (Reference Modarress, Tan and Elghobashi1984), Shuen et al. (Reference Shuen, Solomon, Zhang and Faeth1985) have previously attributed this to the presence of fully or partly responsive particles in a turbulent gas jet, whereby there is a mean slip of particles that leads to momentum transfer to the gas from the particle phase. Since $St_d\gt 1$ near the atomiser for many droplets, it is understood that the primary momentum transfer occurs here from the gas phase to the liquid phase, in addition to part of the gas kinetic energy being utilised by the spray formation mechanisms.

Figure 3.

Gas-phase mean axial velocity ( $u_x$ ) characterisation in spraying (PDPA) and non-spraying conditions (hotwire anemometry, HWA). Non-swirling flow conditions are indicated by open markers from PDPA measurements (conditioned on only droplets whose size is below $ 5$ $\unicode{x03BC}$ m) and swirling conditions are shown in filled markers. $(a)$ Comparison of the PDPA (symbols) and hotwire anemometer (lines) radial profiles of the mean axial velocity at $x/d_g=10$ ( $\triangle$ and - -) and 25 ( $\circ$ and –) for ${\textit{Re}}_g={102\,400}$ . For PDPA, red filled symbols indicate swirled sprays. $(b)$ Longitudinal decay of the centreline mean axial velocity along the axial distance, identifying one virtual origin ( $x_{0,ns}=-2.4d_g$ ) for ${\textit{SR}}=0$ and using the same for all other conditions ( ${\textit{SR}} \gt 0$ ). The gas flow statistics obtained from all ${\textit{Re}}_l$ and ${\textit{Re}}_g$ (listed in table 1) are added with different colours depending on their ${\textit{SR}}$ value. The same convention is used for the rest of the panels in this figure and figure 5. The blue line is from the round jet experiments of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993). $(c)$ Spreading of the annular gas jet displayed using the non-dimensional half-width ( $\circ$ ) and the $10\,\%$ width ( $\triangle$ ). Here, $x_{0,ns}^\alpha$ represents the virtual origin corresponding to $\alpha =0.5$ or $0.1$ for $r_{0.5}$ and $r_{0.1}$ . $(d)$ Normalised mean axial velocity as a function of the non-dimensional radial coordinate $r/r_{0.5}$ . The displayed conditions span all ${\textit{Re}}_g$ , ${\textit{Re}}_l$ and ${\textit{SR}}$ values from table 1, as well as all the explored longitudinal locations $x$ . The dashed and solid underlying lines represent data obtained using hotwire anemometer at both $x/d_g = 10$ and $25$ .

A summarised version of the profiles is illustrated through the jet centreline decay and spread. Figure 3 $(b)$ shows the inverse centreline axial velocity of the gas jet, $u_{x,0} (x) =$ $u_x(x,r=0)$ normalised with the exit gas velocity, $U_g$ . In single-phase gas-jet measurements $u_{x,0} \propto ( x-x_0^{u_x} )^{-1}$ beyond the developing region ( $x/d_g \geqslant 7$ ), where $x_0^{u_x}$ is the virtual origin based on the decay rate of the jet’s centreline velocity. In order to quantify this evolution and also compare the changes in the presence of spray, we calculate the centreline profile as defined by Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993), Lau & Nathan (Reference Lau and Nathan2014), Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022):

(4.1) \begin{equation} \dfrac {U_g}{u_{x,0}}=\frac {1}{B} \dfrac {x-x_0^{u_{x}}}{d_g}, \end{equation}

where $1/B$ is the average decay rate and $x_0^{u_{x}}$ is the virtual origin associated with it. Different values of the gas jet’s parameters from the literature are provided in table 2. We observe a small deviation of the single-phase gas jet (black line) from the data of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993) (blue line). This difference may be due to the fact that while the present configuration is an annular gas jet, the experiments of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993) consider a round jet. A higher decay rate of the annular gas jet agrees with the results of Kuhlman (Reference Kuhlman1987) as shown in table 2. Moreover, measurements of gas flow from the spray droplets point to a decrease in its decay rate in the case of a non-swirling spray, indicating a modification of the carrier phase flow. The observations of Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022) regarding the decay of the gas jet, therefore, are closer to the observations of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993); however, this is an outcome of a trail of physical phenomena. Combining the information that an annular jet yields a bigger decay rate value than a round jet (Kuhlman Reference Kuhlman1987) and that, for particle-laden coaxial jets for similar mass loading, adding particles to a jet decreases its decay rate (Park & Chen Reference Park and Chen1989), our observations, with different decay rates in single- or two-phase conditions are in good agreement with the literature accounting all the physical traits.

Table 2.

Comparison of the jet decay and spread constants for a single-phase jet and in the presence of spray from (4.1)–(4.3). Note that here $d_g=1$ cm.

For a swirling spray, there are significant deviations from the non-swirling case. The possible deviations are in virtual origin, decay rate and spreading rate. While the limitation of the data does not reveal the virtual origin ( $x_{0,SW}$ ), it indeed points to an increase in the decay rate of velocity with an increase in ${\textit{SR}}$ . Such an increase is a consequence of the increased length scale associated with the swirling motion for a given axial transport of both the dispersed and continuous phases. More consequences of this behaviour will be elaborated in the sections ahead.

The conservation of momentum of a turbulent jet necessitates a linear increase in the jet width as it evolves downstream. To represent this spread, the half-width ( $r_{0.5}$ ), which is the radial location where $u_x = 0.5u_{x,0}$ , is identified. This radius is related to the axial distance through

(4.2) \begin{equation} r_{0.5}=S_{0.5}\big (x-x_0^{0.5}\big ), \end{equation}

where $S_{0.5}$ is the spreading rate at the half-width and $x_0^{0.5}$ is the virtual origin corresponding to the jet half-width. In addition, with a similar definition, the ten-per-cent width ( $r_{0.1}$ ) is also calculated. This radius is considered the jet edge on many occasions and is the location where entrainment plays a significant role. Figure 3 $(c)$ illustrates both radii. It is observed here that due to the increase in the decay rate with the increase of swirl in the gas jet, there is also an increase in its spreading rate. This is somewhat inferred from the $r_{0.5}$ plots and vividly from the $r_{0.1}$ plots. The jet angles, which are direct representatives of spread, are obtained from the evolution of $r_{\alpha }$ , with $\alpha =$ 0.5 and 0.1, as

(4.3) \begin{equation} \theta =2 \tan ^{-1} (S). \end{equation}

The different values obtained for the annular gas jet with spray are close to the reported constants of Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022). Therefore, the present results extend the investigation to swirling sprays. A quantitative assessment of this angle ( $\theta _{0.1}^{u_x}$ ) for the swirling gas jet of ${\textit{Re}}_g={102\,400}$ with ${\textit{SR}}=0.8$ indicates an approximate increase by $25\,\%$ from the corresponding non-swirling condition (from $11.59^\circ$ to $14.55^\circ$ half-cone angle). As there is very small difference in the spread of ${\textit{SR}}=0.8$ and $1$ , figure 3 $(c)$ confirms that the 25 $\%$ increase of $\theta$ is reached gradually along increments of ${\textit{SR}}$ values (except for ${\textit{SR}}=0.2$ which resembles a non-swirling spray, in agreement with previous work). A similar behaviour is also observed for $\theta _{0.5}$ , the jet angle based on $50\%$ centerline velocity. While the spreading rate of swirled spray may slightly vary with the gas Reynolds number, it is first and foremost determined by the value of ${\textit{SR}}$ .

For a round gas jet, the radial and axial variation of $u_x$ is well accommodated using a self-similarity curve often given by

(4.4) \begin{equation} \dfrac {u_x}{u_{x,0}}=\mathrm{exp}(-C\eta ^2), \end{equation}

where $\eta =r/(x-x_0^{u_{x,0}})$ . The self-similar variable $\eta$ has a corresponding value in $r/r_{0.5}$ with a scale ( $S_{0.5}$ ) to recover the same shape. In light of this understanding, a similar nature for different conditions with respect to the $r/r_{0.5}$ coordinate is illustrated in figure 3 $(d)$ . It includes non-swirling as well as swirling conditions for different axial locations. There is an appreciable collapse across all the cases of table 1. This collapse is indicative of the axial momentum-conserving nature of the jet, even in the presence of a swirl. However, critical information for a swirling spray also underlies other velocity components of the carrier phase, pertinent to determining the drop size and liquid mass distributions in a particular spray plane.

4.2. Mean radial ( $u_r$ ) and tangential ( $u_\theta$ ) velocity of the jet

Figure 4.

The swirl number ${\textit{SN}}$ calculated using (4.5) and plotted with the swirl ratio ${\textit{SR}}$ for two ${\textit{We}}_g$ values $(a)$ and along ${\textit{We}}_g$ at ${\textit{SR}} = 0.8$ $(b)$ .

It is important to discuss here that the swirl ratio ${\textit{SR}}$ is a proxy parameter for quantifying the swirl. As the values of $u_\theta$ and $u_x$ are not a constant value over the annular gap of the nozzle, an integrated approach until the maximum measurement radius, $r_{max}$ is used to better characterise the swirl intensity by calculating the swirl number ( ${\textit{SN}}$ ), given by

(4.5) \begin{equation} SN=\dfrac {\int _{0}^{r_{\textit{max}}} 2 \pi \rho_{g} u_x u_{\theta } r^2 \mathrm{\,d} r}{r_{\textit{max}} \int _{0}^{r_{\textit{max}}} 2 \pi \rho_{g} u_{x}^2 r \mathrm{\,d} r}. \end{equation}

Here, ${\textit{SN}}$ represents the ratio of the axial flux of the angular momentum to the axial flux of the axial momentum. This definition follows that by Beér & Chigier (Reference Beér and Chigier1972) and has been mentioned to be applicable for the characterisation of swirl generated through different mechanisms, such as vanes, helical structures, and also by inlet-based methods (like in the present studies). The appropriateness of this parameter can be clearly observed in figure 4, which plots ${\textit{SN}}$ at the nozzle exit with respect to ${\textit{SR}}$ . The measurements are conducted 1 mm away from the atomiser using PDPA with olive-oil-based smoke with sizes of the order of 1 $\unicode{x03BC}$ m. Figure 4 $(a)$ showcases the fact that irrespective of gas Weber number, ${\textit{SR}}$ characterises the flow as well as ${\textit{SN}}$ , and therefore can be used interchangeably for parameterisation. The broad span of ${\textit{We}}_g$ values reported here encompasses the range of the spray operations of this article. Figure 4 $(b)$ shows the relation between ${\textit{SN}}$ and ${\textit{We}}_g$ for ${\textit{SR}}=0.8$ . It further reinforces the independence of ${\textit{SN}}/SR$ with the change in ${\textit{We}}_g$ . In what follows, ${\textit{SR}}$ is used throughout, being the controlled parameter, as it conveys the same level of information about the swirl intensity as ${\textit{SN}}$ .

Figure 5.

Gas-phase radial velocity ( $u_r$ ) and tangential velocity ( $u_\theta$ ) characterisation. All the non-swirling conditions are shown with black open circles, and swirling conditions are shown with filled circles of different colours depending on ${\textit{SR}}$ . $(a)$ Radial velocity profiles normalised with their centreline axial velocity shown as a function of the radial coordinate non-dimensionalised with their half-widths of the axial velocity profiles for all the conditions in table 1. $(b)$ Radial profiles of the tangential velocity for all cases in table 1. $(c)$ The tangential velocity normalised with centreline axial velocity and ${\textit{SN}}$ (from (4.5)) as a function of non-dimensional radial coordinate for ${\textit{SR}}\geqslant 0.4$ at $x/d_g = 25$ for all ${\textit{Re}}_l$ and ${\textit{Re}}_g$ . $(d)$ Non-dimensional radial profiles of the normalised velocity in the $r{-}\theta$ plane for ${\textit{SR}}\gt 0$ in table 1 for all ${\textit{Re}}_l$ and ${\textit{Re}}_g$ at $x/d_g = [25, 50]$ . The colour map indicates the angle subtended by the $u_\perp$ with the $u_r$ .

Figure 5 demonstrates the mean radial ( $u_r$ ) and tangential ( $u_\theta$ ) velocity components of the gas jet in the presence of spray, which are roughly one order of magnitude less than $u_x$ . The self-similar behaviour of the radial velocity profile and its radial increase followed by gradual decrease is given by Pope (Reference Pope2001), calculated from the data of Hussein et al. (Reference Hussein, Capp and George1994) using the continuity equation. A similar profile is also observed here for the non-swirling gas jet (figure 5 a), although with a difference in scale as the continuity equation used by Pope (Reference Pope2001) will not provide exact values due to entrainment effects. In addition, we also observe the same profile shape for the radial velocity, even in the presence of some tangential motion in the gas flow. The measurements were restricted to the radial locations where there were enough droplets available for statistical convergence (for acquisition times limited to 25 min). Nonetheless, the radial extrapolation of radial velocity profiles leads to a region of negative value, marking the entrainment region. This behaviour is observed not only for the non-swirling gas jet but also for the swirling jet.

In a swirling flow field, along with $u_x$ and $u_r$ , $u_\theta$ also contributes significantly to the droplet distribution due to its contribution towards the centrifugal force. Figure 5 $(b)$ , therefore, demonstrates the radial behaviour of the tangential velocity, which establishes the fact that the spray encounters a meaningful tangential velocity only when ${\textit{SR}}\geqslant 0.4$ . Hence, the comparisons of the characteristics of the swirling spray with the non-swirling spray are conducted primarily with ${\textit{SR}}=0.8$ . The mean tangential velocities for conditions with ${\textit{SR}}\geqslant 0.4$ at a particular axial location can be non-dimensionalised with a product of the axial location’s centreline velocity and swirl number ( ${\textit{SN}}$ ) given by (4.5). This non-dimensional tangential velocity is plotted along the radial distance normalised by $r_{0.5}$ . Figure 5 $(c)$ presents this profile, which is self-similar across different ${\textit{SR}}$ and flow conditions but not across axial locations. The definition of ${\textit{SN}}$ establishes that the relative angular velocity of the gas jet at different radial locations concerning the centreline axial velocity is nearly a constant factor defined by the ratio of axial flux of angular momentum and axial flux of axial momentum.

Although most of the momentum is in the axial direction, with a minor contribution from $u_r$ and $u_\theta$ , together they can affect migration of the droplets which may result in a change in the dispersion behaviour of the droplets. To cater the lateral flow field in a combined manner, the velocity values in the perpendicular plane of the spray axis, given by $u_\perp =\sqrt {u_r^2+u_\theta ^2}$ , are normalised by their maximum value and plotted with the radius non-dimensionalised with $r_{0.5}$ in figure 5 $(d)$ . The profile is similar to non-dimensional $u_r$ and $u_\theta$ profiles. In addition, the angles subtended by $u_\perp$ with $u_r$ are high only for $r\gt \,r_{0.5}$ . However, the converse may not always be true, as observed from the plot. It is observed that $r_{crit}$ , the radial location where $u_\perp$ is maximal, roughly coincides with $r_{0.5}$ . The self-similarity of $u_\perp$ and $u_x$ establishes a conserved nature of the spray, independent of swirling intensity, whereby appropriate identification of the similarity parameters can result in the unique description of the spray, irrespective of condition and location.

Figure 6.

Swirl value of the gas phase calculated from the data for ${\textit{SR}} = 0.8$ and ${\textit{Re}}_g= 102\,400$ at $x/d_g=25$ .

Until now, the mean velocity profiles have mostly been normalised with centreline axial velocity. However, a better essence of the swirling motion comes from its normalisation with the axial velocity for that location, which is shown in figure 6 along ${\textit{We}}_g$ and ${\textit{Re}}_l$ . It indicates that, for a constant ${\textit{SR}}$ value, the swirling intensity of the gas decreases with increasing ${\textit{We}}_g$ as a higher swirling velocity at the exit of the nozzle is associated with high shear and a more considerable effective distance covered by the fluid particle, resulting in higher dissipation with respect to the axial motion. On the other hand, with an increase in ${\textit{Re}}_l$ , more kinetic energy is required for breakup and dispersion, which leads to a smaller contribution to the tangential velocity. It is important to note here that the mechanism of loss of the swirling motion with increasing ${\textit{We}}_g$ and ${\textit{Re}}_l$ are different. Consequently, the droplet dispersion at higher ${\textit{We}}_g$ and ${\textit{Re}}_l$ are affected differently, which will be elaborated in further sections.

4.3. Droplet slip

The gas phase on interaction with the droplets determines the latter’s dispersion. A classical way of representation has been to investigate the slip of the big droplets in the surrounding gas phase. While particles with $St\lt 0.01$ are typically considered to behave as tracer particles (Balachandar & Eaton Reference Balachandar and Eaton2010), for a more optimistic determination of the gas phase, we consider $St\lt 0.002$ as a limit for identifying particles as tracers of the gas jet. On the other hand, the bigger particles with Stokes number, $St \gt 0.01$ , are known to accumulate at regions of high strain rate (Eaton & Fessler Reference Eaton and Fessler1994). As a result, particles in a high-speed jet exhibit some slip relative to the carrier flow. In a polydispersed spray, the size of droplets varies widely, resulting in $St_d\sim 10^{-4}$ at the nozzle exit for the present investigation, where the subscript $d$ indicates droplet here. Considering that the droplets for which $St_d\gt 0.005$ are either partially responsive or non-responsive, we investigate the slip behaviour of these droplets in figure 7. To find this slip velocity ( $U_s$ ), we first compute the size class of droplets that yields $St_r\gt 5\times 10^{-3}$ , defining a local Stokes number where the flow time scale is based on the radius of the location ( $r$ ) as length scale divided by the mean local gas axial velocity. This formulation leads to a radial increase in the average drop size used in the definition as the location is farther from the jet centre which is shown in figure 7(a,b). Therefore, $U_s$ is limited to the radial location for which $d_{|St_r\gt 0.005}\lt 100$ $\unicode{x03BC}$ m, since this corresponds to the biggest drop measured. Then, we find the average axial velocity of these droplets for which $d_{|St_r\gt 0.005}\lt d\lt 100$ $\unicode{x03BC}$ m and calculate the difference from the gas mean axial velocity $U_s(r)=\langle {v_x}_{|St_r\gt 0.005}\rangle (r)-u_x(r)$ . Note that from here, plots with open and filled markers will represent ${\textit{SR}}=0$ and ${\textit{SR}}\gt 0$ , respectively.

Figure 7.

Radial profiles for the computed average size of drops whose local Stokes number respects $St_r\gt 0.005$ with varying $(a)$ ${\textit{We}}_g$ and $(b)$ ${\textit{Re}}_l$ for swirling (open markers) and non-swirling (filled markers) conditions. The horizontal 100 $\unicode{x03BC}$ m line indicates the limit of droplet size measurement in this study. Radial profiles of slip velocity defined by the velocity difference of droplets with $St_r\gt 0.005$ and $u_{g,x} (r)$ for different $(c)$ ${\textit{We}}_g$ and $(d)$ ${\textit{Re}}_l$ .

Compared with the gas flow, the big droplets in a non-swirling spray accelerate, overshoot the gas flow, and then decelerate as they transport downstream (Lasheras & Hopfinger Reference Lasheras, Villermaux and Hopfinger1998; Wu et al. Reference Wu, Zhang, Zhang and Hou2022; Zhou et al. Reference Zhou, Hawkes, Lau, Chin, Nathan and Wang2022). At $x/d_g = 25$ , $U_s\gt 0$ at the centreline, as noticed in figure 7 $(c\kern-0.5pt,{\kern-2pt}d)$ , which is either in the overshoot or deceleration region. This is in confirmation of the results on PLJs (Hardalupas, Taylor & Whitelaw Reference Hardalupas, Taylor and Whitelaw1992) and sprays (Lasheras et al. Reference Lasheras, Villermaux and Hopfinger1998). The slip velocity, $U_s$ , increases with radius from the centreline and decreases after reaching a maxima. The radial profiles observed here are similar to the trends observed by Wu et al. (Reference Wu, Zhang, Zhang and Hou2022) in sprays and can be interpreted from the results of Zhou et al. (Reference Zhou, Hawkes, Lau, Chin, Nathan and Wang2022) on PLJs. The reason for an off-centre maximum is the higher particle inertia of the large droplets, which leads to high velocity far downstream while the gas jet spreads more rapidly, thereby decaying faster (Gounder et al. Reference Gounder, Kourmatzis and Masri2012). Eventually, when $r/r_{0.5} \gt 1$ , the Stokes number experienced by the big droplets becomes small enough to attain a velocity value close to that of the gas phase. These results are also complemented by the fact that $U_s$ decreases with higher ${\textit{Re}}_l$ closer to the centreline (figure 7 c) and is indicative of the late response of the big droplets, which occurs at farther downstream distance from the exit of the atomiser. Therefore, the inertia attained is also low.

With swirl added to the gas jet, the increase in spreading leads to a reduced mean axial velocity of the carrier flow, however, much smaller than that found for the droplet velocities. As a result, figure 7(c,d) demonstrates that $U_s\lt 0$ for most radial locations accessible here, reaching a minimum before increasing again with the radius. In addition, no specific correlation was observed with ${\textit{We}}_g$ or ${\textit{Re}}_l$ . Considering droplets that are centrifuged away from the jet’s centreline, carrying high axial momentum accumulated there and carried over to regions of $r\gt 0$ , one may expect droplet velocities larger than that of the carrier flow. On the contrary, droplets that are formed by high-amplitude flapping events, commonly found in swirled sprays, are ejected directly outside of the gas jet and may find their way back to the jet only due to entrainment, explaining their low axial velocities. Combining these statements, the averaged droplet velocity is found largely to be smaller than in non-swirling conditions, indicating that there may be a larger proportion of initially detrained droplets that are then entrained back into the jet than droplets centrifuged from the jet’s axis. An exception is the ${\textit{Re}}_l = 8480$ condition displaying a lower negative amplitude of $U_s$ . In comparison with the other conditions, the high-momentum liquid jet may in that case produce more large droplets in the vicinity of the centreline whose high inertia better preserves their axial momentum as they get centrifuged radially, leading to positive slip velocity close to $r=r_{0.5}$ .

5.1. Local drop size characteristics

A noteworthy feature of coaxial sprays is the production of drops with a wide range of sizes, which one can assess from the droplet distribution plots shown in figure 8. While the peaks of the distributions are found below $10$ $\unicode{x03BC}$ m, the mean droplet size can vary with radial location by typically tens of $\unicode{x03BC}$ m, depending on the condition (not shown here). The 70–100 $\unicode{x03BC}$ m range is found to approximately plateau for most conditions and locations as shown in figure 8.

The spray formation mechanism explained for high ${\textit{We}}_g$ (Marmottant & Villermaux Reference Marmottant and Villermaux2004; Huck et al. Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022) leads to larger mean droplet size in the centre of the spray than at the edge in the absence of swirl. We show the probability distribution functions (p.d.f.s) of drop sizes for such a condition for a fibre-type atomisation regime at different radial locations for a non-swirling spray in figure 8 $(a)$ , which confirms the established understanding. An addition of swirl (shown in figure 8 b) can be observed to reverse the spatial sampling of droplets in the spray, with smaller droplets found at the centre. This change indicates the manipulative influence of the centrifugal force in rearranging droplets in a spray plane, based on their mass ( $F_c = m_d\omega ^2r$ , where $m_d$ is the mass of droplets and $\omega = u_\theta /r$ ), and hence size. Such behaviour contributes to an increase in the spray angle beyond the gas jet angle for a swirling spray as the large droplets are flung far due to $F_c$ (Préaux et al. Reference Préaux, Lasheras and Hopfinger1998). This explanation is complemented by the smaller eddy turnover time near the toroidal region close to the atomiser for high swirling flows, due to which only the small droplets that have a small response time are entrained. This argument also holds good for conditions with low ${\textit{We}}_g$ . However, as the mechanism of droplet breakup and entrainment in low- ${\textit{We}}_g$ conditions favour the same organisation of droplets with and without swirl, no change is noticeable with such a representation and will require a different description, which we provide later in the following subsection.

Figure 8.

The p.d.f. at several radial locations for a given longitudinal distance $x/d_g=25$ , for ${\textit{Re}}_l = 1120$ and ${\textit{We}}_g=830$ at (a) ${\textit{SR}}=0$ and (b) ${\textit{SR}}=0.8$ .

Figure 9.

The mean droplet size ( $d_{10}$ ) plotted with radius non-dimensionalised with $r_{0.5}$ for ${\textit{Re}}_l=1120$ and ${\textit{We}}_g=830$ at various axial distances $x/d_g.$ $(a)$ ${\textit{SR}} = 0$ and $(b)$ ${\textit{SR}}=0.8$ .

On the other hand, for low- ${\textit{We}}_g$ conditions, the origin of droplets of different size ranges varies, and hence their dynamics differs in the carrier phase. Chigier & Farago (Reference Chigier and Farago1992) have reported that liquid jet breakup with low ${\textit{We}}_g$ , and hence low ${\textit{Re}}_g$ , is a regime where atomisation predominantly occurs through the formation and bursting of bags, contributing to forming much finer droplets than expected at these ${\textit{We}}_g$ values (if bags were absent). Its rims and ligaments form large droplets, following similar processes as for the breakup of a single droplet subjected to a gas flow. In addition, due to the flapping behaviour of the liquid jet at these low ${\textit{We}}_g$ , the outward ejections of large droplets result in a droplet distribution peak ${\sim}100$ $\unicode{x03BC}$ m at the spray edge. Calculating the Weber number of these big droplets based on slip velocity, the value obtained is less than 1, indicating no further breakup of these drops is possible. It is well known that the spread of a gas jet results in mass entrainment. While droplets with $St\lt 0.01$ can certainly respond to entrainment and attain an inward radial velocity, the droplets with $St\gt 10$ have a ballistic motion. This behaviour is distinguishable primarily near the atomiser, where the breakup occurs. Combining these understandings points to the fact that only finer droplets can get influenced by entrainment ( $\propto u_x$ ) and have a higher probability of accumulating near the centre of the spray. Therefore, the droplet distribution favours small droplets at the centre and large ones at the edge (not shown) for low ${\textit{We}}_g$ . This understanding confirms the results obtained by Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022).

Figures 9(a) and 9( $b$ ) shows the radial profiles of the mean drop size at different axial locations for high- ${\textit{We}}_g$ non-swirling and swirling conditions, respectively. These measurements reveal two aspects of spray development as droplets travel downstream in a turbulent environment. In the case of non-swirling and swirling sprays, we observe that at high ${\textit{We}}_g$ for $x/d_g = 25$ and $50$ , the drop size profiles are almost identical, indicating the fully developed nature of the spray. Comparing these profiles with the profiles at $x/d_g = 10$ , two completely different behaviours can be observed for non-swirling and swirling conditions. For the non-swirling condition, one can observe a slight increase in the drop size throughout the spray with almost the same shape as observed at $x/d_g = 25$ and $50$ . This indicates a very minor rearrangement due to entrainment, but mostly coalescence of the droplets throughout the spray. On the other hand, for a swirling spray, the profile changes drastically, crossing the other two profiles at $r = r_{0.5}$ , with drop size at the spray’s edges more than double the values found farther downstream. The profiles change from the centrifugal sorting near the atomiser to homogenisation due to turbulence far from the nozzle. Therefore, we conduct further investigations on the radial profiles of the drop size at $x/d_g=25$ , by which time the radial profiles have evolved to a fully developed profile for all conditions.

Figure 10.

The mean droplet size ( $d_{10}$ ) plotted with radius non-dimensionalised with $r_{0.5}$ for different flow conditions at $x/d_g=25.$ $(a)$ ${\textit{We}}_g=830$ , ${\textit{Re}}_l=1120$ for different ${\textit{SR}}$ values. (b,c) Fixed ${\textit{Re}}_l=1120$ at different ${\textit{We}}_g$ values. (d,e) Fixed ${\textit{We}}_g=830$ at different ${\textit{Re}}_l$ values. (b,d) ${\textit{SR}}=0$ . (c,e) ${\textit{SR}}=0.8$ .

Figure 10 demonstrates the influence of different parameters on the radial variation of the mean drop size for a non-swirling and a swirling spray. A logical way forward is first to understand the influence of various swirl intensities and then downsize the investigation to a few conditions. Therefore, figure 10 $(a)$ highlights the changing nature of the mean droplet size radial profiles with increased swirl intensity. Such a radial variation of the drop size (albeit for the Sauter mean diameter $d_{32}$ ) is also observed by Préaux et al. (Reference Préaux, Lasheras and Hopfinger1998) and (for $d_{10}$ ) by Fong et al. (Reference Fong, Xue, Osuna-Orozco and Aliseda2024). The invariance of the profile for ${\textit{SR}}\lt 0.4$ is in confirmation to the inference obtained from figure 5 $(c)$ that there is a limited effect of swirl in such cases. The radial $d_{10}$ profile changes from a ‘ $\sim$ ’ shape at negligible swirl values to a ‘ $\cup$ ’ shape when the swirl effects exist. To quantify this change we describe a two-location identifier of the shape $d_{10}$ at $r/r_{0.5}=1:2$ (normalised by the value at $r=0$ ) given by [ $a$ $b$ ]. Measuring the shape identifier for the conditions in figure 10 $(a)$ , we find that this value changes from [0.74 0.8] for ${\textit{SR}}=0$ to [1.24 1.61] for ${\textit{SR}}=1$ . When the radial profile shape is ‘ $\sim$ ’, as in the former condition, the shape identifier value decreases slightly (from 1 to 0.74 here) and then increases (from 0.74 to 8 here). For a ‘ $\cup$ ’ profile, as in the later condition, the value increases monotonously with larger relative amplitudes (1, 1.24 and 1.61 here). For ${\textit{SR}}\geqslant 0.4$ , one can observe in addition a decrease in the drop size with increasing swirl throughout the radial extent of measurements. This behaviour can be attributed to the increased probability of the small droplets closer to the centre and an indication of a decrease in the overall size of the droplets with increasing swirl for high ${\textit{We}}_g$ . These results are in agreement of the findings of Fong et al. (Reference Fong, Xue, Osuna-Orozco and Aliseda2024).

Figures 10(b) and 10 $(c)$ compare the effect of swirl on drop size for different ${\textit{We}}_g$ without and with swirl. The drop size decreases throughout a particular spray plane with an increase in ${\textit{We}}_g$ , irrespective of the swirl. However, a closer look at the radial drop size profiles for non-swirling sprays (figure 10 b) suggests that the shape identifier value changes from [0.92 1.4] for ${\textit{We}}_g=210$ , indicating almost a ‘ $\cup$ ’ shape with very high values at the edge, to [0.75 0.83] for ${\textit{We}}_g=1280$ , indicating a ‘ $\sim$ ’ shape. For swirling conditions (figure 10 c), this profile retains the ‘ $\cup$ ’ shape for all ${\textit{We}}_g$ due to the explanation given earlier. Nonetheless, the curvature of the profile, i.e. the steepness of the radial increase in mean droplet size, can be observed to decrease with the increase in ${\textit{We}}_g$ for the later conditions. The change is especially striking for the two higher- ${\textit{We}}_g$ values, such that the drop sizes do not deviate significantly in all the spatial domains and are in the closer range found for non-swirling sprays. The shape identifier value varies from [1.88 4.14] for ${\textit{We}}_g=210$ to [1.22 1.51] for ${\textit{We}}_g=1280$ , indicating the drastic change in the magnitudes of the radial variations making the spray much more homogeneous in terms of droplet size (going from a fourfold to a 50 % increase only). These results lead to the inference that three factors contribute to defining the shape of the droplet size radial profiles: (i) the higher gas kinetic energy with increasing ${\textit{We}}_g$ leading to the formation of smaller droplets, which are influenced by entrainment, (ii) the increase in the unfavourable pressure gradient with increase in $u_\theta$ and (iii) the independence of critical swirl number, $S_{c}$ from $M$ at high $M$ (Hopfinger & Lasheras Reference Hopfinger and Lasheras1996). Altogether, these factors reduce the preferential segregation of droplets for higher ${\textit{We}}_g$ . Nevertheless, a subtle difference in the radial profile from the ‘ $\cup$ ’ shape is the change from a monotonic increase without noticeable inflection point $ (({\mathrm{d}^2d}/{\mathrm{d}r^2})\geqslant 0 )$ at low ${\textit{We}}_g$ to the presence of profiles with inflection point $ (({\mathrm{d}^2d}/{\mathrm{d}r^2})=0 )$ at higher ${\textit{We}}_g$ , either side of which the curvature changes sign.

While a similar investigation for two values of ${\textit{We}}_g$ on radial drop size profiles has been demonstrated recently by Fong et al. (Reference Fong, Xue, Osuna-Orozco and Aliseda2024), an investigation into the effect of ${\textit{Re}}_l$ on a non-swirling spray is elaborated here. Figure 10 $(d)$ shows that although there is a change in the magnitude of radial drop size profiles with increase in ${\textit{Re}}_l$ , there is barely any change in its shape, contrary to the effect of ${\textit{We}}_g$ . For the non-swirling spray, the ‘ $\sim$ ’ radial drop size profile varies from the shape identifier value of [0.81 0.87] at ${\textit{Re}}_l=8480$ to [0.74 1.1] at ${\textit{Re}}_l=1120$ . This examination substantiates the significant role that ${\textit{We}}_g$ plays in the initial liquid jet breakup mechanisms over ${\textit{Re}}_l$ (Lasheras & Hopfinger Reference Lasheras and Hopfinger2000; Tolfts, Deplus & Machicoane Reference Tolfts, Deplus and Machicoane2023), leading to the drop size profiles. On swirl addition to the gas flow (figure 10 e), the radial profiles are ‘ $\cup$ ’ shaped. Specifically, this shape has a similar inflection for conditions with small- ${\textit{Re}}_l$ and large- ${\textit{We}}_g$ values. The associated shape factors are [1.87 2.92] for ${\textit{Re}}_l=8480$ and [1.32 1.76] for ${\textit{Re}}_l=1120$ , indicating a significant change in the size between the edge and the centre. Focusing on both parameters indicates that the gas-to-liquid dynamic pressure ratio $M$ may have a role in determining the drop size profiles for swirling sprays. However, our efforts in combining the effect of ${\textit{Re}}_l$ and ${\textit{We}}_g$ through $M$ does not provide any direct correlation. This behaviour confirms that the relationship between drop size and the flow parameters is non-trivial. With more results, this aspect is elucidated further in § 7.

5.2. Global drop size characteristics

Most of the literature has focused on a radial change of drop size, which is a point-wise measurement with a variable contribution to the global spray based on the radial location. Elaboration of this representation is shown through figure 1 $(d)$ , where each cross represents the radial location and the annular space represents the area that each of these locations (blue annulus for one of the locations is highlighted as an example) is assigned to. Therefore, in order to make the contributions of each radial location a justifiable proportion of the global spray, we define a p.d.f. as follows (Tratnig & Brenn Reference Tratnig and Brenn2010; Dhivyaraja et al. Reference Dhivyaraja, Gaddes, Freeman, Tadigadapa and Panchagnula2019; Osuna Orozco Reference Osuna Orozco2021; Fong et al. Reference Fong, Xue, Osuna-Orozco and Aliseda2024):

(5.1) \begin{equation} p_i=\dfrac {\sum _{\!j} N_{\boldsymbol{ij}} S_{\!j} / w_i T_{\!j}}{\sum _i \sum _{\!j} N_{\boldsymbol{ij}} S_{\!j} / w_i T_{\!j}}, \end{equation}

where $N_{\textit{ij}}$ is the number of particles in droplet size class $i$ at location $j$ , $S_{\!j}$ is the area which the location $j$ represents (figure 1 d), $w_i$ is the weight of the droplet size class $i$ depending on the diameter of the probe volume, which is required for rectifying the bias due to the Gaussian nature of the laser beam and $T_{\!j}$ is the time of collection of data at location $j$ . The weighting function $w_i$ is determined by fitting a logarithmic curve fitting between the drop size and the path length of the droplets in the probe volume. The cumulative p.d.f. is plotted for different ${\textit{We}}_g$ and ${\textit{Re}}_l$ in figures 11 $(a)$ and 11 $(b)$ , respectively, for non-swirling and swirling sprays. One can notice that at low ${\textit{We}}_g$ and high ${\textit{Re}}_l$ , there is a higher probability of bigger droplets for the swirling spray compared with the non-swirling one, and the converse is true for high ${\textit{We}}_g$ and low ${\textit{Re}}_l$ . However, when swirl assists the spray (e.g. at high ${\textit{We}}_g$ and low ${\textit{Re}}_l$ ) the reduction in the probability of large droplets is massive, with for instance a probability decrease of a factor of roughly 3–5 in the 50–80 $\unicode{x03BC}$ m range. The resulting drop size ( $d_{10}$ ) from (5.1) is shown in figure 12. One can observe a decrease of drop size with the increase in ${\textit{We}}_g$ and the converse with increasing ${\textit{Re}}_l$ due to the increased inertia compared with the surface tension forces with increasing $M$ . While the spread of the spray due to swirl is well known, this analysis gives a global assessment of the drop size. We conclude that swirl reduces the drop size for high $M$ . However, it may have a detrimental effect in terms of drop size at lower $M$ . The recently presented results on the mean drop size by Fong et al. (Reference Fong, Xue, Osuna-Orozco and Aliseda2024) claims that there is a monotonous decrease in the size for the whole spray droplet size with the introduction of swirl for $M=25$ and $56$ , and hence for all regimes of spray. However, the present exploration to further lower $M$ and also different ${\textit{Re}}_l$ values indicates a complicated relation between the global drop size depending on ${\textit{Re}}_l$ , ${\textit{We}}_g$ and ${\textit{SR}}$ . In this way, the present result extends the existent findings on findings of Fong et al. (Reference Fong, Xue, Osuna-Orozco and Aliseda2024) on the effects of gas swirl on drop size.

Figure 11.

Global drop size distributions for different ${\textit{We}}_g$ for ${\textit{Re}}_l=1120$ $(a)$ and ${\textit{Re}}_l$ for ${\textit{We}}_g=830$ $(b)$ , at $x=25d_g$ . Empty symbols correspond to non-swirling conditions while filled ones are for ${\textit{SR}}=0.8$ .

Figure 12.

Variation of the global mean drop size $d_{10}$ with ${\textit{We}}_g$ for ${\textit{Re}}_l=1120$ $(a)$ and ${\textit{Re}}_l$ for ${\textit{We}}_g=830$ $(b)$ , at $x=25d_g$ . Empty symbols correspond to non-swirling conditions while filled ones are for ${\textit{SR}}=0.8$ .

6.1. Axial flux

The preceding discussion on gas-phase velocity and drop size have elucidated two important features of the swirling spray: (i) high axial gas velocity at the centre of the spray (figure 3) and (ii) large droplets at the edge of the spray (figure 10). Individually, they do not inform whether the mass flux will be high at the centre due to high velocity or at the edge due to the large droplet size, which is essential to assess liquid mass distributions. Therefore, we calculate the volume flux of the liquid at a particular location by

(6.1) \begin{equation} \dot {g_x}=\frac {\pi }{6T}\sum _{i=1}^{D} \sum _{j=1}^{N_i} \dfrac {d_{j, i}^3\cos(\theta _{\!j})}{\mathcal{A}_i}, \end{equation}

where $T$ is the collection time (at the measurement location), $d_{j, i}$ is the size of the $j$ th drop in the $i$ th bin and $\theta _{\!j}=\text{tan}^{-1}(u_r/u_x)$ . Here, $\mathcal{A}_i$ is the size-dependent probe area given in (2.1). Figure 13 illustrates the influence of different flow parameters on the radial profiles of axial volume fluxes ( $\dot {g}_x$ ), non-dimensionalised by their centreline values. For a non-swirling flow, the narrowing of the volume flux profiles with increasing ${\textit{We}}_g$ has been reported earlier by Hardalupas & Whitelaw (Reference Hardalupas and Whitelaw1994), Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022), which is also observed here. In addition, we observe opposite trends with an increase in ${\textit{Re}}_l$ , confirming the role of $M$ in determining the axial flux profiles.

On the other hand, although the volume flux profiles for the swirling flows follow a similar relationship with ${\textit{We}}_g$ and ${\textit{Re}}_l$ , they deviate from the bell-shaped radial profile. Especially for low- $M$ values, the radial decrease is non-monotonic, and the maxima shifts away from the centreline, which is similar to the observations made by Fong et al. (Reference Fong, Xue, Osuna-Orozco and Aliseda2024) for the swirling conditions they investigated. In addition to these observations, the present investigation also highlights that depending on the liquid Reynolds number, in corresponding high- ${\textit{We}}_g$ conditions, the centreline can also have the maximum flux at a radial distance away from the spray axis. Note that our assessment does not associate a definitive $M$ for this transition across varying ${\textit{Re}}_l$ , leading to a non-trivial relation of the change of shape of the radial flux profiles with $M$ for swirling sprays.

Figure 13.

Radial profiles of axial flux with $(a)$ ${\textit{Re}}_l=1120, (b)$ ${\textit{We}}_g=830$ measured at $x=25d_g$ . Open and filled symbols represent the non-swirling and ${\textit{SR}} = 0.8$ conditions, respectively.

The liquid jet breakup near the atomiser involves its orbiting about the axis of the atomiser when swirl is introduced (Machicoane et al. Reference Machicoane, Ricard, Osuna-Orozco, Huck and Aliseda2020; Kaczmarek et al. Reference Kaczmarek, Osuna-Orozco, Huck, Aliseda and Machicoane2022). In the presence of this swirling gas, a large portion of the big droplets, and hence mass, are expected to centrifuge outside. However, the influence is predominant when the initial droplet formation happens sufficiently away from the axis, so adequate angular momentum is available. At low- $M$ conditions, the orbiting liquid jet has a long length and will have a substantial propensity to break farther away from the axis and attain higher angular momentum, thereby shifting the maximum flux away from the centreline (found in the vicinity of $0.5\lt r/r_{0.5}\lt 1$ ). Extrapolating this understanding to high- $M$ conditions, due to very short breakup length in the case of high- $M$ conditions, such a possibility is difficult due to the pre-existing high confinement effects close to the atomiser ( $\propto U_g$ ). Nevertheless, the spread of the gas jet farther downstream and the corresponding transport of droplets facilitate the liquid mass to deviate away from the centreline due to the swirling motion of the gas, resulting in a broader bell curve, whose maximum is still located at $r=0$ . Whether swirled high- $M$ conditions bear axial flux whose radial profiles can retain the shape of their non-swirling counterpart is unclear and would require more data, but the radial spread increases when swirl is impeded to the gas jet. This explanation can be complemented with the observations of Fong et al. (Reference Fong, Xue, Osuna-Orozco and Aliseda2024). Their results indicate that at a given gas-to-liquid momentum ratio $M$ (with 25 and 56 sampled in their study), roughly similar shapes of radial volume flux profiles are observed through the gradual increase of swirl when its effect is noteworthy, with a clear broadening of the profiles. On closer inspection, high-swirl conditions present profiles that remain constant over a small subset of radial values (up to $2d_g$ ), before they decrease. We also observe similar results for low $M$ in the same range of the parametric values as they examined. For higher- $M$ conditions, adding swirl to the gas jet makes the profile wider but approximately retains the non-swirl shape across different ${\textit{SR}}\gt 0.2$ values explored here. The combined understanding, therefore, is that the shape of the radial profile of axial fluxes are determined by the value of $M$ even in the presence of swirl. Depending on the value of $M$ , swirl may then change the shape of the profiles.

Figure 14.

Droplet time scale based on mean size compared with the Eulerian time scale at the spray edge. The symbols $\square$ , $\triangle$ and $\circ$ represent ${\textit{SR}}=0,{\gt}0$ and ${=}0.8$ conditions, respectively.

6.2. Droplet response at the edge

The spray edge is often considered an identifier of the spray formation method (Acharya et al. Reference Acharya, Deevi, Dhivyaraja, Tangirala and Panchagnula2021; Huck et al. Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022). In addition, the response of specific-sized particles to particular eddies has been demonstrated, e.g. for shear layer by Lazaro & Lasheras (Reference Lazaro and Lasheras1992) and for gas jet by Chung & Troutt (Reference Chung and Troutt1988), Longmire & Eaton (Reference Longmire and Eaton1992), Sbrizzai et al. (Reference Sbrizzai, Verzicco, Pidria and Soldati2004). The Eulerian time scale for these eddies is based on the large-scale helical structures existing in a gas jet and given by

(6.2) \begin{equation} T_E = 2r_{0.1}/u', \end{equation}

where $u'$ is the centreline turbulent axial velocity fluctuation. The length scale $2r_{0.1}$ associated with this time scale for the jet is due to the correlation of the eddies with its width (Tso & Hussain 1989; Yoda, Hesselink & Mungal Reference Yoda, Hesselink and Mungal1992), which we analyse at $r_{0.1}\lt r\lt r_{0.05}$ as its validity has been demonstrated throughout the jet cross-section by Prevost et al. (Reference Prevost, Boree, Nuglisch and Charnay1996). The mean droplet diameter $\langle d_p \rangle$ at this radial location is used to calculate the time scale by

(6.3) \begin{equation} \tau _{p}=\rho _l\langle d_{\!p}\rangle ^2/(18\nu _g\rho _g). \end{equation}

This relation is plotted in figure 14. The corresponding Stokes number is given by

(6.4) \begin{equation} St = \tau _{\!p}/T_E, \end{equation}

which is ${\sim} 0.1$ or greater for all cases. These values indicate that the large eddies at the edge of the spray, which are evidenced both in non-swirling and swirling gas jets, will affect at least partially the droplets’ trajectories, e.g. with the occurrence of radial ejections of the inertial droplets and loitering. In swirling conditions, these eddies are accompanied by helical structures that may play an additional role in spray dispersion. For non-swirling sprays (red squares), the relation between large eddy time scale and mean droplet time scale is linear with varying conditions. This observation indicates that as the mean droplet time scale increases due to an increase in its size, so does the eddy size to which such drops can respond. The analysis of Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022) considers the peak droplet size in the distribution at the edge for varying axial locations for a single condition in a spray and the resulting Stokes number ${\sim}1$ to claim the role of the helical eddies in transporting the big droplets by ejection. Their analysis was restricted for $M\lesssim 50$ when the ejections occurred. In contrast, the present analysis spans all ${\textit{We}}_g$ and ${\textit{Re}}_l$ values given in table 1. The present result underlines a universality of the droplet behaviour at the edge, irrespective of the regime.

Extending such an analysis to swirling sprays, we observe that with a gradual increase in ${\textit{SR}}$ , the large droplets at the edge respond to eddies of increasing time scales (as observed in figure 10) as traced from figure 14: the red squares representing ${\textit{SR}}=0$ are followed by black triangles going from left to the right of the plot representing ${\textit{SR}}=0.2$ to $0.6$ until the blue circles. Subsequently, the circles are indicative of ${\textit{SR}}=0.8$ , and, again, a triangle to its right is for ${\textit{SR}}=1$ . The droplet versus the eddy time scale plot for ${\textit{SR}}=0.8$ (blue circles) is also observed to be almost linear like their non-swirling counterpart, although with a higher dispersion. Note that the helical structures in the case of swirling spray also have a contribution from the tangential component of velocity. The consequence is an increase in the time scale of axial fluctuations due to the eddies at the edge, which is larger than the droplet response time scale due to bigger drops.