5.1. VFD and VF definitions

The VFD is calculated for each diameter class, i, containing a total number $N_{i}$ of droplets

(5.1)\begin{equation} \dot{\mathcal{G}}(d_i)=\frac{\rm \pi}{6T_s{\mathcal{A}_{i}}}\sum_{j=1}^{N_{i}}{d_{j,i}^3} , \end{equation}

where $T_s$ is the total sample time, $\mathcal {A}_{i}$ is the probe cross-section (2.1) of the $j$th droplet of the $i$th size class with diameter $d_{j,i}$. We can calculate the VF assuming a single droplet occupies the probe volume at a time for the $i$ droplet class

(5.2)\begin{equation} \varPhi(d_i)=\frac{\rm \pi}{6}\frac{\tilde{t}_i}{T_s{\mathcal{V}_i}}\sum_{j=1}^{N_{i}}{d_{j,i}^3}, \end{equation}

where $\tilde {t}_i$ is the residence time of a droplet in the size-class probe volume $\mathcal {V}_i$ (2.2). Both (5.1) and (5.2) are defined over a given binned droplet size class. Between 15 and 21 binned size classes (index $i$) are used, with fewer bins used for higher momentum ratios $M$. A more general quantity obtained by integrating over all $D$ droplet size classes,

(5.3)\begin{equation} \dot{g}=\sum_{i=1}^D\dot{\mathcal{G}}(d_i), \end{equation}

is the VFD for all droplet size classes and,

(5.4)\begin{equation} \phi=\sum_{i=1}^D\varPhi(d_i), \end{equation}

the VF for all droplet size classes. These quantities are understood to be time averages of instantaneous values of VFD and VF.

It is important to note that not every drop passing through the PDI probe volume is captured. Due to the Gaussian nature of the laser beam, smaller droplets scatter less light than large droplets at the beam's edge. We correct for the bias that arises in the flux and VF measurements by introducing size-dependent probe areas (2.1) and volumes (2.2). Other biases such as multiple droplets and non-spherical droplets in the probe volume as well as multi-mode scattering are corrected for (Bachalo Reference Bachalo1994) but lead to a sub-sampling of the droplet population. When the VFD (5.4) is integrated over the spray cross-section for different downstream locations and the total volume flux is measured, values are found around [12.5–25] % of the nominal value at the nozzle for each momentum ratio. Similarly, the droplet-size distributions averaged over the spray cross-section to give an arithmetic mean diameter, $d_{10}$, vary by at most 8 % of the average value over all downstream locations. Consistency in the area-averaged volume flux and diameter measurements in the far field indicate that, despite the fact that the PDI subsamples the droplet population in the spray, these measurements are unlikely to introduce a bias in the droplet populations.

5.2. VFD and VF profiles

The VF and VFD, normalized by their maximum values for different momentum ratios, are plotted against the self-similar coordinate $\eta =r/(x-x_0^{\scriptscriptstyle \langle U\rangle })$ in figure 5 for different distances downstream of the nozzle. Increasing the gas flow rate (increasing $M$) narrows both the VFD (figure 5a) and the VF (figure 5b) profiles, as observed with similar co-axial atomizers (Hardalupas & Whitelaw Reference Hardalupas and Whitelaw1993, Reference Hardalupas and Whitelaw1994; Engelbert et al. Reference Engelbert, Hardalupas and Whitelaw1995). An important difference between the VFD and VF is that the former is narrower than the latter over the range of momentum ratios investigated, in line with earlier observations (Hardalupas, Taylor & Whitelaw Reference Hardalupas, Taylor and Whitelaw1989) in a particle-laden jet. While the VFD is always narrower than the average velocity profile (figure 5a), the VF profile straddles the velocity profile (figure 5b) depending on the momentum ratio. For a critical momentum ratio, $M\approx 56$, the average concentration profile roughly follows the average velocity profile.

Figure 5.

Comparison of VFD, $\dot {g}$, and VF, $\phi$. (a,b) VFD and VF normalized by value at centreline plotted against the self-similar coordinate $\eta =r/(x-x_0)$ for $x/d_g=[9, 12, 18, 24]$. The momentum in the liquid phase is constant while varying gas-phase momentum $M=[25, 56, 177]$.(c,d) VF and VFD profiles are normalized by the ten per cent width coordinate defined in (5.5) at $x/d_g=[18]$.

For each momentum ratio plotted in figure 5, profiles from different downstream locations collapse onto a single curve, when the self-similar coordinate $\eta$ is used. We denote this self-similar region the ‘far field’, occurring roughly from $7< x/d_g<45$, in agreement with the PLJ observations in Picano et al. (Reference Picano, Sardina, Gualtieri and Casciola2010). Although this parameter accounts for the self-similarity of the profiles with downstream distance, it does not account for variation in profile shape as the momentum ratio is varied. Using the 10 per cent width defined as

(5.5)$$\begin{gather} \dot{g}(r_{0.1}^{\dot{g}})=0.1\dot{g}_{max}, \end{gather}$$

(5.6)$$\begin{gather}\phi(r_{0.1}^\phi)=0.1\phi_{max}, \end{gather}$$

(5.7)$$\begin{gather}\langle U\rangle(r_{0.1}^U)=0.1\langle U\rangle_{max}, \end{gather}$$

we normalize the VF and VFD in figure 5(c,d). We focus on the 10 per cent width because it was found to be more sensitive to momentum-ratio-dependent changes in the tails of the curves (e.g. figure 5a,b) than the more common half-width metric. The normalized VF and VFD profiles display a satisfactory collapse at $x/d_g=18$ in figure 5(c,d). This collapse indicates that the momentum-ratio-dependent physics governing the shape of the VF and VFD profile is captured by an appropriate choice of a self-similar variable in agreement with observations in the literature (Picano et al. Reference Picano, Sardina, Gualtieri and Casciola2010; Lau & Nathan Reference Lau and Nathan2016).

For all $M$, the 10 per cent widths (both VF and VFD, figure 5a,b) evolve linearly in the far field of the jet. In general, as the momentum ratio increases, the width of the spray (either by VFD or VF) is narrower, similar to Engelbert et al. (Reference Engelbert, Hardalupas and Whitelaw1995). Near the critical momentum ratio, $M\approx 56$, we find that $r_{0.1}^\phi \approx r_{0.1}^U$. For $M>56$, we find that $r_{0.1}^\phi < r_{0.1}^U$ and for $M<56$ that $r_{0.1}^\phi > r_{0.1}^U$, in accordance with the self-similar VFD profiles in figure 5(a–d), $r_{0.1}^\phi >r_{0.1}^{\dot {g}}$ for all $x/d_g$.

Both the VFD and VF are governed by the spreading rate in the far field of the turbulent jet and are defined in a similar manner to the spreading rate of the velocity profile in (3.3)

(5.8)$$\begin{gather} S_{0.1}^{\dot{g}}={\textrm{d} r_{0.1}^{\dot{g}}}/{\textrm{d}\kern0.06em x}, \end{gather}$$

(5.9)$$\begin{gather}S_{0.1}^{\phi}={\textrm{d} r_{0.1}^\phi}/{\textrm{d}\kern0.06em x}, \end{gather}$$

(5.10)$$\begin{gather}S_{0.1}^{U}={\textrm{d} r_{0.1}^U}/{\textrm{d}\kern0.06em x}. \end{gather}$$

We take the spreading rate to be the value of $\eta$ where the seventh-order polynomial interpolation of $\phi /\phi _{max}$ and $\dot {g}/\dot {g}_{max}$ in figure 5(a,b) reaches 10 %. Similar values are obtained from a linear fit of figure 6(a). The opening angle of the VFD, VF and velocity with respect to the 10 per cent width is given by

(5.11)$$\begin{gather} \theta_{0.1}^{\dot{g}}=2\tan^{{-}1}(S_{0.1}^{\dot{g}}), \end{gather}$$

(5.12)$$\begin{gather}\theta_{0.1}^{\phi}=2\tan^{{-}1}(S_{0.1}^{\phi}), \end{gather}$$

(5.13)$$\begin{gather}\theta_{0.1}^{U}=2\tan^{{-}1}(S_{0.1}^{U}).\end{gather}$$

The evolution of the opening angles of each metric with respect to the opening angle of the jet $\theta _{0.1}^U=20.6^\circ$ is plotted in figure 6(b) as a function of momentum ratio. Data for constant liquid flow rate and variable gas flow rate (blue) and constant gas flow rate and variable liquid flow rate (shades of grey) are given. For all momentum ratios, we observe that the opening angle of the VFD profiles is smaller than the VF. The opening angle decreases with increasing momentum ratio and, past $M\approx 10$, follows a logarithmic trend. The critical momentum ratio $M\approx 56$ is observed to indicate the threshold beyond which the VF profiles spread less than the gas phase. Due to the agreement of the observations of critical behaviours in spreading rates of liquid presence and VF in both near and far field, respectively, we define an overall critical momentum ratio of $M_c=50$.

Figure 6.

Evolution of VFD and VF profiles. (a) Normalized 10 per cent width compared against the self-similar jet solution (dashed line) for $M=[25, 56, 177]$. (b) Opening angles calculated of the VF/VFD profiles normalized by the opening angle (5.11) of the gas phase. Points in blue correspond to $Re_L=1050$, in light grey to $Re_L=2900$ and $M=5.1$,and dark grey to $Re_L=4770$ and $M=1.9$. Solid (dashed) line describes the logarithmic trend as $\theta _{0.1}^{\phi }/\theta _{0.1}^U=1.625-0.153\log (M)$ ($\theta _{0.1}^{\dot {g}}/\theta _{0.1}^U=1.625-0.153\log (M)$); (inset) difference in opening angle of VF and VFD with respect to the opening angle of the gas.

The logarithmic dependency of $\theta _{0.1}^{\dot {g}}$ and $\theta _{0.1}^\theta$ will not continue to arbitrarily large $M$ because the opening angles cannot be negative. The opening angles of the VF profiles tend toward that of the VFD (figure 6b, inset) suggesting that an asymptotic regime where $\theta _{0.1}^{\dot {g}}/\theta _{0.1}^{\phi }\rightarrow 1$ and $\theta _{0.1}^{\dot {g}}/\theta _{0.1}^{U}<1$ is likely. This is because radial transport of the droplet phase is sustained by the radial transport of gas momentum. Thus, for arbitrary $M$, the radial expansion of the VFD profile may approach, but not exceed, the radial expansion of the gas-phase profile.

6.1. Liquid ligament ejection

In figure 3(c,d), two sprays are imaged, one where $M< M_c$ and the other with $M>M_c$. In the far field of the former, at $x/d_g\approx 24$, droplets are clearly detected beyond the edge of the gas jet ($r_{0.1}^U/d_g=4.8$) given by the blue lines, with some even observed near $r/d_g=\pm 10$. For $M< M_c$, large droplets can be found on the jet's edge, however, the finite resolution of the images and weak light scattering by small particles may obscure their presence in these images.

To confirm the dominance of large drops near the edge of the spray for $M< M_c$, probability density functions (p.d.f.s) were calculated for $M=25$ at four downstream locations for $r\sim 1.5r_{0.1}$ in figure 7(a). With the exception of $x/d_g=36$, each p.d.f. displays a peak for diameters much larger ($d>66\,\mathrm {\mu }\mathrm {m}$) than the typical peak of the spray droplet-size distribution near the centreline ($d= {\textit{O}}(10\,\mathrm {\mu }\mathrm {m})$, figure 7a, inset). We refer to the droplets constituting the secondary peaks at the spray's edge as ejections. This peak diameter increases in size with downstream distance until $x/d_g=36$, where it shifts back to a smaller droplet diameter. Beyond $M\approx M_c$, no peak corresponding to an ejection is observed.

Figure 7.

Characterization of droplet-size p.d.f.s on the spray's edge: (a) p.d.f.s for $M=25.3$, radial position $r\sim 1.5r_{0.1}$, downstream position $x/d_g=[9,18,24,36]$, ([$\bullet$, $\blacksquare$, $\blacktriangle$, $\blacklozenge$] respectively). The p.d.f.s display a predominant mode that is characteristic of $M=[25.3, 39.2, 56.0]$ except for $x/d_g=36$; (inset) p.d.f.s at $r/d_g=0$ with modes of $ {\textit{O}}(10\,\mathrm {\mu }\mathrm {m})$. Symbols correspond to downstream position. (b) The droplet size corresponding to the mode when $r\sim 1.5r_{0.1}$, $d_{p,peak}$, is assigned the characteristic droplet time scale $\tau _{p,peak}=\rho _\ell d_{p,peak}^2/(18\nu _g\rho _g)$ and compared with the integral scale $T_E$ (6.1).

This phenomenon can be explained by the interaction of ejections with the largest eddies of the turbulent jet. The role of the eddies in selectively transporting droplets was proposed by Chung & Troutt (Reference Chung and Troutt1988) and subsequently experimental (Lazaro & Lasheras Reference Lazaro and Lasheras1992a; Longmire & Eaton Reference Longmire and Eaton1992) and numerical (Sbrizzai et al. Reference Sbrizzai, Verzicco, Pidria and Soldati2004) investigations in different shear-driven flows have largely supported this hypothesis. These eddies are characterized by an Eulerian time scale

(6.1)\begin{equation} {T_E={2r_{0.1}}/{u^\prime},} \end{equation}

where $u^\prime$ is the longitudinal velocity fluctuations on the centreline and $2r_{0.1}$ approximates the diameter of the gas jet at a given downstream position. This length scale is chosen because the literature suggests the presence of large axisymmetric and helical structures (Dimotakis, Miake-Lye & Papantoniou Reference Dimotakis, Miake-Lye and Papantoniou1983; Mungal & Hollingsworth Reference Mungal and Hollingsworth1989) that persist into the far field of the jet and are correlated over its width (Tso & Hussain Reference Tso and Hussain1989; Yoda, Hesselink & Mungal Reference Yoda, Hesselink and Mungal1992). Similar definitions of $T_E$ have been used (Prevost et al. Reference Prevost, Boree, Nuglisch and Charnay1996) to characterize large-scale motions over the entire jet cross-section.

The ejections for $M=[25,~39,~56]$ are used to calculate a time scale based on the characteristic diameter at the peak $d_{p,peak}$ in figure 7(a)

(6.2)\begin{equation} {\tau_{p,peak}=\rho_\ell d_{p,peak}^2/(18\nu_g\rho_g)}, \end{equation}

and are plotted as a function of their local Eulerian time scale $T_E$ in figure 7(b). The ejections collapse on a single line whose slope gives a Stokes number characteristic of the ejections

(6.3)\begin{equation} St_{peak}={\tau_{p,peak}}/{T_E}. \end{equation}

A slope of $St_{peak}=1.9$ is measured and is of order one, strongly suggesting that large eddies are responsible for the presence of liquid ejections on the edge of the spray.

Experimental evidence in PLJs (Hardalupas et al. Reference Hardalupas, Taylor and Whitelaw1989) and sprays (Engelbert et al. Reference Engelbert, Hardalupas and Whitelaw1995) suggests that the initial conditions seen by a particle at injection determine a ballistic trajectory until the local Stokes number with respect to the large energy containing scales becomes of order one. In the case when $M< M_c$, the radial velocity associated with the flapping instability sends droplets on ballistic trajectories as they are ejected from the jet. Then, droplets travelling downstream for which $St< St_{peak}$ would be less probable at the spray edge because they have been re-entrained on the upstream side of the eddy where the entrainment process is strongest (Lampa & Fritsching Reference Lampa and Fritsching2013). Such an entrainment process would culminate in predominantly smaller droplets on the spray's edge, explaining the shift to smaller droplet modes in the range $x/d_g=[24 - 36]$ in figure 7(a).

6.2. Ejection relaxation to the gas phase

To determine if droplets capable of interaction with large eddies exist within the spray, we have calculated the normalized and radially integrated VFD conditioned on droplet size. As opposed to the size-conditioned VFD at a given radial location (5.1), an integral VFD was calculated over successive annuli of the spray centred on the position of the PDI probe volume and weighted by the relative area of each annulus. The probability of finding droplets within the annulus is assumed to be statistically homogeneous. Finally, the conditioned and weighted VFD was normalized by the sum over all sizes. We call this normalized metric the VFD function (v.f.d.f.) and it is implied in the following section that it is a quantity integrated over a cross-section of the spray although it can also be evaluated locally (§ 7). The v.f.d.f. relates the VFD (volume per unit area and unit time) carried by a droplet with diameter between $d$ and $d+\mathrm {d}(d)$. In fact, the v.f.d.f. contains the same information as the number flux density (number per unit area and unit time), commonly referred to as the p.d.f., and the two are analytically related (Appendix A).

Once atomization has occurred, and assuming coalescence and evaporation are negligible in the short residence times in the near- to far-field droplet trajectories, the v.f.d.f. remains essentially unchanged as the spray evolves downstream. Preserved values of arithmetic and volume-averaged diameters were taken to be an indication of high-quality PDI measurements in § 5.1 for this reason. The v.f.d.f.s are presented in figure 8(a) for $M=[25.3 - 176.6]$ at $x/d_g=24$ but other locations collapse onto the same curve. Narrowing of the v.f.d.f. with increasing $M$ is characterized by a decrease in the $d_{43}^i$ (mass-flux-weighted) diameter which is the first moment of the v.f.d.f. (B3). The superscript $i$ indicates that the v.f.d.f. is integrated over the spray cross-section and its first moment ($d_{43}^i$) is a global characteristic of the spray at a given downstream location $x/d_g$. Absence of the superscript indicates that the v.f.d.f. is evaluated at a given downstream and radial location and its first moment ($d_{43}$) is a local characteristic of the spray. The $d_{43}^i$ follow a power law $d_{43}^i\propto M^{-0.79}$ and are plotted in figure 8(b).

Figure 8.

Radially integrated measurements taken at $x/d_g=24$ characteristic of the entire spray cross-section. (a) The v.f.d.f. for different momentum ratios. Solid lines correspond to (6.5) with $n=3.9$. (b) First moment of the v.f.d.f. evaluated over the spray cross-section, $d_{43}^i$, at $x/d_g=24$ for various $M$.

Interestingly, the v.f.d.f. can be described by a single parameter gamma function

(6.4)\begin{equation} \varGamma(n,x=d/d_{43}^i)=\frac{n^n}{\varGamma(n)}x^{n-1}\exp({-}nx), \end{equation}

which gives the following approximation:

(6.5)\begin{equation} \mathrm{v.f.d.f.}(d)=\frac{1}{d_{43}^i}\varGamma(n,d/d_{43}^i), \end{equation}

as observed by solid lines in figure 8(a). The v.f.d.f.s can be collapsed onto a single master curve given by (6.5) using $n=3.9$ and $d_{43}^i$ in figure 8(b). This implies that, for both $M< M_c$ and $M>M_c$, sprays are governed by the same atomization mechanism, which has been attributed to the dynamics of ligaments formed by the co-axial gas jet (Marmottant & Villermaux Reference Marmottant and Villermaux2004; Villermaux, Marmottant & Duplat Reference Villermaux, Marmottant and Duplat2004).

The mass-flux-weighted average diameter $d_{43}^i$ is indicative of the characteristic droplet carrying the overall mass flux in the spray. The characteristic time scale based on this droplet size class is

(6.6)\begin{equation} \tau_{43}={\rho_\ell}({d_{43}^{i}})^2/({{\rho_g}18\nu_g}). \end{equation}

A comparison of $T_E$ and $\tau _{43}$ at $x/d_g=18$ for several momentum ratios is plotted in figure 9(a). If a droplet is resonant with the large eddies, then $\tau _{43}\sim T_E$. If $\tau _{43}> T_E$, there must be some droplets for which $\tau _{43}\sim T_E$ and ejections at the spray's edge are likely. However, if $\tau _{43}< T_E$, it is assumed that there are no droplets in the spray resonant with the large eddies and no, or few, ejections exist.

Figure 9.

Evolution of area-averaged droplet time scale with $M$ and $x/d_g$. (a) The droplet response time $\tau _{43}$ based on the $d_{43}^i$ diameter and the large-eddy time scales $T_{E}$ vary as a function of $M$. Dashed line is $\tau _{43}\approx 5.3\times 10^3 M^{-1.58}$ and the solid line is $T_E\approx 70.4\times M^{-0.50}$. (b) Evolution of the Stokes number for the peak probability droplet, as a function of the momentum ratio. Different symbols represent different downstream locations. Values above $St_{43}=1.9$ represent combinations where ejections are possible, values below $St_{43}=1.0$ are combinations where ejections are unlikely.

Evolution in droplet interactions with the large-scale eddies at a given $x/d_g$ is captured by a global Stokes number

(6.7)\begin{equation} St_{43}=\tau_{{43}}/T_E. \end{equation}

For a given momentum ratio $M$ in figure 9(b), $St_{43}$ decreases with downstream location because $\tau _{43}$ is constant and $T_E\propto (x/d_g)^2$. The range $St_{43}>St_{peak}$ ($St_{peak}=1.9$) corresponds to strongly inertial droplets and high ejection probability on the spray edge. The area bounded by dashed lines, $1< St_{43}< St_{peak}$, corresponds to a transitional regime where ejections become less probable on the spray edges. Below the solid line, $St_{43}<1$, droplets are weakly inertial with respect to the large eddies and ejection presence on the spray edges is unlikely. These observations are consistent with the momentum ratio and location where ejections were observed in figure 7.

7.1. The case $M< M_c$

For the momentum ratios below $M_c$, the $d_{43}$ profiles retain a ‘$\cup$’ shape where the smallest droplets are found near the centreline and the largest near the edge (figure 10a–c). Within this range of $M$, slight differences are observed between $5.1< M< M_c$ (figure 10b,c) and $M<5.1$ (figure 10a).

Figure 10.

Evolution of $d_{43}/\tilde {d}$ profiles where $\tilde {d}=(T_E18\nu _g\rho _g/\rho _\ell )^{1/2}$ corresponds to a droplet with a characteristic time scale equal to the local Eulerian time scale $T_E$. Radial position is normalized by the $10\,\%$ width of the gas jet as an estimate of its radius. Solid lines correspond to $(St)^{1/2}=(1.9)^{1/2}=1.4$, the Stokes number corresponding to ejections; (a) $M=1.9$, (b) $M=5.1$, (c) $M=25.3$, (d) $M=56$, (e) $M=81.2$, (f) $M=176.6$, (f,inset) $M=176.6$, reduced ordinate range emphasizing central maximum.

When $M<5.1$ there is a non-monotonic increase in $d_{43}$ from the centreline to the edge. There appears to be a local peak in $d_{43}$ near the centreline which falls off until $r\sim r_{0.1}$ and the droplet diameter increases again beyond the edge of the jet, where a maximum occurs. Similar profiles were observed by Engelbert et al. (Reference Engelbert, Hardalupas and Whitelaw1995) and were attributed to a delayed atomization characteristic of low momentum ratios. When break-up of the liquid core eventually occurs, it takes place further downstream in an environment with weaker shear. Large detached liquid ligaments form, which are subsequently atomized inefficiently, leading to large droplets on the centreline. Smaller droplets towards the edge are a result of their faster response to the jet turbulence. However, droplets even larger than those on centreline are found near the edge. These are likely the result of a flapping mechanism similar to the one established for larger $M$ sprays.

For $5.1< M< M_c$, there is a monotonic increase in $d_{43}$ towards the edge of the spray (figure 10c). As suggested by Hardalupas et al. (Reference Hardalupas, Taylor and Whitelaw1989) and Engelbert et al. (Reference Engelbert, Hardalupas and Whitelaw1995), initial injection conditions lead to ballistic trajectories which persist until the droplets reach resonance with large eddies, typically for $d_{43}/\tilde {d}<(St_{peak})^{1/2}$. From figure 10(b,c) flattening in the profiles occurs when the droplets on the edge of the spray resides in the range $1< d_{43}/\tilde {d}<1.4$ similar to the range of droplet ejections that were hypothesized to interact with large-scale eddies in figure 9(b).

7.2. The case $M\approx M_c$

Close to $M=M_c$, a confluence of factors affects the shape of the droplet diameter profiles. The liquid presence probability profiles (figure 4d) approach values determined by the gas-phase velocity profile, ($\theta _{2\sigma }\approx \theta _{0.1}^U$) which is thought to be due to suppressed flapping as $M$ approaches $M_c$. As a consequence, there are fewer large droplets on the spray edge and the VF profile in the far field (figure 6b) tends toward the gas-phase velocity profile ($\theta _{0.1}^\phi \approx \theta _{0.1}^U$). In figure 10(d), the diameter profile for $M=56$ is plotted and only at $x/d_g=12$ do the diameter profiles both present the signature of ejections with a prominent central minimum. For $x/d_g>12$, the integrated Stokes number is subcritical ($St_{43}< St_{peak}$), and nearly all of the radial values fall below the ejection threshold $d_{43}/\tilde {d}=(1.9)^{1/2}$. As the droplets continue downstream the profiles begin to flatten which is indicative of droplets relaxing to the gas phase ($St_{43}<1$).

7.3. The case $M\gg M_c$

The highest momentum ratio regime is characterized by a $d_{43}$ profile with a central maximum and minima on the edges (figure 10f). As opposed to the $M< M_c$ regime, the flapping phenomenon does not significantly contribute to the dynamics of atomization nor does it impart initial radial momentum onto the droplets. In this regime, enhanced radial transport would be possible if droplets were resonant with the large eddies. However, the time scale of the droplets produced by atomization decreases with increasing $M$ ($\tau _{43}\propto M^{-1.58}$) at a faster rate than eddy time scales decrease ($T_{E}\propto M^{-0.5}$) with increasing $M$ (increasing $Re_g$). This results in Stokes numbers for the droplet cloud well below unity. Due to the recirculating gas cavity present at $M\gg M_c$, larger droplets are confined to the centreline. The smallest droplets can follow the radial expansion of the gas phase closely. Thus, they are found more frequently near the edge of the jet, explaining the marked central maximum of $d_{43}$ and the minima on the edges.

7.4. Spray regime diagram ($We - Re_\ell$)

The data have been presented as a function of $M=\rho _gu_g^2/(\rho _\ell u_\ell ^2)$ either by experimentally varying $u_g$ or $u_\ell$, independently. However, in § 7.3 it was seen that a convex (‘$\cap$’) profile with a central maximum could be observed both for low ($M=1.9$, figure 10a) and high momentum ratios ($M=[81.2, 176.6]$, figure 10e,f). Instead, the $We - Re_\ell$ phase space is introduced to account for gas- and liquid-phase momentum separately in the wider context of experimentally observed sprays.

PDI data from the literature presenting radial profiles of droplet size were surveyed (table 4). Results for the Sauter mean diameter ($d_{32}$) we found, but not for the mass-flux-weighted diameter ($d_{43}$). From the present data, the evolution of $d_{43}$ is found to have less extreme differences between minima and maxima but, in general, the evolution described in §§ 7.1–7.3 for $d_{43}$ is similar for $d_{32}$. Since the profiles evolve downstream, the data displayed in the phase space are restricted to $x/d_g<13$.

Table 4.

Experimental parameters from the literature, used in figure 11. Momentum ratio: $M=\rho _g U_g^2/(\rho _\ell U_\ell ^2)$. Weber number: $We = \rho _g(U_g-U_\ell )^2d_\ell /\sigma$. Mass ratio: $m=\rho _\ell A_\ell U_\ell /(\rho _g A_g U)$. Liquid-phase Reynolds number: $Re_\ell =U_\ell d_\ell /\nu _\ell$. Gas-phase Reynolds number: $Re_g=U_g d_g/\nu _g$. Distance from nozzle where profile was measured: $x/d_g$. Diameter classes correspond to the Sauter mean diameter $d_{32}$ and mass-flux-averaged diameter $d_{43}$.

The phase-space diagram (figure 11) is divided into five regimes, colour coded based on the observed droplet diameter radial profile. Note that the solid symbols correspond to the PDI data in this paper. Regime I (red) corresponds to ‘∪’-shaped profiles. Regime II (yellow) corresponds to flat profiles. Regimes III (green), IV (blue) and V (grey) correspond to ‘$\cap$’-shaped profiles. The white area is typically inaccessible to PDI measurements due to the inability of the spray to create spherical droplets in this regime.

Figure 11.

Phase-space diagram ($We - Re_\ell$) indicating the shape of the radial droplet profiles for the present experiments and those found in the literature, symbols are the same as in table 4, solid symbols are the present experiments. All of the points shown correspond to measurements within $x/d_{g}\leq 13$ and correspond to $d_{32}$ measurements except the present experiments ($d_{43}$). Colours are an indication of regime. Regime I (red) corresponds to ‘$\cup$’-shaped profiles. Regime II (yellow) corresponds to flat profiles. Regimes III (green), IV (blue) and V (grey) correspond to ‘$\cap$’-shaped profiles. The white area is typically inaccessible to PDI measurements due to the inability of the spray to create spherical droplets in this regime.

For $Re_\ell \lesssim 4000$ and $50< We<300$, the profiles fall in regime I where flapping is dominant and gives rise to other morphologies such as ‘ladle’ (Eroglu & Chigier Reference Eroglu and Chigier1991) structures (figure 3a,c) which launch drops beyond the gas jet creating ‘$\cup$’-shaped profiles. Increasing the momentum in the gas phase for $Re_\ell \lesssim 4000$ and $We\gtrsim 300$ reduces the role of the flapping instability, marking the transition from regime I to II. This transition corresponds to $M=M_c$ for our data set and a flattening of the mass-flux-weighted diameter profile (figure 10d). If the momentum in the gas phase is increased beyond $We\sim 800$ (and $Re_\ell \lesssim 4000$), a transition between flat and ‘$\cap$’-shaped profiles occurs (II to III). This regime is characterized by the emergence of a recirculating gas cavity just downstream of the liquid core.

The present data ($M<5.1$, figure 10a,b), help confirm the transition from regime II to IV that occurs for $200< We<800$ and $Re_\ell \approx 4000$. In regime IV there are large droplets near the spray edges and also on the centreline, giving ‘W’-shape profiles close to the transition at $Re_\ell \approx 4000$ (figure 10a). Further into regime IV, the gas phase lacks sufficient momentum to initiate the flapping instability and the instability transitions to the ‘dilational waves’ observed by Engelbert et al. (Reference Engelbert, Hardalupas and Whitelaw1995). These waves are sufficiently long lived for successive vortices to establish recirculation regions in the wake of the KH wave. In Zandian et al. (Reference Zandian, Sirignano and Hussain2018), vortex stretching leads to the formation of hairpin vortices in the wake region that deform these recirculating vortices which in turn deform the liquid core. This cascade gives rise to the droplets constituting ‘$\cap$’-shaped profiles.

Interestingly, these profiles in regime IV ($Re_\ell \gtrsim 8000$) resemble profiles in regime III which are also ‘$\cap$’-shaped and also do not exhibit flapping due to the recirculating gas cavity. While at lower $We$ surface tension resists the formation of droplets, in the high $We$ regime the hairpin vortices described above are able to perforate the ligaments forming smaller droplets as described by Zandian et al. (Reference Zandian, Sirignano and Hussain2018). This mechanism helps us to understand the difference in droplet sizes between regime III and IV despite similarities in the profile shape.

Regime V is not explored in this paper and deserves comment. The boundary between regimes III and V is difficult to delimit as both display ‘$\cap$’-shaped profile. Increasing the liquid momentum, and with it the liquid–gas mass ratio, in the spray when the gas momentum is high ($We>800$) results in increased transfer of momentum from gas to the liquid phase as more liquid is required to accelerate to the gas-phase velocity. This may change the nature of atomization that occurs after liquid separates from the intact core, known as secondary atomization (SA). Lasheras, Villermaux & Hopfinger (Reference Lasheras, Villermaux and Hopfinger1998) and Lasheras & Hopfinger (Reference Lasheras and Hopfinger2000) identify two mechanisms of SA-driven break up: mean shear and turbulence induced. The former occurs when a strong enough relative velocity occurs between the droplet and the gas while the latter is due to destabilizing turbulent fluctuations occurring over the droplet's surface. As more gas momentum is transferred to the accelerate the liquid phase, velocity fluctuations are dampened (Modarress, Wuerer & Elghobashi Reference Modarress, Wuerer and Elghobashi1984; Engelbert et al. Reference Engelbert, Hardalupas and Whitelaw1995) resulting in increasingly weak turbulent SA. This is thought to be the mechanism governing the spray transitions from regime III to V. However, the precise role of turbulent and shear SA in regimes III and V deserves further research.