3.2.1 Droplet number density

Figure 5. (a) Normalized mean droplet number density; (b) intensity of fluctuations of droplet number density for the three droplet size classes at different measurement locations for the acetone and the water sprays.

Figure 5(a) shows the average droplet number density, $\overline{N}$ , of the considered drop size classes at different measurement locations for both the acetone and water sprays. $\overline{N}$ is normalized by the corresponding value for evaporative spray at $R=0$  mm. The uncertainty of $\overline{N}$ was approximately $\pm 4\,\%$ at a 95 % confidence interval. The average number density decreases towards the spray edge. The radial gradient of $\overline{N}$ across the spray is always higher for the water spray, while evaporation reduces the gradient considerably for the acetone spray. Also, in the case of the acetone spray, due to droplet evaporation, significant reduction in the number of 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplets (by approximately 2–3 times) in the central spray region ( $R=0$  mm) can be observed in comparison to the water spray. So, the vapour mass fraction is expected to be higher in this region, though the number density of droplets of other size classes reduces proportionately such that their probability remains the same for both sprays. The intensity of fluctuations of droplet number density (ratio of r.m.s. of droplet number density fluctuations to the mean value, $n_{r}/\overline{N}$ ) is shown in figure 5(b). The statistical uncertainty of $n_{r}$ was approximately $\pm 9\,\%$ with 95 % confidence interval. It should be noted that for a non-evaporating spray (the water spray in the present case), the fluctuations of droplet number density are due to droplet dispersion, which can lead to preferential accumulation of droplets and formation of droplet clusters as a consequence of interaction with the carrier phase (e.g. Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2012). However, for an evaporating spray (the acetone spray) the number density fluctuations are additionally governed by droplet evaporation (i.e. the rates at which droplets of a particular intermediate size class evaporate to become smaller droplets and larger droplets evaporate to become droplets of that intermediate size class resulting in, respectively, decrease and increase in droplet number density of that intermediate size class). It can be observed in figure 5(b) that the fluctuations (relative to mean) are higher for the larger droplets, and increase towards the edge of the spray, in contrast to trends in variations of the mean values (figure 5 a). Also, importantly, evaporation causes increased fluctuations of droplet number density at all measurement locations within the spray. Near the spray edge these fluctuations can even be higher than their corresponding mean values.

3.2.2 Droplet velocity

Figure 6 shows the vector plots of the average droplet velocity of 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplets for the acetone spray, which indicate that the droplet velocity is mostly axial and with a downward direction (away from the nozzle), and its magnitude decreases away from the spray centre. The mean droplet velocity is observed to be almost spatially invariant within each measurement region. The same is true for droplet velocity fluctuations. Similar trends were observed for water droplets, so are not presented here. For the reason mentioned above, the spatially averaged velocity of droplets (of any size class) across each measurement area (i.e. $8\times 12~\text{mm}^{2}$ ) is considered to be representative of individual droplet velocity of that size class. Figure 7(a,b) compares respectively the spatially averaged mean axial velocity ( $\overline{U}$ ) and intensity of axial velocity fluctuations ( $u_{r}/\overline{U}$ ) for droplets of the considered size class for acetone and water sprays. The uncertainties (with 95 % confidence interval) of $\overline{U}$ and $u_{r}$ were approximately $\pm 1\,\%$ and $\pm 15\,\%$ respectively. Although no significant difference is present in the mean velocity between the droplet size classes, the droplet velocity is consistently higher for larger droplet size classes. It is observed that evaporation caused the mean and fluctuations of droplet velocity to become more uniform across the spray radial profile. This requires some explanation, which is provided below.

Figure 6. Vector plots of mean droplet velocity for acetone droplets of size class of 15–30  $\unicode[STIX]{x03BC}\text{m}$ at different measurement locations.

Figure 7. (a) Mean droplet velocity and (b) intensity of fluctuations of droplet velocity at different measurement locations for acetone and water droplets for the three droplet size classes.

In comparison to the water droplets, the mean velocity of acetone droplets appears to be smaller near the spray axis and higher towards the spray edge (by approximately 10 %). However, it can be shown to a first approximation that evaporation is expected to always cause reduction of droplet velocity. From (1.4), the rate of change of size of a droplet can be expressed as

(3.2) $$\begin{eqnarray}\frac{\text{d}D}{\text{d}t}=-2\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{l}}\frac{D_{v}\ln (1+B)Sh}{D}.\end{eqnarray}$$

Since the droplet Reynolds number (based on its mean velocity) is low ( ${\approx}0.2$ ), assuming Stokes’ drag law for a single droplet, and considering the effect of evaporation on droplet drag (Spalding Reference Spalding1951),

(3.3) $$\begin{eqnarray}\frac{\text{d}U}{\text{d}t}=\frac{18\unicode[STIX]{x1D707}(U_{g}-U)}{\unicode[STIX]{x1D70C}_{l}D^{2}}\frac{\ln (1+B)}{B}.\end{eqnarray}$$

From the above equations it can be shown that

(3.4) $$\begin{eqnarray}\frac{\text{d}U}{\text{d}D}=\frac{9\unicode[STIX]{x1D707}}{2\unicode[STIX]{x1D70C}D_{v}Sh}\frac{(U-U_{g})}{D}\frac{1}{B}.\end{eqnarray}$$

Assuming that for a given measurement location the transfer number $B$ and the gas velocity $U_{g}$ are constants within the measurement volume, and since $Sh\rightarrow 2$ as $Re\rightarrow 0$ , integration of the above equation implies that droplet velocity reduces proportionately with droplet size. We consider the droplet size range of 15–30  $\unicode[STIX]{x03BC}\text{m}$ as an example to explain the trends of figure 7(a). For the location at $R=0$  mm, the velocity of 15–30  $\unicode[STIX]{x03BC}\text{m}$ acetone droplets, which before evaporation had a larger size (say 30–45  $\unicode[STIX]{x03BC}\text{m}$ ; this droplet size range is selected only to demonstrate our argument) with a larger velocity, is now smaller compared to water droplets of the same size. Equation (3.4) also shows that the rate $\text{d}U/\text{d}D$ is smaller for larger $B$ . Since $B$ increases towards the spray edge (as the vapour mass fraction away from a droplet is smaller), the evaporation rate ( $\text{d}D/\text{d}t$ ) is higher but $\text{d}U/\text{d}D$ is smaller. Thus, for the location at $R=45$  mm, the velocity of 15–30  $\unicode[STIX]{x03BC}\text{m}$ acetone droplets, which used to be of even larger size (say 45–60  $\unicode[STIX]{x03BC}\text{m}$ ; again this droplet size range is selected only to demonstrate our argument) with an even larger velocity, is now larger compared to water droplets of the same size. The above discussion does not consider the response of the different droplet sizes to the induced mean airflow in the spray, which will be discussed below.

For both sprays, the axial velocity fluctuations were smaller than the respective mean values at all measurement locations although they increased away from the spray centre, and were smaller for larger droplet size classes (figure 7 b). Also, fluctuations of droplet velocity were higher for the acetone droplets compared to water droplets for both axial directions (approximately 15 %) and radial direction (approximately 25 %). The anisotropy of the droplet velocity fluctuations, defined as the ratio of $u_{r}/v_{r}$ , was approximately 1.5–2 for all cases, and was slightly reduced away from the spray axis. These characteristics are according to expectations for a jet flow, which is approximately the flow behaviour induced by the spray.

Figure 8. Probability of instantaneous droplet–gas slip velocity in the axial direction for 30–45  $\unicode[STIX]{x03BC}\text{m}$ and 45–60  $\unicode[STIX]{x03BC}\text{m}$ droplet size classes for the acetone spray at the measurement location $R=0$  mm.

The droplet–gas slip velocity is an important parameter, which not only influences droplet evaporation and drag (equations (3.2) and (3.3)) but also the correspondence between droplet properties and neighbouring vapour mass fraction. Therefore, the droplet slip velocity was determined by considering the instantaneous relative velocity of droplets with respect to gas velocity (approximated by the velocity of 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplets, which is justified by the Stokes number evaluation presented below) for the evaporative spray. The corresponding probability distributions for the 30–45  $\unicode[STIX]{x03BC}\text{m}$ and 45–60  $\unicode[STIX]{x03BC}\text{m}$ droplet size classes are shown in figure 8 for the axial velocity component of the droplets. The slip velocity for a given droplet size class is normalized by the respective mean velocity of the droplet size class. It is observed that the mean droplet–gas slip velocity is close to zero, thus the droplet Reynolds number for droplets of all sizes is very small ( $Re<1$ ). However, the slip velocity of the droplets has both positive and negative sign, indicating that droplets may lead or lag the gas motion. This refers to fluctuations of slip velocity indicating the role of gas-phase turbulence. Therefore, the vapour evacuation due to convective effects (mean slip) is not expected to be significant in comparison to the turbulence effects. This highlights the differences in the droplet evaporation experiments based on isolated pendant droplets (typically in wind tunnels), where the convective effects are always dominant, and sprays. Nevertheless, as will be explained later, the instantaneous slip velocity is responsible for spatial disparity between the locations of droplets and vapour, which results in reduced correlation between the two phases.

Table 1. Estimated turbulent characteristics of the air flow in the acetone spray and corresponding droplet Stokes numbers for the 15–30  $\unicode[STIX]{x03BC}\text{m}$ , 30–45  $\unicode[STIX]{x03BC}\text{m}$ and 45–60  $\unicode[STIX]{x03BC}\text{m}$ droplet size classes at the measurement locations $R=0$ and 45 mm.

Table 1 shows the estimated turbulent characteristics of the air flow for the acetone spray at measurement locations $R=0$  mm and 45 mm. Since the air velocity around the droplets was not measured in our experiments, the gas velocity is assumed to be the same as the velocity of small droplets of size 15–30 $\unicode[STIX]{x03BC}\text{m}$ . These droplets are expected to faithfully follow the gas flow since the measurement location is situated far downstream of the injector exit, and so the ‘ballistic’ motion of atomized droplets that strongly determines droplet motion near the atomizer (Hardalupas et al. Reference Hardalupas, Taylor and Whitelaw1990) is attenuated. This is supported by the Stokes number analysis presented below. The integral length scale ( $L$ ) of the turbulent air flow is determined by measuring spatial velocity correlation coefficients between fluctuations of velocity of 15–35  $\unicode[STIX]{x03BC}\text{m}$ droplets following the method described in Sahu et al. (Reference Sahu, Hardalupas and Taylor2014a ). The value of $L$ at $R=0$  mm is approximately 13 mm, which is in agreement with the value of $L$ estimated as $1/5$ of the spray radius ( ${\approx}10$  mm), as suggested by Tennekes & Lumley (Reference Tennekes and Lumley1972). The characteristic time scale of the entrained air flow by the spray ( $\unicode[STIX]{x1D70F}_{g}$ ) is chosen as the ratio of the integral length scale to the axial r.m.s. velocity of the 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplets. The Stokes number ( $St$ ) of a droplet size class is defined as the ratio of the droplet aerodynamic time constant or the ‘droplet relaxation time’ ( $\unicode[STIX]{x1D70F}_{d}$ ) to an appropriate turbulence time scale of the flow. Based on the characteristic time of the entrained air flow by the spray, the Stokes number of the 15–30  $\unicode[STIX]{x03BC}\text{m}$ , 30–45  $\unicode[STIX]{x03BC}\text{m}$ and 45–60  $\unicode[STIX]{x03BC}\text{m}$ droplet size classes at the spray centre are estimated as $St_{L}=0.05$ , 0.16 and 0.30, respectively. This means that while droplets of 15–30  $\unicode[STIX]{x03BC}\text{m}$ follow the gas motion, larger droplets show partial response to the large-scale eddies of the gas phase. The estimated values in table 1 were nearly the same for the water spray, hence are not presented here.

Since the estimated $St_{L}$ for the smallest droplet size class is very low ( $=$ 0.05), this justifies our assumption of those droplets following the air flow faithfully. Also, the fact that the droplet–gas slip velocity decreases significantly for all droplet sizes far downstream of the injector exit (see Gounder, Kourmatzis & Masri Reference Gounder, Kourmatzis and Masri2012; Sahu et al. Reference Sahu, Hardalupas and Taylor2016) further supports the above assumption. However, in our earlier work (Sahu et al. Reference Sahu, Hardalupas and Taylor2014a ) we have shown that the integral length scale of the droplet flow is smaller than that of the gas flow, and also, the gas velocity fluctuations are smaller than the droplet velocity fluctuations, though the differences are smaller for droplets of low Stokes number. Thus, the energy-containing eddy time scale is expected to be slightly overestimated, and the Stokes numbers are expected to be smaller than the estimated values presented in table 1. Nevertheless, we emphasize that the turbulent characteristics and Stokes numbers are good estimates in an order of magnitude sense.

3.2.3 Vapour mass fraction

The contour plots of average vapour mass fraction ( $\overline{Y}$ ) are presented in figure 9. As compared to the saturated condition ( $Y_{sat}=0.32$ at $15\,^{\circ }\text{C}$ ), the mean vapour mass fraction was low ( ${\approx}5$ –15 % of the saturation value) throughout the measurement region. This corresponds to equivalence ratio of approximately 0.07–0.3 implying a lean mixture. Also, as expected, $\overline{Y}$ reduces considerably for measurement locations towards the edge of the spray in accordance with the reduction of acetone droplet number density (compared to the water droplets), as presented in figure 5(a). This confirms that the assumption of neglecting laser energy absorption by the vapour in the vapour mass fraction evaluation from the PLIF measurements is valid. In general, $\overline{Y}$ is expected to increase in the direction of mean flow downstream as the droplets continue to release vapour during their motion away from the nozzle exit. However, this effect is expected not to be significant in the present case, since the droplet evaporation time, $\unicode[STIX]{x1D70F}_{v}\approx 50$ –40 ms (obtained from (3.2) and based on droplet SMD, as explained later), is larger than the droplet residence time within the measurement window ( ${\approx}5$ –10 ms), where the limits of the time interval correspond to the spray centre and edge, respectively.

Figure 9. Contour plots of mean mass fraction of acetone vapour at different measurement locations: $R$   $=$  0 mm (a), 15 mm (b), 30 mm (c) and 45 mm (d) for the acetone spray.

Figure 10. Radial profiles of (a) mean vapour mass fraction and (b) intensity of fluctuations of vapour mass fraction in the acetone spray.

For the rest of the paper, the spatial average of the vapour mass fraction within the measurement area of $8\times 12~\text{mm}^{2}$ across an ‘instantaneous’ realization is considered to be representative of the vapour mass fraction at that instant. This is done to allow estimation of the correlation between instantaneous vapour mass fraction, and droplet number density and velocity, which will be presented later. Also, this is justified since no notable spatial gradient of vapour concentration is observed in figure 9 for either axial or radial directions at each of the measurement locations. The average values of the instantaneous vapour mass fraction over all realizations is obtained and presented in figure 10 along with the corresponding normalized fluctuations of vapour mass fraction ( $y_{r}/\overline{Y}$ ). This is consistent with figure 9 and, as expected, the mean vapour mass fraction reduces away from the spray axis. However, the fluctuations (relative to the mean) increase away from the spray axis, as a consequence of the local fluctuations of droplet number density and velocity for which similar trends were observed (figure 5 b). In fact, near the spray boundary, the fluctuations of vapour mass fraction are of the same order as the mean value.

Figure 11. Probability distribution of fluctuations of vapour mass fraction at $R=0$  mm (a) and 45 mm (b).

Figure 11 presents the probability distributions of the instantaneous vapour mass fraction within the acetone spray at the measurement locations $R=0$  mm and 45 mm respectively. These probability distributions signify temporal variations of vapour mass fraction around the droplets, hence the transfer number $B$ is expected also to be a time-varying parameter and therefore may not be assumed constant. While at the spray centre, the probability distribution function of the vapour mass fraction is approximately Gaussian, near the spray edge the probability function is positively skewed with a peak appearing at the minimum vapour mass fraction values (figure 11 b). This can be explained as follows. For measurement locations close to the spray edge ( $R=30$ and 45 mm), an instantaneous PLIF image may not contain any droplets. The vapour mass fraction in such cases was found to be very low and can be either due to vapour transported from elsewhere or background noise (even though a background noise image is subtracted from the measured LIF intensity, some shot-to-shot variation of laser energy is possible). Thus, the peak probability, corresponding to nearly zero vapour mass fraction, is attributed to the flow intermittency near the spray edge.

3.3.1 Correlation between droplet number density and vapour mass fraction

At first sight, it may appear that the momentum and inertia of the droplets can easily separate them from the vapour that they release due to evaporation, so that the measured droplets at any location in the flow are not responsible for the local vapour. However, it is important to note that the correlation between droplet and surrounding vapour depends on the droplet evaporation rate as well as the droplet Stokes number $St$ (in other words how well the droplets follow the surrounding gas flow) and density difference between the vapour and ambient gas. Although the density of acetone vapour is nearly twice that of the air, due to the very low vapour mass fraction (figure 9) the mixture density can be assumed to be the same as that of the air. For our experimental conditions, as mentioned before, $St_{L}\approx 0.1$ , which means medium and large droplet sizes partially follow the large scales of the air flow. This means the generated vapour is transported with the droplets. Hence, a certain level of correspondence between local droplets and surrounding vapour mass fraction is expected. This is confirmed in figure 12, which presents the scatter plot of the instantaneous droplet number density ( $N$ ), measured by ILIDS, and the corresponding local vapour mass fraction, ( $Y$ ) measured by PLIF, for measurement locations at $R=0$  mm and 45 mm. The plots signify proportionate increase of acetone vapour (within the measurement region) with droplet number density in an average sense (as shown by the linear fit of the data). This linear dependence of the two quantities is expected if the droplets are far apart from each other, so that the total generated vapour is simply a multiple of the vapour released by single droplets: these characteristics indicate the relevance of the $d^{2}$ -law for the modelling of droplet evaporation. However, considerable scatter is observed around the linear dependence of figure 12 for both number density $N$ and vapour mass fraction $Y$ axes. The spreading along the $N$ -axis (horizontal) indicates the presence of the same amount of vapour for different droplet numbers in the measurement region, while the spreading in the $Y$ -axis (vertical) indicates variation of vapour mass fraction for the same number of droplets present in the measurement region. This can be attributed to dispersion of droplets due to interaction with carrier-phase turbulent flow, which leads to droplet clustering resulting in smaller inter-droplet spacing than the average, which consequently reduces the droplet evaporation rate which is then determined by the group evaporation of droplet clouds. On the other hand, scalar mixing of droplet vapour with the surrounding air results in local dissipation of vapour fluctuations. Hence, the scattering of the data can be attributed to competition between droplet dispersion and scalar mixing. It can also be observed in figure 12 that the extent of spreading for both $N$ and $Y$ directions is reduced for the case of location $R=45$  mm in comparison to the central spray location, and the slope of the linear fit is larger. This indicates that the group evaporation tendency of droplets reduces towards the spray boundary. It should be noted that the dependence between $N$ and $Y$ , as presented above, is important for the evaluation of the Eulerian joint probability density function of mixture fraction and vapour source term, which can fully describe the unclosed source terms appearing in the transport equations of mixture fraction in (1.1) and (1.2) (Reveillon & Vervisch Reference Reveillon and Vervisch2000). Experimental evaluation of these terms is not available in the literature.

Figure 12. Scatter plot of instantaneous droplet number density and vapour mass fraction at $R=0$  mm (a) and 45 mm (b).

The above discussion explained the need to evaluate the correlation between fluctuations of droplet number densities and vapour mass fraction $\overline{n\times y}$ from our measurements, which uniquely provides this ability. Since the definition of the mixture fraction in sprays (containing vapour sources) is not straightforward (Reveillon & Vervisch Reference Reveillon and Vervisch2000), we present $\overline{n\times y}$ . The corresponding correlation coefficient, denoted as $R_{n\ast y}$ , can be expressed for droplet size class $D$ as

(3.5) $$\begin{eqnarray}R_{n\ast y}(D)=\frac{\overline{n(D)\times y}}{n_{r}(D)\times y_{r}},\end{eqnarray}$$

where $n_{r}$ and $y_{r}$ are the r.m.s. of the number density fluctuations for size class $D$ and mass fraction fluctuations respectively. $R_{n\ast y}$ , evaluated for each droplet size class and for different measurement locations in the acetone spray, is presented in figure 13. The uncertainty in $R_{n\ast y}$ was approximately $\pm 0.05$ with 95 % confidence interval.

Figure 13. Correlation coefficient ( $R_{n\ast y}$ ) between fluctuations of number density (for different droplet size classes) and vapour mass fraction at different measurement locations in the acetone spray. The uncertainty in $R_{n\ast y}$ was approximately $\pm 0.05$ with 95 % confidence interval.

It can be observed in figure 13 that the correlation is always positive, which means that positive/negative fluctuations of droplet number density results in higher/lower vapour mass fraction than the mean value at any instant in time. Also, the correlation is stronger away from the spray axis. This is in agreement with the scatter plots of figure 12, and the discussion thereafter. At any measurement location, $R_{n\ast y}$ varies inversely with droplet size. Even if the droplet evaporation rate is higher for larger droplets (see (1.4)), since the smaller droplet size class dominates the droplet size distribution, the respective contribution to vapour mass fraction from smaller droplet sizes is higher. For any droplet size class, it can be argued that the relative magnitude of $R_{n\ast y}$ at different radial measurement locations suggests the mode of droplet evaporation: while larger values indicate individual droplet evaporation, smaller values of the correlation signify group evaporation.

Figure 14. Correlation coefficient of the fluctuations of droplet number density and vapour mass fraction, $R_{n\ast y}$ , for different window sizes and measurement locations for droplet sizes of (a) 15–30  $\unicode[STIX]{x03BC}\text{m}$ , (b) 30–45  $\unicode[STIX]{x03BC}\text{m}$ and (c) 45–60  $\unicode[STIX]{x03BC}\text{m}$ .

Due to their partial response to the gas motion, the droplets are associated with the slip velocity (see figure 8) such that the released vapour from droplets may lead or lag the droplet motion. In such a case, it is important to recall the effect of droplet–gas slip on the correlation $R_{n\ast y}$ , which, however, cannot be discerned from figure 13 since the current dimensions of the measurement area ( $8\times 12~\text{mm}^{2}$ ) are of the order of the large-scale eddies of the carrier-phase flow. Hence, $R_{n\ast y}$ was also calculated for smaller window sizes, one-half and one-quarter of the measurement window dimensions (i.e. $4\times 6~\text{mm}^{2}$ and $2\times 3~\text{mm}^{2}$ , respectively) for droplets of the three size classes. The statistical uncertainty of $R_{n\ast y}$ increases for smaller window size, and was found to be $\pm 0.06$ and $\pm 0.15$ for the $4\times 6~\text{mm}^{2}$ and $2\times 3~\text{mm}^{2}$ windows respectively. Figure 14 shows the comparison of $R_{n\ast y}$ for different measurement locations for the three window sizes. It can be observed that for any window size, the correlation is smaller for larger droplet size at all measurement locations, as observed before. Importantly, the magnitude of $R_{n\ast y}$ decreases for smaller window sizes (by approximately 30 % and 50 % for windows of $4\times 6~\text{mm}^{2}$ and $2\times 3~\text{mm}^{2}$ , respectively), implying the role of droplet–gas slip, which results in spatial disparity between the location of droplets and vapour. Due to the larger statistical uncertainty for the smallest window size, the radial increase of $R_{n\ast y}$ cannot be observed, unlike for the other window sizes.

3.3.2 Correlation between droplet velocity and vapour mass fraction

As mentioned earlier, since in our experiments small droplets (15–30  $\unicode[STIX]{x03BC}\text{m}$ ) can be assumed to follow fairly well the gas velocity fluctuations, the correlation between the 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplet velocity and vapour mass fraction can elucidate the effects of gas turbulence on the droplet evaporation rate. Figure 15 presents the scatter plot between instantaneous fluctuations of droplet velocity and vapour mass fraction for both axial and radial velocity components of 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplets for a measurement location at $R=0$  mm. It is interesting to observe that instantaneous fluctuations of droplet velocity are positively correlated to fluctuations of vapour mass fraction. This means increase/decrease of gas velocity fluctuations results in similar variation of the droplet evaporation rate. The correlation is much stronger for axial droplet velocity (figure 15 a) in comparison to the radial component (figure 15 b), which is expected since there is strong anisotropy at the spray centre.

Figure 15. Scatter plot of fluctuations of velocity of 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplets (approximating the gas flow velocity fluctuations) and vapour mass fraction at $R=0$  mm for (a) axial velocity and (b) radial velocity.

Next we calculate the normalized correlations for the considered droplet size classes and different measurement locations. The correlation coefficients, denoted as $R_{u\ast y}$ and $R_{v\ast y}$ , can be expressed for droplet size class $D$ as

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle R_{u\ast y}(D)=\frac{\overline{u(D)\times y}}{u_{r}(D)\times y_{r}} & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle R_{v\ast y}(D)=\frac{\overline{v(D)\times y}}{v_{r}(D)\times y_{r}}, & \displaystyle\end{eqnarray}$$

where $u_{r}$ and $v_{r}$ are the r.m.s. of axial and radial velocity fluctuations for size class $D$ . Figure 16 presents the results. The statistical uncertainty for the correlation coefficients was approximately $\pm 0.05$ with 95 % confidence interval. Unlike the correlation between droplet number density and vapour mass fraction ( $R_{n\ast y}$ ) as presented earlier (figure 13), both $R_{u\ast y}$ and $R_{v\ast y}$ are nearly independent of droplet size, since their $St_{L}$ are not significantly different, although $R_{v\ast y}$ reduces slightly with droplet size. $R_{u\ast y}$ is nearly invariant ( ${\approx}0.6$ ) across the spray and higher than $R_{v\ast y}$ for all measurement locations. $R_{v\ast y}$ is nearly zero at the spray centre but increases towards the spray edge as radial transport of droplets becomes more important.

Figure 16. Correlation coefficient between fluctuations of droplet velocity and vapour mass fraction for different droplet size classes and measurement locations for (a) axial velocity and (b) radial velocity components.

Although in the past considerable research work has been reported concerning the effect of turbulence on evaporation rate of suspended droplets (for example, see Gökalp et al. Reference Gökalp, Chauveau, Simon and Chesneau1992; Birouk et al. Reference Birouk, Chauveau, Sarh, Quilgars and Gökalp1996; Birouk & Toth Reference Birouk and Toth2015), there is no general consensus yet on the role of air turbulence in a spray. According to Gökalp et al. (Reference Gökalp, Chauveau, Simon and Chesneau1992), a way to evaluate the role of turbulence in droplet evaporation is by considering the vaporization Damköhler number ( $Da_{v}$ ), which is the ratio of the time scale of turbulent eddies ( $\unicode[STIX]{x1D70F}_{g}$ ) and droplet evaporation time ( $\unicode[STIX]{x1D70F}_{v}$ ). For two droplets of different volatility but the same size subjected to the same turbulent flow field, smaller $Da_{v}$ implies a stronger influence of turbulence on droplet evaporation. A single droplet evaporation study by the same authors focused on very low values of $Da_{v}$ ( $\ll 1$ ) due to their large initial droplet size (approximately 1 mm), which was of the same order as the integral length scale. However, this is not true in technical applications of sprays. For the present case, $Da_{v}$ was calculated considering the characteristic time scale of energy-containing eddies of the gas flow turbulence (see table 1) and the theoretical droplet evaporation time, since in the present experiments the evaporation rate of individual droplets of the spray cannot be measured as this requires Lagrangian tracking of the droplets. The theoretical evaporation time of a droplet (whose initial size $D_{o}$ is equal to the measured SMD in our experiments) was calculated based on the $d^{2}$ -law according to (3.2):

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{v}=\frac{{D_{o}}^{2}}{K_{v}},\end{eqnarray}$$

where the evaporation constant $K_{v}=8(\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{l})D_{v}\ln (1+B)Sh$ . This leads to $Da_{v}\approx 0.4$ and 1.3 at locations $R=0$  mm and 45 mm, respectively, which means the evaporation time of isolated droplets is comparable to the large eddy time scale of the surrounding turbulent flow. Thus, the gas-phase turbulence can influence the droplet evaporation rate. In order to evaluate the role of turbulence on droplet evaporation one can also compare the Stefan velocity ( $v_{s}$ ) and turbulent velocity fluctuations of the carrier-phase velocity ( $u_{r}$ ). For an initial droplet size equal to the measured SMD, the Stefan velocity can be calculated as

(3.9) $$\begin{eqnarray}\displaystyle v_{s} & = & \displaystyle {\dot{m}}_{v}/\unicode[STIX]{x1D70C}{D_{o}}^{2}\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{K_{v}}{8D_{o}}\frac{\unicode[STIX]{x1D70C}_{l}}{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

The ratio $v_{s}/u_{r}$ was found to be approximately equal to 0.15 and 0.22 at $R=0$  mm and 45 mm respectively, which values are less than the corresponding $Da_{v}$ . However, it conveys the same information, as described above in the context of the $Da_{v}$ , that the evaporation speed of the droplets is comparable to turbulent velocity fluctuations, and turbulence effects on the evaporation of the droplets are higher at the spray centre as $v_{s}/u_{r}$ is lower compared to that close to the spray edge. Here, we note that in sprays, unlike in single droplet evaporation studies, turbulence has an indirect effect on droplet evaporation through droplet dispersion leading to interaction between vapour from neighbouring droplets. The influence is more significant when droplets tend to form clusters due to their interaction with the flow turbulence, resulting in reduction of inter-droplet distance and therefore evaporation rate. Thus, $\unicode[STIX]{x1D70F}_{v}$ increases and $Da_{v}$ is smaller. This explains the strong correlations observed in the above figures.

In summary, the droplet–vapour correlations presented above provide useful information regarding the interaction between the two phases in sprays. However, the question remains as to what factors govern the correlation between local vapour mass fraction and droplet properties: Should this be due to local evaporation, mixing or advection from upstream? We attempt to explain this below. In our experiments the residence time of droplets within a measurement window is approximately 5–10 ms, while the droplet evaporation time is typically 40–50 ms, as predicted by the convection corrected $d^{2}$ -law. The actual droplet evaporation time may be somewhat higher due to droplet clustering resulting in evaporation of the droplet cloud rather than individual droplets. In addition, the time scale of vapour diffusion across the measurement area is much larger (approximately 1 s). Hence, the correlation between instantaneous droplet number density and the local evaporation of droplets and/or mixing must be very weak. On the other hand, since the density difference between vapour and air is not significant and the droplet Stokes number (based on the large eddy scale) is small, the vapour around droplets is carried along with the carrier air flow. Hence, the instantaneous correlation between the droplet number density and velocity with surrounding local vapour concentration is attributed to advection of vapour.

3.3.3 Correlation between droplet number density of different size classes

The correlation coefficient between fluctuations of droplet number density of different droplet size classes, denoted as $R_{n_{i}\ast n_{j}}$ , is useful for understanding droplet dispersion and clustering in isothermal sprays (Sahu et al. Reference Sahu, Hardalupas and Taylor2016) and can provide an insight into the dispersion of evaporating droplets. The correlation coefficient of the fluctuations of droplet number density of two droplet size classes $D_{i}$ and $D_{j}$ can be expressed as

(3.11) $$\begin{eqnarray}R_{n_{i}\ast n_{j}}=\frac{\overline{n(D_{i})\times n(D_{j})}}{n_{r}(D_{i})\times n_{r}(D_{j})}.\end{eqnarray}$$

Large values of $R_{n_{i}\ast n_{j}}$ imply that any increase or decrease in droplet concentration of one size class is accompanied by a similar variation for the other, which means that droplets belonging to both size classes respond similarly to the gas motion and therefore may both contribute to droplet clustering affecting droplet evaporation. In the present case, $R_{n_{i}\ast n_{j}}$ is obtained for both acetone and water sprays, and the results are plotted in figure 17 for different measurement locations, $R$ . The statistical uncertainty in the correlation was approximately $\pm 0.05$ –0.07 with 95 % confidence interval.

Figure 17. Correlation coefficient ( $R_{n_{i}\ast n_{j}}$ ) between fluctuations of number density of different droplet size classes as a function of distance from spray axis. The uncertainty in the correlation was approximately $\pm 0.05$ –0.07 with 95 % confidence interval. The open and closed symbols refer to water and acetone droplets, respectively.

Figure 17 shows $R_{n_{i}\ast n_{j}}$ for the three combinations of droplet size classes considered in the present case (i.e.  $R_{n_{1}\ast n_{2}}$ , $R_{n_{2}\ast n_{3}}$ and $R_{n_{1}\ast n_{3}}$ , where 1, 2 and 3 denote increasing order of droplet size classes), which are positive for all cases. It can be observed that for all measurement locations, $R_{n_{1}\ast n_{2}}>R_{n_{2}\ast n_{3}}>R_{n_{1}\ast n_{3}}$ , which is due to increasingly poor response of the large droplet size class to gas motion. For water droplets $R_{n_{i}\ast n_{j}}$ increases considerably towards the spray edge due to the better response of the droplets (smaller $St_{L}$ ). However, for acetone droplets, the correlation coefficients for any combination of droplet size classes are almost spatially uniform across the spray, and higher in comparison to water droplets, especially close to the spray centre. This is attributed to evaporation of acetone droplets, which caused increased fluctuations of droplet number density compared to the water spray (figure 5 b). Thus, $R_{n_{i}\ast n_{j}}$ can be considered as an indication of existence of droplet clustering, which is more prominent in the acetone spray than the water spray.

3.3.4 Group evaporation number ( $G$ )

As mentioned earlier, Chiu & coworkers (Chiu & Liu Reference Chiu and Liu1977; Chiu et al. Reference Chiu, Kim and Croke1982; Chiu & Kim Reference Chiu and Kim1983) characterized group burning of droplets by the group combustion number, $G$ . For a mono-dispersed spherical cloud of droplets, the magnitude of $G$ specifically depends on droplet diameter ( $\overline{D}$ ), number of droplets ( $n_{T}$ ) and inter-droplet distance ( $l_{d}$ ), and can be expressed as

(3.12) $$\begin{eqnarray}\displaystyle G & = & \displaystyle 1.5(1+0.276Re^{0.5}Sc^{0.33})Len_{T}^{2/3}\overline{D}/l_{d}\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & = & \displaystyle 0.15Len_{T}^{2/3}/S,\end{eqnarray}$$

where $Re$ is the droplet Reynolds number, $Sc$ and $Le$ are the gas Schmidt and the Lewis numbers respectively and $S$ is the non-dimensional separation distance such that $S=(l_{d}/10D)/(1+0.276Re^{0.5}Sc^{0.33})$ . With a progressive increase in the magnitude of $G$ , different modes of group combustion may prevail (Sirignano Reference Sirignano1999), for example, single droplet combustion ( $G\ll 1$ ), internal group combustion ( $G<1$ ), external group combustion ( $G>1$ ) and external sheath combustion ( $G\gg 1$ ).

For a reacting spray, a flame surrounding a cloud of droplets results in the group burning mode of the droplet cloud. In the present work we identify droplet clustering as one of the reasons behind droplet cloud formation, though in the context of an evaporating, non-reacting spray. Away from the injector exit, dynamic interaction of droplets with the surrounding turbulent air flow leads to formation of clusters of droplets. The droplet clusters are dynamic entities and the droplets within a cluster remain close to each other for a time scale comparable to the characteristic turbulence time scale of the surrounding air flow (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2012). Although the convection of the vapour away from the droplet vicinity is the controlling factor for droplet evaporation, however, in our case, since the droplet Reynolds number is small ( $Re\approx 0.1$ ), the diffusion plays an important role in addition to turbulence. The diffusion of vapour from the droplet surface is controlled by the gradient of the vapour concentration. Due to smaller separation distance between droplets within a cluster, the gradient in vapour concentration at the droplet surface is smaller than that for an isolated droplet. Hence, the overall evaporation rate of a droplet cluster is expected to be smaller than that for an isolated droplet. Earlier, several works and review articles (for instance, Reveillon & Vervisch Reference Reveillon and Vervisch2000; Kronenburg Reference Kronenburg2007; Reveillon & Demoulin Reference Reveillon and Demoulin2007; Jenny, Roekaerts & Beishuizen Reference Jenny, Roekaerts and Beishuizen2012) identified that the droplet segregation strongly affects the evolution of the vapour mixture fraction and insisted on lack of models to account for such effects. However, experimental work highlighting the above effect is rare. The measurement of the local group evaporation number is a way to quantify the change of evaporation rate due to the above effects. For non-reacting and evaporating sprays $G$ can be termed the group evaporation number, which indicates the significance of the gross droplet evaporation rate relative to the inward diffusion rate of air. Since for our flow conditions the droplet Reynolds number is small, and for mixture of acetone vapour and air $Sc\approx 1$ and $Le\approx 1$ , equation (3.12) becomes

(3.14) $$\begin{eqnarray}G=1.5n_{T}^{2/3}\overline{D}/l_{d}.\end{eqnarray}$$

Since the expression for $G$ (3.12) was derived for droplets uniformly spaced within a cloud of droplets, the volume occupied by the cloud approximately equals to $n_{T}\times \unicode[STIX]{x03C0}{l_{d}}^{3}/6$ . Thus, for a spherical droplet cloud of size $L_{c}$ ,

(3.15) $$\begin{eqnarray}L_{c}={n_{T}}^{1/3}l_{d}.\end{eqnarray}$$

Thus, equation (3.14) can be expressed as

(3.16) $$\begin{eqnarray}G=1.5{L_{c}}^{2}\overline{D}/{l_{d}}^{3}.\end{eqnarray}$$

The group evaporation number is evaluated using (3.16) instead of (3.14); the reason behind this will be explained later. However, in such a case, two challenges are encountered: (i) an estimation of the size of droplet clouds or clusters is essential, and (ii) the droplet cluster passing through the measurement region must be identified, since (3.16) is applicable to droplet clouds only. Both of these aspects are discussed below.

3.3.5 Radial distribution function (RDF)

In the present case, the droplet cluster dimension ( $L_{c}$ ), required for (3.16), was obtained by evaluating the radial distribution function (RDF) of the droplets for different measurement locations in the spray. RDF essentially measures the probability of finding a second droplet at a given separation distance from a reference droplet, compared to a case where the droplets are homogeneously distributed (Sundaram & Collins Reference Sundaram and Collins1999; Salazar et al. Reference Salazar, de Jong, Cao, Woodward, Meng and Collins2008). It is computed from a field of $M$ number of droplets by binning the droplet pairs according to their separation distance ( $r$ ), and calculating the function

(3.17) $$\begin{eqnarray}\text{RDF}(r_{i})=\frac{P_{i}/\unicode[STIX]{x1D6FF}A_{i}}{P/A},\end{eqnarray}$$

where $P_{i}$ is the number of droplet pairs separated by a distance $r_{i}\pm \unicode[STIX]{x1D6FF}r/2$ , the parameter $r_{i}$ refers to the discrete values of separation distance, where the RDF is calculated, $\unicode[STIX]{x1D6FF}A_{i}$ is the area of the discrete shell located at $r_{i}$ , $P=M(M-1)/2$ is the total number of pairs and $A$ is the total area of the measurement region. Effectively, the values of $\text{RDF}>1$ refer to the degree of preferential accumulation or clustering of droplets due to their interaction with turbulent eddies, while values of $\text{RDF}<1$ represent voids. Thus, the value of radial separation ( $r$ ) for which $\text{RDF}(r)\approx 1$ provides an estimate of the length scale of the clusters ( $L_{c}$ ). In the present work, measurement of RDF was achieved by ILIDS measurements of droplets. We have verified in our previous work (Sahu et al. Reference Sahu, Hardalupas and Taylor2016) that calculation of $L_{c}$ by an RDF based on ILIDS agrees well with the ‘droplet counting in a cell’ approach (Fessler et al. Reference Fessler, Kulick and Eaton1994) based on focused spray droplet images. The expression for $G$ in (3.16) is derived for uniform size of droplets in a cloud. However, a droplet cluster may contain droplets of different sizes. Thus, $L_{c}$ should be determined for an average droplet size. However, before presenting this we will explain the tendency of droplets of different sizes to form clusters, and in such a case whether the corresponding length scales of droplet clusters are different.

Figure 18. Evaluation of RDF for the radial measurement locations in the acetone spray at (a)  $R=0$  mm ( $L=12$  mm) and (b)  $R=45$  mm ( $L=17$  mm) conditional on droplet size.

Figure 18 presents RDFs for radial measurement locations in the acetone spray at $R=0$  mm and 45 mm respectively. The RDFs are calculated conditional on droplet size classes, which means that only droplets of that size class are assumed to be present in an ILIDS image. The RDFs are plotted for different $r$ values normalized with the local integral length scale, $L$ , which is larger towards the spray edge (table 1). The error bars indicate uncertainty in the RDF with 95 % confidence interval. The increments in $r_{i}$ , and $\unicode[STIX]{x1D6FF}r_{i}/2$ were chosen to be 1 mm and 2 mm respectively, as a compromise between losing the spatial resolution and having enough droplets to obtain appropriate statistics. Similar trends were found for RDFs at other measurement locations, and are not presented here. It can be observed that the RDF values are higher for larger droplet sizes due to the smaller droplet number count per unit area in an image (low $P/A$ ). This trend is in agreement with fluctuations of droplet number density (relative to the mean) as shown in figure 5(b). The length scale of the droplet clusters is approximately 0.1 $L$ –0.3 $L$ , as shown in figure 18, which means the energy-containing eddies influence dispersion of droplets similarly to particle-laden jet flows (Longmire & Eaton Reference Longmire and Eaton1992). Also, the cluster dimension is larger for higher droplet size classes. Thus, the larger droplets have a greater tendency to preferentially accumulate in some regions of the flow. It is also observed in figure 18 that the RDF values increase for the location of $R=45~$ mm for all droplet size classes in comparison to those for the spray centre. This is justified due to the better response of the droplets towards the edge of the spray as the flow is slower and $St_{L}$ is smaller.

Figure 19. Evolution of radial distribution function (RDF) of the spray droplets for increasing radius of separation for the acetone and water sprays at measurement locations $R=0$  mm and 45 mm.

In order to estimate the droplet size-averaged $L_{c}$ , RDF is calculated by considering droplets of all sizes in the ILIDS images. Figure 19 presents such RDFs for the acetone spray at the measurement locations $R=0$  mm and 45 mm. The functions are plotted for different radii of separation. The RDF values are close to those of the 15–30  $\unicode[STIX]{x03BC}\text{m}$ droplets in figure 18, as expected, due to domination of smaller droplets in the droplet population. Figure 19 also shows the RDFs for the water spray for the respective measurement locations in order to assess the effect of droplet evaporation on droplet clustering. For both sprays, the function shifts upward for the $R=45$  mm location implying increase in RDF values and droplet cluster length scale towards the spray boundary. Also, it can be observed that the RDF values are higher (though not significantly) for the acetone spray for all radii of separation – hence the tendency of droplets to form clusters and the cluster dimension increase due to evaporation. As the droplets tend to form clusters, the temporal variation of droplet number density is expected to increase at any spatial location. This is in agreement with the increase in number density fluctuations in figure 5(b), and number density correlation ( $R_{n\ast n}$ ) in figure 17.

The length scale of the droplet clusters, $L_{c}$ , in the acetone spray was estimated to be approximately 2–3 mm, which is approximately one-third of the measurement window dimension as depicted in the PLIF images shown in figure 20. As mentioned earlier, since $L_{c}$ is of the order of $L$ , the large scales of the turbulent flow are expected to influence the cluster size. In addition, the large eddies are responsible for the transport of these clusters, and thus the frequency of passage of droplet clusters and inter-cluster spacing at any location in the spray are expected to be governed by the large-scale eddies of the flow (Fessler et al. Reference Fessler, Kulick and Eaton1994; Sahu et al. Reference Sahu, Hardalupas and Taylor2016).

3.3.6 Measurement of group evaporation number

For the present experiments, the droplet residence time at the measurement region ( ${\approx}5$ –10 ms) is smaller than the energy-containing eddy time scale ( ${\approx}21$ –51 ms), which is of the order of the time scale of transport of droplet clusters. Thus, at any time instant about one droplet cluster is expected to be present within the measurement area at any measurement location (the image dimensions are similar to the large eddy length scales of the flow). This is also supported by the visual inspection of the PLIF images at $R=0$  mm in figure 20. The group evaporation number ( $G$ ) is evaluated for different time instants corresponding to passage of a droplet cluster through the measurement window at any radial measurement location in the spray. For instantaneous ILIDS images where droplet clusters are identified, $G$ is calculated under the assumption that the characteristic droplet size and inter-droplet distance within a cluster is the spatially averaged droplet size and average droplet separation distance calculated based on all droplets appearing in an instantaneous image.

Figure 20. Instantaneous PLIF images of the spray depicting (a) droplet clusters and (b) no clustering at the measurement location $R=0$  mm. The images are presented in greyscale (0–256) for better visualization.

Thus, in (3.16), the parameter $\overline{D}$ is considered as the arithmetic mean diameter (AMD) of droplets identified in an instantaneous ILIDS image such that

(3.18) $$\begin{eqnarray}\text{AMD}=\frac{\displaystyle \mathop{\sum }_{i=1}^{n_{Tv}}D_{i}}{n_{Tv}},\end{eqnarray}$$

where $D_{i}$ is the size of $i$ th droplet and $n_{Tv}$ is the total number of droplets identified in the ILIDS image.

The parameter $l_{d}$ is the average minimum droplet separation distance defined as

(3.19) $$\begin{eqnarray}l_{d}=\frac{\displaystyle \mathop{\sum }_{i=1}^{n_{Tv}}l_{di}}{n_{Tv}},\end{eqnarray}$$

where $l_{di}$ is the minimum separation distance of the $i$ th droplet with respect to any other droplet in the ILIDS image. Since the validation rate of ILIDS image processing (i.e. the ratio of number of droplets identified in the image ( $n_{Tv}$ ) to the total number of droplets actually present within the measurement volume) is approximately 30–50 %, $n_{Tv}<n_{T}$ , and hence (3.14) cannot be used directly; however, equation (3.16) can be reliably used, since $l_{d}$ can be uniquely measured by ILIDS.

In order to identify the time instant when droplet clusters pass though the measurement region, the maximum inter-droplet distance ( $l_{d_{max}}$ ) is estimated using (3.15) for a given cluster size of $L_{c}$ at any measurement location and for the identified number of droplets, $n_{Tv}$ . The presence of droplet clusters ensures $l_{d_{max}}$ is larger than $l_{d}$ (measured directly from the ILIDS image), and so $G$ is evaluated using (3.16). If for an instantaneous ILIDS image $l_{d_{max}}<l_{d}$ , then no clusters are present, and $G$ is not evaluated. The result of the above method is visually depicted in figure 20, as an example which shows some sample PLIF images where droplet clusters could be identified (top images) and when no clusters are observed (bottom images). The boundaries are drawn around groups of droplets to emphasize clustering. Figure 21 shows histograms of ratio of inter-droplet distance to average droplet size (i.e. $l_{d}/\text{AMD}$ ) measured for the instantaneous images corresponding to passage of droplet clusters at a measurement location. The results are presented at the locations $R=0$  mm and 45 mm. Although the ratio of mean droplet separation to mean droplet size is approximately 20–30, locally the inter-droplet distance can be smaller ( ${<}5$ times the AMD) as the droplets preferentially accumulate in some region of the flow, as a consequence of the interaction between surrounding turbulent air flow. While the studies on mono-sized droplet streams (for instance, Depredurand & Castanet Reference Depredurand, Castanet and Lemoine2010) reported that beyond a separation distance of approximately eight droplet diameters the evaporation of adjacent droplets does not influence each other, in such a case any droplet is accompanied by only two neighbouring droplets; this limit is much larger when a droplet is embedded within a cluster of droplets and collective evaporation of all droplets prevails (for instance, see the experiments for an array of droplets in Imaoka & Sirignano Reference Imaoka and Sirignano2005). An order of magnitude variation in inter-separation distances between droplets can be observed in figure 21, especially at the spray edge. This implies similar variation in the value of $G$ and hence the mode of group evaporation.

Figure 21. Probability of the ratio of mean droplet separation ( $l_{d}$ ) to arithmetic mean diameter of droplets identified from instantaneous images of droplet clusters present in the measurement window at locations (a)  $R=0$  mm and (b)  $R=45$  mm.

Figure 22. (a) Evolution of mean group evaporation number $G$ and the standard deviation of its fluctuations (as error bars) at different measurement locations. (b) Distribution of group evaporation number of droplet clusters on an $n_{T}$ – $S$ plot for measurement locations $R=0$  mm and 45 mm. Iso- $G$ lines derived from (3.13) are shown and indicate different group evaporation regimes.

Figure 22(a) presents the evolution of mean group evaporation number $G$ at different radial measurement locations in the evaporative spray following (3.16), while the r.m.s. of fluctuations of $G$ are indicated by error bars. The statistical uncertainty in the calculation of $G$ was approximately $\pm 5\,\%$ with 95 % confidence interval. The magnitude of the mean $G$ is between 0.5 and 1, implying the mode of droplet cloud evaporation falls in the regime of internal group evaporation. This regime is characterized by an external ring of individually evaporating droplets centred around a central core of droplets, which evaporate collectively as a group. $G$ decreases from the central spray region towards the spray edge, where the droplets tend to evaporate individually (due to larger droplet spacing) without interacting with each other. Figure 22(b) shows the plot of $n_{T}$ versus the non-dimensional separation distance $S$ (see (3.13)) at the measurement locations $R=0$  mm and 45 mm. The number of droplets present in a droplet cloud ( $n_{T}$ ) is estimated from (3.15) by considering the inter-droplet separation at that time instant. Figure 22(b) also indicates constant $G$ lines, which indicate various regimes of group evaporation. For $R=45$  mm, a horizontal shift of the instantaneous values of $G$ towards the line of $G=0.01$ can be seen. Significant variations in the magnitude of $G$ can be observed for both measurement locations, which is also evident from the error bars in figure 22(a). The above result signifies the consequence of droplet dispersion on droplet evaporation, indicating not only the fact that even far away from the injector exit collective evaporation of droplets is promoted, but also the mode of group evaporation considerably varies even close to the spray centre. Also, the trend in $G$ is opposite to that of $R_{n\ast y}$ (figure 13) for any droplet size class. Thus, the greater the tendency of droplets to evaporate as a group, the lower is the correlation coefficient between droplet number density and local vapour mass fraction. Hence, it is possible that the same amount of vapour is generated even when the droplet number density increases five times (see figure 12). This happens due to presence of vapour from neighbouring droplets reducing the droplet evaporation rate.

Now we discuss some factors which may possibly influence evaluation of the group evaporation number. As mentioned earlier, the thickness of the laser sheet is approximately 1 mm, which is approximately 25 times the average droplet size. Thus, droplets at different depths could have been imaged. Unfortunately, we could not verify the effect of reducing the laser sheet thickness on our results. However, there is reasonable doubt that this depth is the full 1 mm for the following reasons. First, the depth of focus was approximately 0.9 mm (for the PLIF camera setting), which is slightly less than the thickness of the laser sheet (1 mm). Hence, most of the droplets and vapour phase in the PLIF images are focused. Although the depth of focus for the ILIDS setting was approximately 0.2 mm, which is much smaller than the laser sheet thickness, in the ILIDS technique, the droplet images are deliberately defocused anyway. Second, we note that the laser sheet has a Gaussian intensity distribution, and instrument thresholds and validation criteria in the image processing will limit the ability to detect droplets far from the central maximum intensity of the laser sheet. (The validation criteria of the ILIDS reject some of the droplets that are at the edges of the laser sheet, which do not satisfy all the criteria. This has been explained in Zarogoulidis Reference Zarogoulidis2016.) The net result of all the above contributions is that the validated droplet images for the ILIDS technique are expected to be from a narrow central area of the laser sheet. The conclusion that the droplets that are close to the central plane of the laser sheet are measured is further justified by the fact that the length of recorded droplet fringe patterns does not vary significantly, which would have been the case if the droplets were imaged at different depths of the laser sheet. We would like to note that the above effect is expected not to be a limiting factor for measurement of vapour–droplet correlations (either droplet number density–vapour mass fraction or droplet velocity–vapour mass fraction correlations). This is because both droplet and vapour phases correspond to the same volume within the spray illuminated by the laser sheet. However, larger thickness of the laser sheet may lead to bias in relative position of the droplets since two droplets at different depths may appear closer in the images. Hence, this may lead to underestimation of the measured inter-droplet distance and thus some overestimation of the $G$ values. We have made an estimate of this for the worst possible scenario, where it is assumed that apart from radial separation between droplets, the separation in the direction of the depth of the laser sheet is also present and equal to the effective thickness of the laser sheet (equal to the $1/e^{2}$ distance). For such a case it is found that the approach followed in the paper results in overestimation of $G$ values by approximately 20–30 % (larger values for inner spray locations). However, the value of $G$ is still of the order of ‘one’ close to the spray centre, indicating internal group evaporation.

In addition, it is noted here that the current estimation of the instantaneous value of $G$ uses some assumptions which are not compatible with the analytical expression for $G$ of Chiu & coworkers, which itself is based on the assumption of a quasi-steady and laminar process, constant droplet size and spacing, spherical symmetry, etc. Hence, some discrepancies between the experimental results and inference from theoretical analysis are found. In figure 22, some instantaneous values of $G$ exceed unity, implying external group evaporation (when individual droplet evaporation seizes to exist), which the theory suggests is not possible for the current measurement locations and flow conditions. In our case, the instantaneous droplet AMD and spatially averaged droplet spacing, $l_{d}$ , in a droplet cluster were used to evaluate $G$ , while within a ‘real’ droplet cluster formed due to the interaction with the flow turbulence, the size of the droplets is not the same and the inter-droplet distance is not uniform. In addition the value of $G$ is affected by the droplet cluster length $L_{c}$ , whose accurate determination is affected by the uncertainty and spatial resolution of the RDF measurement. However, the current results represent the first measurements of instantaneous values of $G$ in evaporating sprays in a turbulent environment and can quantitatively elucidate the validity of group combustion theory, which only considers laminar flows without the presence of slip velocity between the droplet and air. The current results demonstrate that, despite the limiting assumptions of the group combustion theory, it provides reasonable assessment of the droplet group evaporation in sprays interacting with turbulent flows for the current conditions.

Finally, we would like to mention that we have not discussed the scope of comparison of our experimental results with computational modelling (this was not the purpose of our research). However, the measured correlations between vapour mass fraction and droplet number density and velocity provide a basis for evaluating existing models and developing new models for unclosed vaporization source terms appearing in the transport equations for mean and variance of vapour mass fraction, as we explained in the Introduction. Measurements of such correlations have not been available until this work.