2.1 Flow and optical arrangement

All the experiments were conducted in a confined spray chamber rig. The experimental set-up (schematically shown in figure 1) was described in detail by Sahu (Reference Sahu2011). The measurement locations and the flow conditions are the same as those described in Sahu et al. (Reference Sahu, Hardalupas and Taylor2014) and so are only briefly mentioned here.

Figure 1.

Schematic of the experimental rig (a) elevation view and (b) plan view.

The rig allowed coflowing air to enter from the top in the annulus around the atomizer, which was a custom-built air-assisted nozzle placed on the centreline of the cylindrical chamber with diameter of 0.5 m. It produced a solid cone spray with Sauter mean diameter (SMD) of the order of $50~{\rm\mu}\text{m}$ at liquid feed rates of the order of $1.5\times 10^{-3}~\text{kg}~\text{s}^{-1}$ and air feed rate of the order of $0.12\times 10^{-3}~\text{kg}~\text{s}^{-1}$ . The coflowing air was seeded with aluminium oxide particles (diameter range 1– $5~{\rm\mu}\text{m}$ ) before entering the rig. The coflowing air flow rate, carrying the seeding particles, was $4\times 10^{-3}~\text{kg}~\text{s}^{-1}$ , resulting in area-averaged air velocity $3.4\times 10^{-2}~\text{m}~\text{s}^{-1}$ around the spray. The low coflowing air velocity (corresponding Reynolds number, $Re_{coflow}\approx 11$ ) and the conditioning of the inlet air flow ensured the absence of any turbulence in the coflow.

A frequency-doubled, double pulse Nd : YAG laser ( $120~\text{mJ}~\text{pulse}^{-1}$ at 532 nm; beam diameter 5 mm; New Wave Research) was used to illuminate the flow. The thickness of the laser sheet at the measurement location was 1 mm. Two identical cameras were used (PCO; Sensicam QE, 12 bit, $1040\times 1376~\text{pixels}^{2}$ ) and positioned on the same side of the laser sheet. Two lenses (135 mm $f/2.8$ Nikon lens for ILIDS and 135 mm $f/8$ Nikon lens for PIV) were used to collect the scattered light from droplets. For ILIDS operation, the choice of the field of view is a compromise between the size of the area of observation and the smallest measurable droplet diameter. Thus, in order to be able to measure at least $20~{\rm\mu}\text{m}$ droplets, both cameras were adjusted to provide a field of view of approximately $8\times 12~\text{mm}^{2}$ , which is comparatively small with respect to that of usual PIV operation. The spatial resolution was approximately $9~{\rm\mu}\text{m}~\text{pixel}^{-1}$ in both flow directions for both cameras. In all experiments, both cameras were set at an forward scattering angle of ${\it\theta}=69^{\circ }$ with respect to the direction of the laser sheet, which is the optimum scattering angle for ILIDS operation with a vertically polarized laser sheet. For this purpose, the scattered light from droplets and seeding particles was divided into two parts by using a beam splitter. A pair of cylindrical lenses, introduced between the objective and the ILIDS camera (Maeda, Kawaguchi & Hishida Reference Maeda, Kawaguchi and Hishida2000), optically compresses the fringe pattern for each droplet in the vertical direction only and generates an out-of-focus image on the focal plane. The collecting angle ( ${\it\alpha}$ ), centred around the main angle of camera orientation, was $5.35^{\circ }$ for an object distance of 300 mm, resulting in a spatial resolution ${\it\kappa}=6.28~{\rm\mu}\text{m}~\text{fringe}^{-1}$ for the ILIDS system. Both cameras were aligned under the Scheimpflug condition (Prasad & Jensen Reference Prasad and Jensen1995) in order to achieve uniform length of the fringe patterns. The delay time ( ${\rm\Delta}T$ ) between the two laser pulses was chosen to be $150~{\rm\mu}\text{s}$ as a compromise between the accuracy of subpixel interpolation and minimizing the probability of droplets moving out of the plane of the laser sheet.

The combined ILIDS and PIV measurements are reported for five different off-axis locations, 500 mm below the nozzle exit, as presented in figure 1. This measurement location in the spray was selected so that the momentum of the spray was dissipated and the entrained air flow is mainly responsible for the droplet motion. In addition, this ensured that the contribution of the boundary conditions of the droplet motion, known as the fan-spreading effect (Hardalupas et al. Reference Hardalupas, Taylor and Whitelaw1989), was minimized and the ’ballistic’ motion of atomized droplets that strongly determines droplet motion near the atomizer (Hardalupas, Taylor & Whitelaw Reference Hardalupas, Taylor and Whitelaw1990) is attenuated far downstream of the nozzle exit. In this way, the droplet–gas interaction dominated the droplet velocity and any history effects are minimized, although probably not eliminated, if only because this is a recirculating flow. Nevertheless, we believe that some interesting aspects of the turbulence modification can be explained by droplet Stokes number and/or the ratio of droplet size to flow length scales.

Because of constraints in the optical set-up, measurements were performed at an off-axis position of 125 mm away from the spray axis measured perpendicular to the laser sheet as shown in figure 1. Thus, the phrase ‘cross-stream’ is used instead of ‘radial’ direction throughout the text. The notation ‘ $R$ ’ refers to the distance from the plane passing through the injector axis and perpendicular to the laser sheet up to the beginning of a measurement area. The cross-stream measurement locations were located at $R=0$ , 50, 100, 150 and 185 mm, respectively, from the nozzle axis. We note that our choice of the off-axis measurement plane restricts us to obtain the two phase flow information only beyond the inner spray region corresponding to the radial location greater than half of the spray radius. However, according to previous experiments and simulations on confined jets (see Akselvoll & Moin Reference Akselvoll and Moin1996; Risso & Fabre Reference Risso and Fabre1997), for far downstream locations from the jet exit, the radial variation of mean and fluctuations of fluid velocity within the inner jet region are not significant.

For each measurement location, 1700 image pairs were captured through each of the cameras. Since the integral time scale of the air flow turbulence was approximately 0.2 s (as estimated later in this section), the repetition rate of the laser was set to 1 Hz, so that the acquired images remained statistically independent. The mass loading ( ${\it\phi}_{m}$ ) and the volume loading ( ${\it\phi}_{v}$ ) for any measurement location (corresponding to the measurement volume ${\approx}8\times 12\times 1~\text{mm}^{3}$ ) were obtained by considering the average number of droplets detected on the PIV image, and were about 5 % and 0.006 %, respectively. This ensured that the spray is dilute at the considered measurement locations to avoid droplet collisions. The mass and the volume loading, when calculated on the basis of the total mass flow rates of liquid and air supplied to the injector, are approximately 36 % and 0.035 %, respectively. We note that our measurement of volume loading was based on the nominal thickness of the laser sheet (i.e. 1 mm): the mass loading was derived from this. Although the Gaussian distribution of light intensity across the laser sheet gives rise to well-known variations, as a function of droplet size, in the ‘detectable’ depth of the laser sheet and in this case we expect this effect to at least half the observable depth for the largest particles and smaller values with decreasing droplet diameter (because the observable depth is roughly proportional to the surface area of a droplet). Thus, the overestimation of the depth of the laser sheet by about a factor of two, in combination with validation rates in image processing can justify the large discrepancy.

At any given measurement location, the notations ‘ $x$ ’ and ‘ $y$ ’ refer to the local axial and cross-stream directions, respectively, both lying on the plane of the laser sheet. The corresponding instantaneous velocities are denoted by ‘ $U$ ’ and ‘ $V$ ’ and velocity fluctuations by ‘ $u$ ’ and ‘ $v$ ’, respectively. Similarly the instantaneous droplet concentration is denoted by ‘ $C$ ’ and, its fluctuations by ‘ $c$ ’. Throughout the text, subscripts ‘ $d$ ’ and ‘ $g$ ’ denote droplet and gas, respectively. Similarly ‘overbar’ over any quantity indicates time-averaging and the subscript ‘ $r$ ’ denotes root mean square (r.m.s.) of that quantity.

2.2 Image processing

The algorithm for image processing is illustrated for a pair of ILIDS and PIV images in figure 2. The details can be found in Hardalupas et al. (Reference Hardalupas, Sahu, Taylor and Zarogoulidis2010) and Sahu (Reference Sahu2011).

Figure 2.

Illustration of the image processing details of the combined ILIDS and PIV technique. Boundaries of the removed glare points from the PIV image are shown as dotted circles. In the plot of simultaneous droplet and gas velocities, the circles represent droplets and the associated bold vectors represent droplet velocity.

Step 1: A pair of typical ILIDS and PIV images of the spray with ‘seeding’ particles is shown in figure 2. The ILIDS images were processed to detect the fringe pattern and obtain the droplet size, velocity and number density thereafter.

Step 2: The PIV images were processed to detect the droplet glare points. The droplets are identified by applying continuous wavelet transform (CWT) along each horizontal line of the image by selecting appropriate scales of the mother wavelet or wavelet basis. The discrimination between droplet glare points and seeding particle is achieved by selecting a suitable threshold for the wavelet spectrum.

Step 3: According to Hardalupas et al. (Reference Hardalupas, Sahu, Taylor and Zarogoulidis2010), straightforward combination of the ILIDS and PIV optical arrangements results in a discrepancy in the location of the geometric centre of a droplet, when imaging through ILIDS and PIV techniques. In the present work, the droplet centre discrepancies in both $x$ and $y$ directions are quantified from the measurements of droplets in a dilute region of the spray (without seeding particles in the surrounding air flow), which was subtracted from the position of the droplet centres identified in ILIDS images from the polydispersed spray with ‘seeding’ particles. This reduced the discrepancy between PIV and ILIDS droplet centres from approximately $1000~{\rm\mu}\text{m}$ to approximately $100~{\rm\mu}\text{m}$ (in terms of pixels, from approximately 100–10 pixels) and hence increased the probability of finding corresponding fringe patterns on the ILIDS image and glare points on the PIV image.

Step 4: For each fringe pattern (belonging to the ILIDS image), the corresponding glare points are identified in the PIV image and associated with the droplet size and velocity obtained from that fringe pattern. Then, the glare points are filtered out of the PIV images following a method based on wavelet transform described by Hardalupas et al. (Reference Hardalupas, Sahu, Taylor and Zarogoulidis2010). Figure 2 shows the PIV image after removal of the glare points, the boundaries of which are shown as the dotted circles.

Step 5: The PIV images after the removal of droplets are processed to obtain the gas velocity. A modified PIV algorithm, based on evaluation of the direct correlation via FFT (Ronneberger, Raffel & Kompenhans Reference Ronneberger, Raffel and Kompenhans1998) in conjunction with a digital mask technique (Gui, Wereley & Kim Reference Gui, Wereley and Kim2003), was found to result in higher accuracy for the considered non-ideal PIV images compared to the conventional FFT-based approach. For an elaborated discussion, the readers are directed to Sahu (Reference Sahu2011). The interrogation window size for PIV processing was $32\times 32~\text{pixel}^{2}$ with 50 % overlapping. The spatial resolution of the instantaneous velocity measurements was approximately 0.3 mm in both directions, larger than the droplet sizes considered here and of the same order as the Kolmogorov length scale, which was of the order of $300~{\rm\mu}\text{m}$ , as estimated in the following section. Due to 50 % overlapping, the distance between the adjacent gas velocity vectors was 0.15 mm. However, since the laser sheet thickness was approximately 1 mm, the measurements are averaged across the depth of the laser sheet. Although, due to the Gaussian distribution of the light intensity along the laser sheet thickness, the averaging may be occurring over two Kolmogorov scales, which is expected to have small influence on the presented results. However, to authors knowledge, only Tanaka & Eaton (Reference Tanaka and Eaton2010) have reported sub-Kolmogorov scale resolution PIV measurement of gas velocity around large monosized particles ( $St>$ 100, when based on Kolmogorov scale) to study the TKE dissipation due to particles. Although the camera viewing area in their experiments is about one-fourth of the viewing area considered in the present work, which considers polydispersed droplets.

Step 6: The results from the ILIDS and PIV measurements are combined to obtain the individual droplet size and velocity and simultaneously the gas velocities around each droplet as shown in the vector plot of figure 2. Due to various validation criteria imposed to detect the droplets, while processing the ILIDS and PIV images, it is not always possible to find the corresponding pairs of fringe patterns and glare points. Hence, the validation rate in the combined technique is low (about 30 %), which necessitates acquisition of large number of image samples to minimize statistical uncertainty. (Here the validation rate is defined as the ratio of the number of droplet glare points on the PIV image for which the corresponding fringe patterns on the ILIDS image could be found to the total number of the droplet glare points detected on the PIV image.)

We should mention here that, in order to obtain droplet concentration or number density measurements of the spray, the ILIDS images are used instead of the corresponding focused (PIV) images, since the latter cannot provide the droplet size. The PIV images are considered for droplet counting only in order to obtain the droplet cluster dimensions independent of droplet size, as will be discussed in the following section. It should be noted that, in any instantaneous ILIDS image, the validation procedure of the image processing does not reject preferentially some droplet sizes. Therefore, the relative droplet number counts of different size classes remain the same compared to the case when all droplets in an image are considered. The droplet concentration was measured by counting the number of detected droplets in the ILIDS image, which corresponds to a volume of $8\times 12\times 1~\text{mm}^{3}$ in the present case (thickness of the laser sheet ${\approx}1$  mm).

3.1 Droplet size distribution in the spray

The probability of the measured droplet size distribution (from ILIDS) is shown in figure 3 for the measurement locations at $R=0$  mm, 100 mm and 185 mm respectively. The minimum measurable droplet size was $20~{\rm\mu}\text{m}$ as determined by the limitations in the optical set-up and ILIDS image processing, Sahu (Reference Sahu2011). The size distributions show that most droplets are in the range of 20– $40~{\rm\mu}\text{m}$ . It can be also observed that away from the axis of the spray, the probability of small droplets (20– $30~{\rm\mu}\text{m}$ ) increases slightly while that of the larger droplets ( ${>}50~{\rm\mu}\text{m}$ ) decreases, although the change is small. Thus, away from the central spray region, the arithmetic mean diameter (AMD) and SMD of the drop size distribution decreases slightly. However, considering the accuracy of the droplet size measurement for the present case ( $\pm 3.25~{\rm\mu}\text{m}$ ), the AMD and SMD of the drop size distribution at any measurement location can be considered to be of the order of $35~{\rm\mu}\text{m}$ and $45~{\rm\mu}\text{m}$ , respectively. All the statistical quantities of the spray were calculated for three droplet size classes (denoted by notation ‘ $D$ ’). The size classes were 20– $35~{\rm\mu}\text{m}$ , 35– $50~{\rm\mu}\text{m}$ and 50– $65~{\rm\mu}\text{m}$ respectively. The size width ( ${\rm\Delta}D$ ) of $15~{\rm\mu}\text{m}$ for each size class was selected as a compromise between higher statistical uncertainty (with smaller ${\rm\Delta}D$ ) and obtaining size-averaged information (with larger ${\rm\Delta}D$ ).

Table 2.

Droplet Stokes number of various size classes based on integral scale, $St_{L}$ ( $=\,{\it\tau}_{d}/{\it\tau}_{g}$ ) and Kolmogorov scale, $St_{{\it\eta}}$ ( $=\,{\it\tau}_{d}/{\it\tau}_{k}$ ) for the cross-stream measurement locations, $R=0$  mm, 100 mm and 185 mm, respectively.

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’ ( ${\it\tau}_{d}$ ) over an appropriate turbulent time scale of the flow. ${\it\tau}_{d}$ is obtained based on the assumption of Stokes flow around the droplet, which is justified since the Reynolds number of the droplets based on mean slip velocity was very small in the present case ( ${\approx}0.1$ ). The average of maximum and minimum droplet sizes of each size class (for instance $27.5~{\rm\mu}\text{m}$ for 20– $35~{\rm\mu}\text{m}$ droplet size class) is considered for calculation of ${\it\tau}_{d}$ . Based on the characteristic time of the entrained air flow by the spray, the values of Stokes number, denoted as $St_{L}$ ( $={\it\tau}_{d}/{\it\tau}_{g}$ ), for the three droplet size classes were calculated for different droplet sizes classes and found to be of the order of 0.01 for different measurement locations (table 2). These values suggest good response of all droplet sizes to the corresponding large-scale fluid motion. The size of all of the droplets in the spray was smaller than the Kolmogorov length scale of the flow. The Stokes numbers based on the Kolmogorov time scale, $St_{{\it\eta}}$ ( $={\it\tau}_{d}/{\it\tau}_{k}$ ), for the 20– $35~{\rm\mu}\text{m}$ , 35– $50~{\rm\mu}\text{m}$ and 50– $65~{\rm\mu}\text{m}$ droplet size classes were of the order of 0.34, 0.88 and 1.48, respectively for the $R=0$  mm location, which decreases towards the spray edge (see table 2). These values signify partial response of the droplets to the smallest length scale of the flow.

In order to estimate the gravitational influence on the droplet motion in comparison to the inertial effects, the terminal velocity ratio of the droplets is estimated, which is defined as the ratio of terminal velocity of droplets to a characteristic velocity of the gas flow. The terminal velocity, $u_{t}$ ( $={\it\tau}_{d}g$ ), were calculated and found to be of the order of 0.02 $\text{m}~\text{s}^{-1}$ , $0.05~\text{m}~\text{s}^{-1}$ and $0.1~\text{m}~\text{s}^{-1}$ for the three droplet size classes respectively. The terminal velocity ratio based on axial mean gas velocity ( $u_{t}/\overline{|U_{g}|}$ ), and based on axial gas r.m.s. velocity ( $u_{t}/u_{gr}$ ) for the three droplet size classes are presented in table 3 for the measurement locations at $R=0$ , 100 and 185 mm. Table 3 shows that, except for the smaller droplets (20– $35~{\rm\mu}\text{m}$ ), the gravitational influence on the motion of larger droplet size classes cannot be considered negligible, and it increases relative to the inertial effects towards the outer spray region. At $R=185$  mm, the ratio of $u_{t}$ to the cross-stream gas r.m.s. velocity ( $v_{gr}$ ) for the three droplet size classes are of the order of 0.2, 0.5 and 0.9, which further signifies the important role of gravity at the spray edge.

Table 3.

Terminal velocity ratio of various droplet sizes based on axial mean gas velocity ( $u_{t}/\overline{|U_{g}|}$ ) and axial gas r.m.s. velocity ( $u_{t}/u_{gr}$ ) for the cross-stream measurement locations, $R=0$  mm, 100 mm and 185 mm respectively.

3.2 Mean flow properties

3.2.1 Droplet and gas velocity

The method of estimating mean and r.m.s. velocity, and the corresponding uncertainties were discussed in detail in Sahu et al. (Reference Sahu, Hardalupas and Taylor2014) and is not repeated here. The mean velocity of droplets of a given size class and the mean gas velocity for both axial and cross-stream velocity components at any measurement location were observed to be quasi-uniform across the measuring area. The corresponding r.m.s. of velocity fluctuations also followed a similar trend. This is possibly because of the small size of the viewing area ( $8\times 12~\text{mm}^{2}$ in the present case), as compared to the length scales of the large eddies of the flow, which were approximately 50 mm. Figure 4(a,b) show the variation of the area-averaged mean and r.m.s. velocity in both axial and cross-stream directions for droplet size class of 20– $35~{\rm\mu}\text{m}$ and gas flow for various measurement positions, $R$ . The droplets, away from the centre of the spray, tend to move upwards (negative velocity) i.e. ‘towards the top of the spraying tower’. This occurs due to the motion of the entrained air flow in the spray, which has a recirculating flow pattern at the outer region (figure 4 a) in order to conserve mass and momentum, similar to the schematic shown in figure 1(a). Towards the outer spray, the droplets are prevented from drifting downward under the action of gravity, as might be expected, by the upward motion of the gas velocity. This explains the small decrease in the probability of large droplets ( ${>}50~{\rm\mu}\text{m}$ ) at $R=185$  mm in figure 3.

Figure 4.

Area-averaged (a) mean velocity and (b) r.m.s. of velocity fluctuations, for droplet (size class 20– $35~{\rm\mu}\text{m}$ ) and gas flow for various cross-stream measurement locations, $R$ . Note that the positive mean velocity is at the lower section of the vertical axis in order to demonstrate the flow motion along the direction of gravity.

The fluctuating velocities of both droplets and gas decreased away from the spray axis implying reduction of the turbulent kinetic energy in the outer spray region (figure 4 b). However, compared to the axial direction, considerable reduction in r.m.s. velocity in the cross-stream direction (approximately 50 %) was observed from $R=0$ to 185 mm. Thus, the flow was nearly isotropic close to the spray centre (the ratio $u_{gr}/v_{gr}\approx 1$ at the location $R=0$  mm). This is further confirmed from the $x$ -velocity spectra as shown in figure 5, which is obtained by taking Fourier transform of the two-point correlation coefficients of axial gas velocity fluctuations ( $R_{u_{g}u_{g}}$ ). However, as the measurement window dimensions are approximately five to six times smaller than the integral length scale of the turbulent carrier phase flow, so we had to extrapolate $R_{u_{g}u_{g}}$ to ‘zero’ (following Sahu et al. Reference Sahu, Hardalupas and Taylor2014). Since nearly 90 % of the overall data is contributed by the extrapolated exponential curve fit, the resulting length scale is sensitive to the fitting parameters and any error they contain. Hence, the spectral data seems to be free from uncertainty and the wavenumbers span a large range. However, we observed a basic agreement between the calculated integral length scale (as derived from the area under the extrapolated $R_{u_{g}u_{g}}$ curve, denoted as, effective length scale $L_{eff}$ ) and the estimated value of $L$ ( ${\sim}50$  mm, as mentioned earlier). The calculated $L_{eff}$ was approximately 65 mm, which is close to $L$ . Thus, the extrapolation procedure has some basis, at least as a first approximation. Figure 5 indicates isotropic turbulence is established for the location $R=0$  mm, while the anisotropy increases towards the edge of the spray ( $u_{gr}/v_{gr}\approx 1.8$ for the location $R=185$  mm).

Figure 5.

One-dimensional axial velocity spectra ( $E_{uu}$ ) with respect to different wavenumbers $k_{u}=2{\rm\pi}/{\it\delta}x$ for the measurement locations $R=0$ , 100, 185 mm. The straight line represents the model spectra according to $E_{uu}\propto k_{u}^{-5/3}$ .

In our experiments, at any measurement location, the r.m.s. of the velocity fluctuations in the axial direction was about two times larger than the mean velocity (figure 4). Similar observations in the trends of mean and r.m.s. velocities have been reported by Risso & Fabre (Reference Risso and Fabre1997) in their single phase turbulence measurements of a confined axisymmetric water jet far from the nozzle. These features are typical to the confined flow systems and have been also reported by Boree, Ishima & Flour (Reference Boree, Ishima and Flour2001). In our measurement locations, since the mean velocity and its gradients in both directions were small, it is also expected that there is negligible turbulence production due to the shear layer. This is confirmed by low values of normalized Reynolds stress ( $\overline{u_{g}v_{g}}/{u_{r}}^{2}<0.2$ ) obtained at different measurement locations (see figure 9 in Sahu et al. (Reference Sahu, Hardalupas and Taylor2014)). We reported the two-point spatial correlation coefficient between droplet–gas, and gas velocity fluctuations ( $R_{dg}$ and $R_{gg}$ , respectively) for different velocity components and also measurement locations (figure 12; Sahu et al. (Reference Sahu, Hardalupas and Taylor2014)). The spatial evolution of $R_{gg}$ at all measurement locations implied strong correlation of the gas velocity field with itself ( $R_{u_{g}u_{g}}>0.8$ and $R_{v_{g}v_{g}}$ ${>}0.6$ ). The magnitude of $R_{gg}$ for both axial and cross-stream velocity remained nearly constant for different lags since the measurement window size is about an order of magnitude smaller than the large-scale eddies. In addition, for all measurement locations our proper orthogonal decomposition (POD) analysis of the gas velocity (see figures 16 and 17 in Sahu et al. (Reference Sahu, Hardalupas and Taylor2014)) signified the dominance of large-scale eddies in the flow dynamics (first three POD modes contributes about 50 % of total TKE of the gas flow). We find that the spatial correlation between droplet and gas axial velocity is also high (the normalized droplet–gas velocity correlation, $\overline{u_{d}u_{g}}/u_{dr}u_{gr}>0.8$ ), and the first POD mode is solely responsible for the strong coupling between droplet and gas phase velocity in axial direction, while contribution of other modes are negligible. This suggests that the fluctuating motion, not the mean velocity, plays a central role in the two phase flow dynamics in the considered measurement locations within the spray.

3.2.2 Droplet concentration

The normalized mean and r.m.s. of the droplet concentration are plotted in figure 6 for the three drop size classes. Normalization was based on the corresponding mean values at $R=0$  mm. Away from the spray axis, the mean droplet concentration of the smallest droplet size class of 20– $35~{\rm\mu}\text{m}$ remains approximately constant, while that of larger size classes decrease. Also, considering the mean velocity of the entrained gas flow (figure 4 a), which implies the presence of a vortical pattern of the mean gas flow, it can be argued that further away from the spray axis, the small droplets (having good response to the integral scale of the flow and negligible gravitational effects) are carried easily towards the top of the tower. This results in their uniform distribution across the spray and smaller fluctuations of droplet concentration (about 30 %) compared to the mean (figure 6 b). Thus, the concentration of smaller droplets is mainly controlled by the convective fluid motion. Due to their partial response to the gas flow and the higher gravitational influence (table 3), the larger droplets resist the upward motion of the gas flow. Hence the corresponding mean droplet concentration consistently decreases towards the spray edge, and no additional influence of the shear layer at $R=100$  mm location is observed. Though more measurements at additional cross-stream locations would strengthen our argument but we believe that our results provide enough evidence to support the arguments for the suggested observations. The larger droplets show larger relative fluctuations of drop concentration ( ${\approx}50\,\%$ of the respective mean) compared to the 20– $35~{\rm\mu}\text{m}$ droplets, which can be attributed to preferential concentration of the large droplets due to their partial response to fluid flow turbulence (i.e. $St_{{\it\eta}}\approx 1$ ) as described in the following section.

Figure 6.

Normalized (a) mean droplet concentration and (b) r.m.s. of droplet concentration fluctuations for the three droplet size classes for various cross-stream measurement locations, $R$ . Normalization was based on the respective mean value at $R=0$  mm.

4.1 Scale of droplet clustering

4.1.1 The ‘droplet counting in a cell’ approach

In order to obtain the characteristic dimensions of droplet clusters in the flow, the probability density function (PDF) of the local concentration is compared with that arising from a purely random process (Fessler et al. Reference Fessler, Kulick and Eaton1994; Aliseda et al. Reference Aliseda, Cartellier, Hainaux and Lasheras2002). For this purpose, at first, the instantaneous droplet concentration was obtained from the two-dimensional focused (PIV) images of the flow at the different measurement locations. However, the droplet sizing technique (ILIDS) was not used for this purpose because it is not possible to identify all droplets appearing on the ILIDS image due to the imposed image processing criteria. At any given instant, the PDF of the droplet concentration was obtained by dividing an image into cells of a certain size and counting the number of droplets inside each cell. If $C$ is the total number of detected droplets and $N_{b}$ is the total number of cells, then the probability of finding $n$ droplets per cell is obtained as the ratio of the sum of the number of droplets in the cells containing ‘ $n$ ’ droplets and the total number of droplets in all of the cells ( $C$ ). The PDF averaged over all images was obtained and compared with the distribution of droplets in cells for a random process, given by a binomial distribution:

(4.1) $$\begin{eqnarray}\displaystyle P_{binomial}(n)=\binom{\bar{C}}{n}\left(\frac{1}{N_{b}}\right)^{n}\left(1-\frac{1}{N_{b}}\right)^{\bar{C}-n}, & & \displaystyle\end{eqnarray}$$

where $\bar{C}$ is the mean number of detected droplets (averaged over all sample images). The comparison between the PDF found for a given cell size and that for a random process provides an indication of how turbulence modifies the droplet concentration field.

Figure 7.

Comparison of PDF of the measured number of droplets per cell with a binomial distribution, representing a random process, for the cross-stream measurement location at $R=0$  mm for six different box sizes with dimensions (a) $1~\text{mm}\times 1~\text{mm}$ , (b) $2~\text{mm}\times 2~\text{mm}$ , (c) $4~\text{mm}\times 4~\text{mm}$ , (d) $4~\text{mm}\times 6~\text{mm}$ , (e) $8~\text{mm}\times 6~\text{mm}$ and (f) $4~\text{mm}\times 12~\text{mm}$ , while the measurement area is $8~\text{mm}\times 12~\text{mm}$ .

The length scale at which preferential concentration is most effective can be identified by computing these statistics for cells of different sizes. Figure 7 shows the PDF comparison for different cell sizes. Considering the horizontal and vertical dimensions of the measurement area ( $=8~\text{mm}\times 12~\text{mm}$ ), sizes of cells were chosen to be approximately $1~\text{mm}\times 1~\text{mm}$ , $2~\text{mm}\times 2~\text{mm}$ , $4~\text{mm}\times 4~\text{mm}$ , $4~\text{mm}\times 6~\text{mm}$ , $8~\text{mm}\times 6~\text{mm}$ and $4~\text{mm}\times 12~\text{mm}$ , respectively, while larger cell sizes were considered to avoid cell overlapping. It can be observed from figure 7 that for the larger boxes, the PDF of the number of droplets per cell substantially deviates from the corresponding random distribution. This indicates that the length scale of droplet clusters is at least of the order of the dimensions of the measurement area.

In order to quantify the deviation of the measured PDF of droplet number density from the random distribution two parameters were used. The first one, $D_{1}$ , was introduced by Fessler et al. (Reference Fessler, Kulick and Eaton1994), and is the difference between the standard deviation of the two distributions ( ${\it\sigma}$ and ${\it\sigma}_{binomial}$ ):

(4.2) $$\begin{eqnarray}\displaystyle D_{1}=\frac{{\it\sigma}-{\it\sigma}_{binomial}}{{\it\mu}}, & & \displaystyle\end{eqnarray}$$

where ${\it\mu}$ is the mean number of particles per cell. Positive values of this parameter indicate the presence of concentrated regions, while zero values represent a quasi-uniform concentration field. The second parameter, $D_{2}$ , was used by Wang & Maxey (Reference Wang and Maxey1993) which represents the square of the difference of probabilities given by the two distributions, and is always positive or zero:

(4.3) $$\begin{eqnarray}\displaystyle D_{2}=\mathop{\sum }_{1}^{C}(P(n)-P_{binom}(n))^{2}. & & \displaystyle\end{eqnarray}$$

Table 4.

The parameters $D_{1}$ and $D_{2}$ for the different cell sizes corresponding to figure 7.

Figure 8.

The variation in $D_{1}$ for the cell size of $4~\text{mm}\times 4~\text{mm}$ at different measurement locations, $R$ . The error bar indicates the uncertainty in $D_{1}$ with 95 % confidence interval.

In the present case the values of $D_{1}$ and $D_{2}$ were calculated for the different cell sizes and are presented in table 4. As expected from figure 7, both values increase initially with the cell size and then decreases afterwards. Considering that the statistical uncertainty for $D_{2}$ was higher (approximately $\pm 25\,\%$ compared to $\pm 5\,\%$ for $D_{1}$ ), the cell size for which $D_{1}$ is maximum i.e. $4~\text{mm}\times 4~\text{mm}$ is considered to correspond to the order of length scale of droplet clusters. Similar results were observed for other measurement locations. Figure 8 shows the evolution of $D_{1}$ for the cell size of $4~\text{mm}\times 4~\text{mm}$ at various measurement locations. $D_{1}$ appears to increase with cross-stream distance $R$ implying greater tendency of droplets to form clusters near the spray edge compared to the central spray region. Again due to larger statistical uncertainty for $D_{2}$ , comparison between the corresponding values at different measurement locations could not be made.

The values of $D_{1}$ for the cell size of 4 mm ( ${\approx}13{\it\eta}$ , ${\it\eta}$ is the Kolmogorov length scale of the gas flow) are comparable to the maximum value of $D_{1}$ for monosized particle laden air flow in a channel as reported by Fessler et al. (Reference Fessler, Kulick and Eaton1994) for $St\approx 0.74$ and 2.2. However, Aliseda et al. (Reference Aliseda, Cartellier, Hainaux and Lasheras2002) have reported lower values of $D_{1}$ in their experiments with polydispersed droplets. These authors argued that, in case of monodispersed distributions, either all droplets (with $St_{{\it\eta}}\approx 1$ ) tend to preferentially accumulate or all tend towards a random distribution (when $St_{{\it\eta}}\ll 1$ or $St_{{\it\eta}}\gg 1$ ). For the case of polydispersed distributions, droplets with $St_{{\it\eta}}\approx 1$ form clusters which entrain droplets of all other size classes. This results in lowering the values of $D_{1}$ . However, the droplet size distribution in the experiments by Aliseda et al. (Reference Aliseda, Cartellier, Hainaux and Lasheras2002) was narrow and the smaller droplets ( ${\approx}10~{\rm\mu}\text{m}$ ) dominated the size distribution with probability of occurrence of nearly 40 %. Also, only few droplets with Stokes number of the order of 1 were present. In the present experimental conditions, the droplet size distribution is broader, and most droplets respond partially to the smallest scales of the gas flow, which justifies larger deviation from random distribution and, therefore, higher values of $D_{1}$ .

For a polydispersed spray, the scale of the droplet clustering can vary for droplets of different sizes, which cannot be distinguished by the presented droplet counting method, which considers PIV images (PIV cannot provide droplet size). In order to quantify the influence of droplet size on the length scale of droplet clusters, the estimation of the radial distribution functions (RDF) for each droplet size is essential, and this has not been reported before. From literature, we mention Saw et al. (Reference Saw, Shaw, Ayyalasomayajula, Chuang and Gylfason2008, Reference Saw, Shaw, Salazar and Collins2012b ) who studied a polydispersed spray in wind tunnel using PDA and measured one-dimensional RDF for different droplet size classes in the spray. They adopted a method equivalent to Taylor’s frozen turbulence hypothesis to convert the time series to droplet spatial distribution along the mean flow direction. However, the hypothesis is valid only for very low turbulent intensities ( $u_{gr}/\overline{U_{g}}\ll 1$ ), and this approach is certainly not suitable for the present experiments where turbulent fluctuations are about two times the mean velocity (see figure 4).

4.1.2 Radial distribution function (RDF) of droplet concentration

RDF is a statistical measure of droplet clustering (Sundaram & Collins Reference Sundaram and Collins1999, Salazar et al. Reference Salazar, de Jong, Cao, Woodward, Meng and Collins2008). It is defined as 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. The RDF is computed from a field of $M$ droplets by ‘binning’ the droplet pairs according to their separation distance, and calculating the function

(4.4) $$\begin{eqnarray}\displaystyle \text{RDF}(r_{i})=\frac{N_{i}/{\it\delta}V_{i}}{N/V}, & & \displaystyle\end{eqnarray}$$

where $N_{i}$ is the number of droplet pairs separated by a distance $r_{i}\pm {\it\delta}r/2$ , ${\it\delta}V_{i}$ is the volume of the discrete shell located at $r_{i}$ , $N=M(M-1)/2$ is the total number of pairs and $V$ is the total volume of the system. Since the present experimental technique is planar, the RDF measurement is restricted to two dimensions only. Thus, in the above equation the discrete volumetric shell is replaced by an annular area with radius $r_{i}$ . This method is similar to the estimation of droplet–gas spatial velocity correlation for different radius of separation, $r_{i}$ , as explained in figure 10 of Sahu et al. (Reference Sahu, Hardalupas and Taylor2014). The two-dimensional RDF was calculated using the droplet positions obtained from ILIDS images, the advantage being the ability to obtain $\text{RDF}(r_{i})$ for different size classes. The increment in $r_{i}$ and, ${\it\delta}r$ 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. Holtzer & Collins (Reference Holtzer and Collins2002) have identified that the lower-dimensional RDFs attenuate the measured values at separations below a characteristic length of the measurement (in the present case, thickness of the laser sheet ${\approx}3{\it\eta}$ ). However, for the purpose of estimating the cluster dimension (which is expected to be at least an order of magnitude larger than ${\it\eta}$ , see for example, Fessler et al. (Reference Fessler, Kulick and Eaton1994), Aliseda et al. (Reference Aliseda, Cartellier, Hainaux and Lasheras2002), Yang & Shy (Reference Yang and Shy2005)) and relative affinity of droplets of different sizes for preferential accumulation, two-dimensional RDF can be reliably used.

Figure 9.

Evaluation of RDF for the cross-stream measurement locations at (a) $R=0$  mm ( ${\it\eta}=0.30$  mm), (b) $R=100$  mm ( ${\it\eta}=0.37$  mm) and (c) $R=185$  mm ( ${\it\eta}=0.45$  mm) conditional on droplet size.

Figure 9 shows the RDF calculated at three different measurement locations $R=0$ , 100 and 185 mm. The RDFs are plotted for different $r_{i}$ values normalized with the local Kolmogorov length scale, ${\it\eta}$ , which is larger towards the spray edge (table 2). The values of RDF ${<}1$ indicate the presence of voids in the flow, i.e. the local concentration becomes lower than the average across the total area of the image. For values of $\text{RDF}>1$ , droplet clustering occurs. Thus, when $\text{RDF}=1$ , the droplet distribution is random, and the corresponding radial separation distance between droplets, $r_{i}$ , can be considered as statistical description of the length scale of droplet clusters (termed as ‘ $L_{c}$ ’). Considering the distribution of RDFs of figure 9, the following trends can be observed.

Figure 10.

A sample PIV image containing droplets without seeding particles. The image also depicts a droplet cluster and the order of the associated length scale.

1. (1) At any measurement location, the length scales of the clusters, i.e. the values of $L_{c}$ , increase for higher droplet size classes, and are approximately 3–6 mm. This is in accordance with the cell size of $4~\text{mm}\times 4~\text{mm}$ (as evaluated by the droplet counting in cell approach) for which the $D_{1}$ factor was maximum at all measurement locations. The cluster dimensions in the present case are approximately $5{\it\eta}$ – $15{\it\eta}$ compared to $10{\it\eta}$ in cases of both Fessler et al. (Reference Fessler, Kulick and Eaton1994) and Aliseda et al. (Reference Aliseda, Cartellier, Hainaux and Lasheras2002). Hence, similar to both of the previous studies, in the present experiments the size of the droplet clusters are expected to be mostly influenced by the viscous effects (at smallest scale) rather than inertial effects (at large scales). A sample PIV image containing droplets without seeding particles is shown in figure 10, which also includes a schematic of a droplet cluster and its associated length scale. The original PIV image was not shown as it contains droplets as well as seeding particles, and so visual evidence of preferential accumulation of droplets is difficult.

2. (2) At any measurement location $R$ and for any radius of separation $r_{i}$ , RDF increases for larger droplet size classes, i.e. with increasing $St_{{\it\eta}}$ of the droplets. So more clustering is evident for 50– $65~{\rm\mu}\text{m}$ droplets, which correspond to $St_{{\it\eta}}$ of the order of 1. Saw et al. (Reference Saw, Shaw, Ayyalasomayajula, Chuang and Gylfason2008) have also shown increased peak value of RDF with increasing $St_{{\it\eta}}$ . However, the present results are in contrast to Wood et al. (Reference Wood, Hwang and Eaton2005), who measured the two-dimensional RDF for monosized particles in homogeneous and isotropic turbulence within a box. These authors found nearly overlapping RDFs for particles with Kolmogorov Stokes number of the order of one ( $St_{{\it\eta}}=0.57$ , 1.06 and 1.33). However, unlike Wood et al. (Reference Wood, Hwang and Eaton2005), gravitational influence on droplets cannot be considered as negligible in our experiments (see table 3). Since, in the present case, clustering is observed at dissipative scales and the Kolmogorov time scale is of similar order as the time required for droplets to fall over a distance ${\it\eta}$ with terminal velocity ( ${\it\tau}_{k}\approx {\it\eta}/u_{t}$ ), gravity plays a role in clustering and is expected to enhance the preferential accumulation of droplets in the gravitational direction. Similar observation of gravitational influence on preferential concentration has also been reported by Ferrante & Elghobashi (Reference Ferrante and Elghobashi2003) and Yang & Shy (Reference Yang and Shy2005).

3. (3) Figure 9 also shows that from $R=0$ to 185 mm, the values of the RDF for closer separation distances increase, especially for the larger droplet size classes, indicating that towards the edge of the spray, where the gravitational influence on droplets is higher, the droplets show greater tendency to preferentially accumulate. This appears more prominently when the $r_{i}$ are not normalized with ${\it\eta}$ (not shown here). The above mentioned trend in RDFs agrees with the increasing value of factor $D_{1}$ from $R=0$ to 185 mm, as discussed in the previous section. The consequence of this is larger cluster dimensions ( $L_{c}$ ) at $R=185$  mm.

We note here that considering a smaller area within the image (for instance, half of the horizontal and vertical dimensions of the original image) for RDF calculation consequently reduces the number of samples for a given separation distance between droplets. This results in higher statistical uncertainty of the measured RDF, which means larger uncertainty in estimating the droplet cluster length scale. We calculated RDF for a smaller window size of $4~\text{mm}\times 6~\text{mm}$ , and for the same separation distances ( $r_{i}$ ) and annular separation ( ${\it\delta}r$ ) as for the full window size. The trends in the RDFs (not shown here) for different droplet size classes were found to be similar for both cases, although some differences in the values were observed. The uncertainty was higher for the smaller window though not significantly. However, the uncertainty in estimation of droplet cluster length scale was higher as one has to extrapolate till the value of $\text{RDF}=1$ .

4.2 Correlation between fluctuations of droplet concentration and droplet/gas velocity

In order to quantify the consequences of the effect of preferential concentration, it is essential to estimate the correlation between fluctuations of droplet concentration and velocities of droplet and gas flow. As mentioned before, such correlation terms appear in the equation for modification of TKE of the carrier phase (1.1) and represent the possibility of turbulence modification due to fluctuations of dispersed phase concentration. Also, the correlation between fluctuations of concentration of different droplet size classes is an important quantity, which can shed light on the relative dispersion of different droplet sizes. In this section, the spatial correlation coefficients are estimated between fluctuations of droplet concentration for different size classes ( $R_{c_{_{I}}\ast c_{_{J}}}$ ), and between fluctuations of droplet concentration and droplet velocity ( $R_{c\ast u_{id}}$ ) and gas velocity ( $R_{c\ast u_{ig}}$ ). Here ‘ $i$ ’ refers to the component of the Cartesian coordinate system. All the correlations are calculated conditional on droplet size class, which are not available in the literature. While $R_{c_{_{I}}\ast c_{_{J}}}$ and $R_{c\ast u_{id}}$ are obtained from ILIDS measurements only, estimation of $R_{c\ast u_{ig}}$ required the simultaneous measurement of the gas velocity from PIV measurement.

It should be noted that the instantaneous droplet concentration ( $C$ ) is measured by counting droplets in ILIDS images (corresponding to the measurement volume $8\times 12\times 1~\text{mm}^{3}$ ) at each measurement location ( $R$ ) in the spray. For any image sample, the local values of concentration for all droplets (of same size class) are assumed to be same and equal to the value of $C$ of that size class. For the calculation of $R_{c\ast u_{id}}$ and $R_{c\ast u_{ig}}$ for each droplet size class, the correlations $cu_{id}$ and $cu_{ig}$ , evaluated for each droplet (of that size class), are averaged over the measurement area and for all image samples. From the previous section, we know that the length scales of the droplet clusters ( $5{\it\eta}$ – $15{\it\eta}$ ) and the dimensions of the measurement area ( $30{\it\eta}$ – $40{\it\eta}$ ) are of similar order. Thus, considering $C$ in the evaluation of the above mentioned correlations would lead to spatial averaging of both random and non-random fluctuations of droplet concentration, which would consequently lower the magnitude of the correlations. In order to avoid this, $C$ should be measured in a smaller area (in comparison to the present measurement area). However, this would in turn lead to higher uncertainty due to the low number of droplets. Since our purpose is to study the relative evolution of the correlations at different measurement locations in the spray, hence, spatial averaging of droplet concentration measurement is accepted as the best option available, while compromising on the magnitude of the correlations. However, the spatial averaging is not of concern for droplet/gas velocity correlation. As mentioned before, the velocity field of the gas is dominated by the large eddies, which is much larger than the dimensions of the viewing area for each experiment (see table 1). Thus, the spatial averaging is expected to filter out velocity fluctuations at the small scales, which are not significant here. This is also reasonable since the mean and r.m.s. of both droplet and gas velocities were nearly uniform across the viewing area. We justify the above discussion in the following section where we present the comparison of the absolute correlations appearing in (1.1) for the current window size and a smaller window.

The mathematical expression for the spatial correlation of droplet concentration ( $R_{c_{_{I}}\ast c_{_{J}}}$ ) for droplet size classes $D_{I}$ and $D_{J}$ , can be written as:

(4.5) $$\begin{eqnarray}\displaystyle R_{c_{I}\ast c_{J}}={\displaystyle \frac{\overline{c_{t_{k},I}(D_{I})\times c_{t_{k},J}(D_{J})}}{c_{Ir}(D_{I})\times c_{Jr}(D_{J})}}, & & \displaystyle\end{eqnarray}$$

where the overbar denotes the averaging operation; $c_{t_{k},I}$ and $c_{t_{k},J}$ denote the fluctuations of droplet concentration of the two size classes $D_{I}$ and $D_{J}$ , respectively, at time instant $t_{k}$ ; $c_{Ir}$ and $c_{Jr}$ are the corresponding root mean square values. Only those $t_{k}$ are considered for which both $C_{t_{k},I}$ and $C_{t_{k},J}>$ 0 so that any effect of droplet rejection due to ILIDS image processing can be avoided.

The mathematical expression for the spatial correlation of droplet concentration and droplet velocity, and gas velocity ( $R_{c\ast u_{id}}$ and $R_{c\ast u_{ig}}$ , respectively) for any droplet size class $D$ can be written as:

(4.6) $$\begin{eqnarray}\displaystyle R_{c\ast u_{id}}(D)=\frac{\overline{c_{t_{k},j}(D)\times u_{t_{k},j,id}(D)}}{c_{r}(D)\times u_{idr}(D)}, & & \displaystyle\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle R_{c\ast u_{ig}}(D)=\frac{\overline{c_{t_{k},j}(D)\times u_{t_{k},j,ig}(D)}}{c_{r}(D)\times u_{igr}(D)}, & & \displaystyle\end{eqnarray}$$

where $c_{t_{k},j}$ and $u_{t_{k},j,id}$ are the instantaneous fluctuations of droplet concentration and velocity for a droplet at location ‘ $j$ ’ in the image at time instant $t_{k}$ ; $u_{t_{k},j,ig}$ is the gas velocity fluctuations ‘seen’ by the droplet at location $j$ ; $c_{r}$ , $u_{idr}$ and $u_{igr}$ are the corresponding r.m.s. values. As mentioned before, $c_{t_{k},j}=c_{t_{k}}$ for all $j$ at the time instant $t_{k}$ , and only those $t_{k}$ are considered for which $C_{t_{k}}>0$ .

Figure 11.

The correlation coefficient between fluctuations of droplet concentration of different size classes, $R_{c_{_{I}}\ast c_{_{J}}}$ , at different measurement locations, $R$ . The error bars indicate statistical uncertainty in $R_{c_{_{I}}\ast c_{_{J}}}$ with 95 % confidence interval. The subscripts 1, 2 and 3 refer to the droplet size classes of 20– $35~{\rm\mu}\text{m}$ , 35– $50~{\rm\mu}\text{m}$ and 50– $65~{\rm\mu}\text{m}$ , respectively.

Figure 11 shows the correlation between fluctuations of droplet concentration of different size classes, $R_{c_{_{I}}\ast c_{_{J}}}$ . The corresponding uncertainty is relatively high (approximately $\pm 0.07$ with 95 % confidence interval) compared to the measured correlation coefficients. Because the correlation is defined only for those time instants at which the droplet concentration of the two size classes are non-zero, the sample size is reduced (compared to the total number image samples of 1700). Large positive values of $R_{c_{_{I}}\ast c_{_{J}}}$ signify that any increase or decrease in droplet concentration of one size class is accompanied with a similar variation in 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. When the droplet response times differ, $R_{c_{_{I}}\ast c_{_{J}}}$ tends to be low. In figure 11, the magnitude of $R_{c_{_{I}}\ast c_{_{J}}}$ is always low ( ${<}0.3$ ) indicating varying degree of droplet response to the gas flow turbulence, which in turn is evident from the range of Stokes numbers of the droplets ( $St_{{\it\eta}}\approx$ 0.4–2.0). It can be observed that the correlation between droplets of size 20–35 and 50– $65~{\rm\mu}\text{m}$ , $R_{c_{1}\ast c_{3}}$ , is very low at all measurement locations. This is due to the greater tendency of larger droplets to preferentially accumulate at certain length scales of the flow (as shown by the RDF plots in figure 9), while the smaller droplets follow the fluid flow better, and thus the correlation between the corresponding variations of droplet concentration is lost. This effect is more pronounced further away from the spray axis in agreement with earlier observations. Therefore, $R_{c_{1}\ast c_{2}}$ is higher close to the spray axis. Since the 35–50 and 50– $65~{\rm\mu}\text{m}$ droplets demonstrate partial response to gas motion, their correlation $R_{c_{2}\ast c_{3}}$ remains relatively high, which agrees with the trends of their respective RDFs. This also implies that the time scale of cluster formation for the droplets of the two size classes are of similar order. These observations suggest that the process causing droplet clustering leads to a redistribution of droplets according to their size in addition to redistribution in space. This means that the instantaneous droplet size distribution in clusters will be different than the average, which can have consequences for industrial process, e.g. evaporation of fuel droplets.

Figure 12.

Spatial correlation coefficient between fluctuations of (a) droplet concentration and droplet velocity, $R_{c\ast u_{d}}$ , and (b) droplet concentration and gas velocity, $R_{c\ast u_{g}}$ in axial direction for various droplet size classes at different measurement locations, $R$ .

The spatial correlation coefficients $R_{c\ast u_{d}}$ and $R_{c\ast u_{g}}$ for axial component of velocity are shown in figure 12 for different measurement locations and droplet size classes. They provide an indication of the degree of coupling between fluctuations of droplet concentration, and droplet and gas velocity in axial direction of the flow. The statistical uncertainties (with 95 % confidence interval) are larger for $R_{c\ast u_{d}}$ ( $\pm 0.02$ –0.05) compared to that for $R_{c\ast u_{g}}$ ( $\pm 0.005$ –0.02 %). This is primarily due to the decrease in the sample record size because the number of validated droplet velocity in any image was much less than the number of validated gas velocity vectors (the ratio being about 10 : 4000 in any image). Also, the uncertainty was larger for higher droplet size classes due to low probability of occurrence. Figure 12 shows that for any droplet size class both correlation coefficients are small (close to zero) near the spray centre, but become negative away from the spray axis. The negative correlation increases towards the edge of the spray, which is supported by the progressive increase of the RDF deviation from the value of 1 for droplets of any size class from $R=0$ to 185 mm (figure 9).

In general, as far as transport of small droplets ( $St_{L}\sim 0.01$ , as in our case) by the large eddies is concerned, no correlation is expected between droplet concentration and gas velocity since the droplets (like flow tracers) are expected to be randomly distributed. However, when droplet clusters are formed as a consequence of interaction of partially responding droplets with small eddies, non-random distribution of droplets results in a correlation. Transport of the droplet clusters by turbulent eddies leads to local fluctuations of droplet concentration, and its correlation with gas velocity. Hence, the trends in figure 12 are attributed to preferential accumulation of droplets, which has been signified in our measurements by the $D$ parameter and RDF, as mentioned before. The observed negative correlation between the droplet concentration and velocity of either phases can be explained as follows. According to Maxey (Reference Maxey1987), as a consequence of preferential concentration, the divergence of particle velocity is positive in the regions of low strain rate and high vorticity. Thus, large number of droplets with low velocity are expected to accumulate at the peripheral regions of vortices. When such a vortex sweeps across the measurement area, either the droplet concentration increases and the corresponding droplet/gas velocity is low ( $c>0$ , $u_{d}<0$ and $u_{g}<0$ ), or a smaller number of droplets with high velocity remain ( $c<0$ , $u_{d}>0$ and $u_{g}>0$ ). Thus, the correlation is negative.

However, as pointed out by a referee, the RDFs for the smallest and intermediate size classes are similar in the spray centre and in its edge (figure 9) and yet this family size also shows negative cross-correlation values in figure 12. Thus an alternative explanation is that the negative values of the cross-correlation could also be linked to the alternate passage of large scale, upward moving structures transporting fluid from the spray centre which exhibit relatively large droplet concentrations and downward moving structures transporting fluid from the recirculating region which is partially depleted from droplets as a result of gravitational settling acting in regions of weak velocities and large residence times. However, as a counter argument, even if the RDFs for 20–35 and 35– $50~{\rm\mu}\text{m}$ droplets are not significantly different for measurement locations (and so also the corresponding absolute length scale of the droplet clusters) as shown in figure 9, the magnitude of the correlations can vary (figure 12), while, for 50– $65~{\rm\mu}\text{m}$ droplets, the RDF increases towards the spray edge and also the correlations are higher. Which of the two views is correct is probably resolvable with time series measurements, which are not available in the current experiments.

The correlation between fluctuations of droplet concentration and cross-stream velocity components of droplet/gas velocity was found to be negligibly low and is not presented here. This is supported by the low correlation between droplet axial and cross-stream velocity fluctuations and low values of Reynolds stress in the gas flow at different measurement locations, as presented by Sahu et al. (Reference Sahu, Hardalupas and Taylor2014). Hardalupas & Horender (Reference Hardalupas and Horender2003) measured $R_{c\ast u_{id}}$ in a horizontal plane shear layer (low-speed air stream on top) laden with glass beads injected at the low-speed side. The correlation was higher for streamwise velocity compared to the cross-stream velocity similar to the present work although in their work gravity acted in cross-stream direction. They found positive values of $R_{c\ast u_{id}}$ in the low-speed side, while negative values were observed in the high-speed side of the flow. The maximum value of magnitude of correlation coefficient was approximately 0.2–0.3 on either side. Thus, the sign of the correlations $\overline{cu_{id}}$ and $\overline{cu_{ig}}$ is not universal, since it depends on a particular flow configuration and gravitational influence on particle dispersion.

We emphasize that the correlations are calculated by considering fluctuations of different quantities after subtracting the respective mean values. Also, the mean flow is weak compared to turbulence for all measurement locations. Hence, the observed correlations are not expected as a consequence of the mean flow. Let us consider the outer edge location $R=185$  mm. The mean axial gas velocity is $-0.08~\text{m}~\text{s}^{-1}$ while the corresponding r.m.s. velocity is $0.2~\text{m}~\text{s}^{-1}$ . Thus, the instantaneous axial gas velocity varies around $-0.08\pm 0.2$ . Since, $R_{c\ast u_{g}}<$ 0 at this location, it is possible that a positive fluctuation of gas velocity, leading to downward gas motion or reduced upward motion of the gas flow, would be accompanied by local depletion of droplet concentration. This cannot be explained by considering mean gas velocity, which is always upward, and thus $R_{c\ast u_{g}}$ is considered as a consequence of preferential accumulation of droplets due to interaction with turbulent eddies.

We point out here that for the present measurement locations within the spray, the droplet volume loading was 0.006 %. Since this is much lower than 0.3 %, droplet–droplet interactions are not expected (Hardalupas et al. Reference Hardalupas, Taylor and Whitelaw1989). Although the interdroplet spacing reduces as a consequence of preferential accumulation of droplets, droplet collision and coalescence are very rare, and so, this does not affect the droplet/gas measurement by the combined technique.

5.1 Droplet–gas slip velocity

Figure 14 shows the variation of the mean slip velocity (for both velocity components) between the droplets and the gas flow for different size classes and measurement locations in the spray. The absolute values of the droplet and gas mean velocities are considered for both velocity components ( $|\overline{U}|$ and $|\overline{V}|$ ) since we intend to evaluate whether the average droplet motion lag or lead the average gas velocity. In order to signify the importance of slip velocity relative to gas velocity fluctuations, the mean slip velocity is normalized by the respective r.m.s. of gas velocity fluctuations ( $u_{igr}$ ) at each measurement location. The uncertainty (with 95 % confidence interval) of the normalized mean slip velocity is estimated to be approximately $\pm 0.07$ –0.12, and is higher for larger droplet size class. It can be observed in figure 14(a) that the average axial droplet–gas slip velocity is low compared to the gas velocity fluctuations. The axial mean slip velocity is positive at the inner spray locations (where gravitational effect on droplets is low) implying that the droplets always precede the downward gas flow (see figure 4 a) due to their finite inertia, while it is negative at the outer spray, i.e. droplets lag the upward gas flow (see figure 4 a) due to higher gravitational contribution. The statistical uncertainty in slip velocity is higher to make comparison between droplets of different size classes. However, the axial slip velocity of 20– $35~{\rm\mu}\text{m}$ droplets is consistently smaller than that for larger droplets for all measurement locations. Due to higher uncertainty for larger droplets (as a result of reduction in sample record size), comparison of trends of 30– $45~{\rm\mu}\text{m}$ and 50– $65~{\rm\mu}\text{m}$ is difficult. However, due to to very good response of droplets of all sizes to large-scale gas flow structures ( $St_{L}\approx 0.01$ , table 2), in general, the difference in average axial slip velocity is not significant.

Towards the spray boundary, the normalized cross-stream slip velocity is higher for the smaller droplet size class (figure 14 b). At the outer spray region, the average gas and droplet flow are largely axial and upward, while the mean cross-stream velocity is small and slightly negative (oriented away from the spray boundary) as shown in figure 4(a). For 20– $35~{\rm\mu}\text{m}$ droplets the mean cross stream velocity is larger ( ${\sim}-0.03~\text{m}~\text{s}^{-1}$ ) in comparison to that for larger droplets and gas velocity, which are very small ( ${\sim}-0.005~\text{m}~\text{s}^{-1}$ ). However, the 20– $35~{\rm\mu}\text{m}$ droplets are expected to follow the gas flow since their Stokes number is small. But the above trend is also evident in the mean velocity vector plots in our earlier publication (figure 6; Sahu et al. Reference Sahu, Hardalupas and Taylor2014), where a very consistent trend can be observed across the cross-stream direction. We cannot find evidence of any source of measurement error (suchas noise).

Figure 15 presents the PDF of the normalized slip velocity between fluctuations of droplets and gas velocities ( $u_{d}-u_{g}$ and $v_{d}-v_{g}$ ) for different size classes at the measurement locations $R=0$ and 185 mm. The PDFs at other measurement locations are similar, and not shown here. The probability distributions clearly indicate the presence of non-zero slip between droplets and gas velocity. A narrow probability distribution implies that droplets are able to follow well the gas velocity fluctuations, while asymmetry of the PDF suggest the direction of momentum transfer between the two phases. As shown in figure 15, the PDFs are broader for the cross-stream slip velocity compared to the axial velocity, which is due to poor droplet response in cross-stream direction. While no significant difference is present at the PDFs for different droplet sizes at $R=0$  mm, the PDFs for axial velocity become broader for larger droplets at $R=185$  mm. In general, the PDFs were found to be somewhat positively skewed, which means higher probability of negative slip or droplets lag the gas velocity fluctuations. This indicates that locally the drag between the two phases cause turbulent energy transfer from the gas to the droplets. We note here that some of the pdfs shown in the above figure are positively skewed (for example 50– $65~{\rm\mu}\text{m}$ droplets for locations $R=0$ and 185 mm). This is due to few erroneous measurement of large droplet velocity, which however has nearly zero probability. However, we have verified that the mean of the instantaneous slip velocity is nearly ‘zero’.

Figure 15.

Probability density functions of slip between fluctuations of droplet and gas velocity for axial (a,c,e,g,i,k) and cross-stream (b,d,f,h,j,l) components of velocity for the three droplet size classes at (a–f) $R=0$  mm and (g–l) $R=185$  mm. Slip velocity is normalized by the corresponding r.m.s. of gas velocity fluctuations at different measurement locations. (a,b,g,h) 20– $35~{\rm\mu}\text{m}$ ; (c,d,i,j) 35– $50~{\rm\mu}\text{m}$ ; (e,f,k,l) 50– $65~{\rm\mu}\text{m}$ .

5.2 Comparison of droplet–gas correlation terms in the TKE equation of the carrier phase

This section presents the comparison of the terms depicting turbulent energy transfer between the droplet and the gas phases (as described by (5.1)–(5.3)) for different measurement locations in the spray. These terms are calculated from different correlations in (1.1) as described previously. It should be noted that for a polydisperse spray, the above terms must be evaluated separately for different droplet size classes. Despite the low validation ratio of the combined technique ( ${\approx}30\,\%$ ), the magnitude of all three terms defined in (5.1)–(5.3) would proportionately decrease due to the rejected droplets. Hence the present approach is suitable for relative comparison between $T_{Eu_{i}1}$ , $T_{Eu_{i}2}$ and $T_{Eu_{i}3}$ .

Figure 16.

Comparison of various terms in TKE equation (i.e. $T_{Eu_{i}1}$ , $T_{Eu_{i}2}$ and $T_{Eu_{i}3}$ ) of the carrier phase for both axial (a,c,e) and cross-stream (b,d,f) velocity components evaluated for three droplet size classes (a,b) 20– $35~{\rm\mu}\text{m}$ , (c,d) 35– $50~{\rm\mu}\text{m}$ and (e,f) 50– $65~{\rm\mu}\text{m}$ for different measurement locations, $R$ . The error bars indicate statistical uncertainty with 95 % confidence interval.

Figure 16 shows the terms $T_{Eu_{i}1}$ , $T_{Eu_{i}2}$ and $T_{Eu_{i}3}$ evaluated for both axial and cross-stream components of velocity for different droplet size classes. The corresponding statistical uncertainties with 95 % confidence interval (indicated by error bars in figure 16) were calculated by considering the uncertainties in individual statistical quantities involved in different terms. Since the uncertainties in the correlation terms (e.g. $\overline{u_{id}u_{ig}}$ , $\overline{cu_{id}u_{ig}}$ etc.) were higher than other quantities such as $\overline{C}$ , $\overline{U}$ , $\overline{V}$ , and the term $T_{Eu_{i}3}$ involves only one correlation quantity (i.e. $\overline{cu_{ig}}$ ), the corresponding uncertainty was less than that of the other two terms. The results are shown for all five cross-stream measurement locations. The following observations are identified.

1. (1) For all measurement locations and droplet size classes, the gas velocity was found to correlate better with itself compared to the droplet velocity ( $\overline{u_{ig}u_{ig}}>\overline{u_{id}u_{ig}}$ ). So, $T_{Eu_{i}1}$ is negative and depicts attenuation of the gas phase turbulence. This also means that on average the droplets tend to lag behind the gas motion (if $u_{ig}>0$ , $u_{id}-u_{ig}<0\rightarrow u_{id}<u_{ig}$ , and if $u_{ig}<0$ , $u_{id}-u_{ig}>0\rightarrow u_{id}<u_{ig}$ ), which is supported by the positively skewed PDFs of droplet–gas slip velocity in figure 15.

   Figure 17.

   Correlation terms (a) ( $\overline{u_{d}u_{g}}-\overline{u_{g}u_{g}}$ ) and (b) ( $\overline{v_{d}v_{g}}-\overline{v_{g}v_{g}}$ ) for the three droplet size classes and different measurement locations, $R$ .

   

   Figure 17(a,b) show the term ( $\overline{u_{id}u_{ig}}-\overline{u_{ig}u_{ig}}$ ) for different droplet size classes and measurement locations. The statistical uncertainties, shown as error bars, were about $\pm 0.002$ –0.005, and larger for larger droplet size classes. It can be observed that the magnitude of these terms is higher for 50– $65~{\rm\mu}\text{m}$ droplets although comparison of trends between 20–35 and 35– $50~{\rm\mu}\text{m}$ droplets is not always possible due to larger uncertainties. Since large droplets have poor response to the gas velocity fluctuations, the corresponding droplet–gas velocity correlation is reduced. So, the difference ( $\overline{u_{id}u_{ig}}-\overline{u_{ig}u_{ig}}$ ) increases with droplet size since $\overline{u_{ig}u_{ig}}>$ 0, and the gas velocity ‘seen’ by the droplets was found to be nearly independent of the droplet size (not shown here). This effect is marginal for axial velocity, while more pronounced for cross-stream velocity component, and also towards the edge of the spray where gravitational influence on droplets is larger. However, the magnitude of $T_{Eu_{i}1}$ for a given droplet size class is proportional to the corresponding mean droplet concentration ( $\overline{C}$ ) and the inverse of droplet relaxation time $(1/{\it\tau}_{d})$ , both of which are smaller for larger droplet sizes. Hence, comparing figure 16(a–c), it can be observed that the term $T_{Eu_{i}1}$ corresponding to drop size class of 20– $35~{\rm\mu}\text{m}$ is around 5 and 15 times larger compared to that of 35– $50~{\rm\mu}\text{m}$ and 50– $65~{\rm\mu}\text{m}$ droplets, respectively.

   Since the mean droplet concentration decreases away from the central spray region (especially for larger droplets, see figure 6), a corresponding reduction in $T_{Eu_{i}1}$ can be observed. However, this variation is not significant for axial velocity. Since towards the spray edge, the cross-stream gas velocity fluctuations reduce (figure 4 b), and also the slip velocity, $(v_{d}-v_{g})_{r}$ , was found to decrease, the corresponding magnitude of $T_{Ev_{1}}$ is low at the measurement locations $R=150$  mm and 185 mm.

2. (2) For all measurement locations and droplet size classes, the term-2 for axial component of velocity was found to be negative, i.e. $T_{Eu2}<0$ . This implies that this term, similar to term-1, causes turbulence reduction although the magnitude of $T_{Eu2}$ is less than that of $T_{Eu1}$ for all droplet size classes. The magnitude of $T_{Eu2}$ increases from $R=0$ to 185 mm. This is in agreement with the increasing trends of RDF (figure 9) and the magnitudes of $R_{c\ast u_{d}}$ and $R_{c\ast u_{g}}$ (figure 12) from spray axis towards the outer spray locations, and indicates the effect of preferential droplet accumulation on energy transfer from the gas to the droplets. For locations $R=150$  mm and 185 mm and droplets of size classes 35– $50~{\rm\mu}\text{m}$ and 50– $65~{\rm\mu}\text{m}$ , $T_{Eu2}$ is about half the value of $T_{Eu1}$ since the larger droplets show greater tendency to form clusters as depicted by the trends of RDF in figure 9, while for 20– $35~{\rm\mu}\text{m}$ droplets $T_{Eu2}$ is approximately 10 times smaller than $T_{Eu1}$ . Hence, even for moderate mass loading cases (such as the present experimental condition), term-2 cannot be considered negligible in comparison to term-1. For the cross-stream direction, $T_{Ev2}$ is always negligibly low, which is expected. Since $\overline{cv_{g}}\approx 0$ , and the droplet and gas velocities are well correlated ( $\overline{v_{d}v_{g}}>0$ ), hence both triple correlation terms ( $\overline{cv_{d}v_{g}}$ and $\overline{cv_{g}v_{g}}$ ) in (5.2) are close to zero for the cross-stream velocity.

3. (3) The third term in TKE equation, $T_{Eu_{i}3}$ , shows the significance of drag between average droplet and gas velocity for interphase momentum transfer in addition to the role of fluctuations of droplet concentration and gas velocity. Since the slip in droplet–gas mean velocity is smaller than the slip in the fluctuations of the respective velocities, it is expected that the contribution of $T_{Eu_{i}3}$ would be less than the other two terms. For cross-stream direction, the correlation $\overline{cv_{g}}\approx$ 0, hence, as shown in figure 16, $T_{Ev_{3}}$ is always negligible, even though the mean slip velocity is considerably larger compared to the gas velocity fluctuations for 20– $35~{\rm\mu}\text{m}$ droplets (figure 14 b). For similar reasons, in axial direction, $T_{Eu_{3}}$ is close to zero at the inner spray measurement locations. However, at the outer spray locations and larger droplets, $T_{Eu_{3}}$ is negative implying attenuation of fluid turbulence and its magnitude is comparable to $T_{Eu_{2}}$ , although always $|T_{Eu_{2}}|>$ $|T_{Eu_{3}}|$ . This is attributed again to preferential accumulation of the larger droplets resulting in increased $\overline{cu_{g}}$ near the spray edge. Thus, the contribution from term-3 cannot be neglected.

Considering the contributions from all three terms, as described above, it is concluded that droplets tend to attenuate the gas phase turbulence at the present measurement locations in the spray. Since in the current experiments, the ratio of droplet size to the integral length scale of the flow was approximately 0.0005–0.001, this agrees with Gore & Crowe (Reference Gore and Crowe1989), who reviewed previous experimental research on particle-laden flows in pipes and channels, and found that turbulence attenuation occurs when the ratio of particle size to a characteristic length scale of the carrier phase flow is below 0.1. A number of previous experimental studies have observed turbulence attenuation by small particles (particle size ${\sim}O({\it\eta})$ ) at various flow configurations, for instance, Tsuji & Morikawa (Reference Tsuji and Morikawa1982), Tsuji et al. (Reference Tsuji, Morikawa, Tanaka, Karimine and Nishida1988), Kulick et al. (Reference Kulick, Fessler and Eaton1994), Kussin & Sommerfeld (Reference Kussin and Sommerfeld2002), Hwang & Eaton (Reference Hwang and Eaton2006), among others. In those studies the particle $St_{{\it\eta}}$ was much greater than 1, so preferential concentration of particles (or contributions from term-2 and term-3) was assumed to be negligible. Also, the droplet–gas velocity correlation ( $\overline{u_{d}u_{g}}$ ) in term-1 is assumed to be close to zero since large droplets are expected to show poor response to the fluid velocity fluctuations. However, the critical value of $St_{{\it\eta}}$ for which such assumptions are valid is still unknown. For instance, in their channel flow experiments, Fessler et al. (Reference Fessler, Kulick and Eaton1994) observed clustering for $50~{\rm\mu}\text{m}$ particles of $St_{{\it\eta}}=8$ and even (up to a certain extent) for $90~{\rm\mu}\text{m}$ particles of $St_{{\it\eta}}=19$ . The degrees of preferential concentration, quantified as the $D$ parameter, are approximately 0.27 and 0.12 for the mentioned particle sizes, respectively, not significantly less than the value of $D=0.38$ for the $25~{\rm\mu}\text{m}$ particles of $St_{{\it\eta}}=0.74$ . Hence, for such cases, the consequences of particle clustering on turbulence modulation cannot be simply discarded. This may be even more important for higher particle mass loading, for example, in the experiments of Kulick et al. (Reference Kulick, Fessler and Eaton1994).

Experimental studies of turbulence modulation by particles exhibiting the tendency to preferentially accumulate are not abundant. We mention Prevost et al. (Reference Prevost, Boree, Nuclisch and Charnay1996), Sakakibara et al. (Reference Sakakibara, Wicker and Eaton1996) and Ferrand et al. (Reference Ferrand, Bazile, Boree and Charnay2003), who observed preferential accumulation of particles in their particle-laden jet experiments, however, reported turbulence attenuation by considering term-1 only. In their experiments in a fan stirred homogeneous and isotropic turbulence laden with particles ( $St_{{\it\eta}}\approx 1$ and particle size ${\leqslant}{\it\eta}$ ), Yang & Shy (Reference Yang and Shy2005) observed augmentation of carrier phase turbulence especially for higher wavenumbers beyond Taylor’s scale. While, these authors did not quantify the correlation terms of (1.1), it can be argued that due to poor response of particles to the smallest eddies, at the dissipative scales, $\overline{u_{id}u_{ig}}$ must be less than $\overline{u_{ig}u_{ig}}$ , which means turbulence attenuation due to term-1. So, the observed turbulence augmentation must be due to larger positive contributions from term-2 and/or term-3. This is supported by the measurements of Yang & Shy (Reference Yang and Shy2005), who found significant increase of the particle settling rate because of preferential accumulation of particles, which aligned with the flow vortices increasing the fluid turbulence. Hardalupas et al. (Reference Hardalupas, Sahu, Taylor and Zarogoulidis2010) used DVM to simulate the particle and fluid flow characteristics in a particle-laden shear layer with Stokes number (based on large eddy time scale) from 1.0 to 4.5. They obtained the correlation terms of Eulerian–Lagrangian form of (1.1), where the last two terms combined as one single term and what is expressed as particle concentration fluctuations is contained in the conditional averaging of the flow variables. They found turbulence enhancement due to preferential concentration, which was, however, much smaller than the turbulence attenuation. Thus, term-1 was dominant, similar to the present work.

We conclude this subsection with two remarks. The first is that our comments and discussion relate to identifying the effects of clustering in a particular example. Apart from critical dimensionless parameters such as the Stokes numbers, it goes without saying (as noted by a referee) that history effects on the gas turbulent kinetic energy budget are relevant in general. This is particularly relevant for flows such as the one studied, where the gas turbulence is first produced by the spray itself. Thus, close to the injector, the presence of the droplets are certain to be a source in the budget for gas turbulent kinetic energy, whereas further downstream the droplets act as a sink. Where gas turbulent kinetic energy production or attenuation by the droplets takes places in general depends on the flow history and its evolution, and therefore the Stokes number at a particular measurement location is not the only parameter involved in whether the gas turbulent kinetic energy will be augmented or attenuated by the droplets. Here, any history effects are taken into account to some extent by using the measured gas phase turbulence. The second remark is that the actual magnitude of modification of TKE of the unladen gas flow cannot be quantified in the present work. This is because: (i) we cannot have the corresponding single phase flow data as the gas flow is induced by the spray, and (ii) limitation in the ILIDS technique to validate all imaged droplets, as mentioned before. However, considering the low droplet mass loading in the present experimental flow conditions (about 5 %), the amount of TKE modification due to droplets is expected to be low. This is also supported by Sahu et al. (Reference Sahu, Hardalupas and Taylor2014), who obtained, for the present flow conditions, spatial correlation coefficients of droplet–gas velocity fluctuations ( $R_{dg}$ ) conditional on droplet size and gas velocity fluctuations ( $R_{gg}$ ). By comparing $R_{gg}$ and $R_{dg}$ for different distances of separation from a reference droplet, which were close to each other (though $R_{gg}>R_{dg}$ ), the authors qualitatively predicted low turbulence attenuation.