3.2.1. Anisotropy and the TKE

The presence of a droplet introduced a significant disruption to the unladen isotropic field. The resultant anisotropy for an example case is visualized in Cartesian and spherical coordinates in figure 6. The spherical presentation in figure 6(b) is logical given the geometry, but the Cartesian approach in figure 6(a) is included to facilitate a comparison with Hoque et al. (Reference Hoque, Mitra, Sathe, Joshi and Evans2016). Based on their planar PIV data, Hoque et al. (Reference Hoque, Mitra, Sathe, Joshi and Evans2016) suggested that the presence of a large fixed sphere in a quasi-HIT field improves the flow isotropy, and that the improvement continues as the sphere diameter increases $(10 \lesssim d/\eta \lesssim 77)$. In Hoque et al. (Reference Hoque, Mitra, Sathe, Joshi and Evans2016), the unladen flow has an average isotropy ratio (defined therein as $\overline {I_r^{-1}}$) of ${\sim }0.82$ which eventually improves to ${\sim }0.96$ for the largest sphere ($d = 8\,{\rm mm}$). The unladen flow, along with each sphere size, was tested at six levels of turbulence (quantified by the Reynolds number of the oscillating grids, $Re_g$) and in no instance did $Re_g$ have any noticeable effect on the isotropy. The conclusion that the sphere improved isotropy is clearly contrary to our results in figure 6, and two possibilities are now suggested. First, the PIV FOV in Hoque et al. (Reference Hoque, Mitra, Sathe, Joshi and Evans2016) does not include the region below the sphere, which removes a large swath of low $I_r^{-1}$ data points. Second, the sphere may reorient the r.m.s. components in an existing anisotropic field toward a more isotropic structure. Neither possibility is observable in the present dataset since the unladen flow has excellent isotropy (table 1) and the FOV does not discard any near-droplet information. Figure 7 plots the r.m.s. components featured in figure 6(b). While both components decrease near the droplet surface, the radial extent of the damping of $\langle u_r^2 \rangle ^{1/2}$ is far greater than $\langle u_\phi ^2 \rangle ^{1/2}$. This conclusion agrees with the PR-DNS findings of Vreman (Reference Vreman2016).

Figure 6. Isotropy field for an example case ($Re_\lambda = 50$, ${\bar {d}/\eta _0} = 1.5$) in (a) Cartesian and (b) spherical coordinates. Note that (a) plots $I_r^{-1}$ for consistency with Hoque et al. (Reference Hoque, Mitra, Sathe, Joshi and Evans2016).

Figure 7. (a) Radial and (b) azimuthal r.m.s. velocities for the example case in figure 6 ($Re_\lambda = 50$, ${\bar {d}/\eta _0} = 1.5$).

Figures 8–10 illustrate the kinetic energy fields – centrally cropped about the droplet to ${\pm }5\,{\rm mm}$ – for the minimum, median and maximum droplet diameters at the three major $Re_\lambda$ levels (50, 100, and 140). For consistency, the maximum colour level is set to the 95th percentile value of the corresponding unladen TKE field. The attenuated region surrounding the droplet is highly symmetric, as expected in a zero-mean HIT environment. It is not surprising that $k$ is universally damped near the droplet, as $(d/L)_{{max}} \approx 0.05$ and $Re_{p,{max}} \approx 28$, both of which fall well shy of the classic demarcation points for kinetic energy enhancement suggested by Gore & Crowe (Reference Gore and Crowe1989) and Hetsroni (Reference Hetsroni1989) ($d/L \gtrsim 0.1$ and $Re_p \gtrsim 400$, respectively). Even if these thresholds were exceeded, it is not clear if/when enhancement would be expected in the absence of a steady wake aligned with a mean flow. However, Hoque et al. (Reference Hoque, Mitra, Sathe, Joshi and Evans2016) reported an increase in the kinetic energy field when $d/L \gtrsim 0.41$, which suggests that the demarcation point between attenuation and augmentation – whatever the specific mechanism for a fixed sphere in zero-mean HIT – is similar in magnitude to the Gore & Crowe (Reference Gore and Crowe1989) limit. While the droplets in the present study could move slightly, they were essentially fixed in place, hence the attenuation of $k$ was assuredly due to the mere presence of a surface and not the exchange of kinetic energy from the turbulence to the droplet.

Figure 8. The TKE field surrounding the (a) minimum, (b) median and (c) maximum droplet diameters at $Re_\lambda = 50$.

Figure 9. The TKE field surrounding the (a) minimum, (b) median and (c) maximum droplet diameters at $Re_\lambda = 100$.

Figure 10. The TKE field surrounding the (a) minimum, (b) median and (c) maximum droplet diameters at $Re_\lambda = 140$.

Preceding a detailed analysis of global attenuation and radial profiles of statistics, Vreman (Reference Vreman2016) presented snapshots of instantaneous flow field quantities surrounding one sphere in the simulated array – these graphical depictions were then repeatedly referenced to help link the findings back to fundamental flow physics. A key discussion point was the demarcation of a front and rear face of the sphere, as determined by the instantaneous flow direction in the sphere vicinity. We take this approach one step further by ensemble averaging the PIV velocity fields based on the approximate approach angle of the instantaneous velocity field. To illustrate, consider a set of vector fields associated with a given condition and diameter. The near-droplet region of each field is first analysed to yield the approach angle of the instantaneous flow, $\phi _a$, and then the entire field is rotated about the droplet centroid through $\phi _a$ such that the approach velocity is horizontal and from the left. This is repeated for each vector field, and the resultant ensemble may be temporally averaged in the usual fashion. A statistic (e.g. mean or r.m.s. velocity) at a given point is no longer the temporal average at a fixed spatial location $(x,y)$ but rather the expected value at location $(r,\tilde {\phi })$, where $\tilde {\phi } = 0$ is the axis extending into the incoming flow. Statistics calculated with respect to the angle-adjusted coordinate system are designated with the subscript $\tilde {\phi }$ (e.g. $\langle U_r \rangle _{\tilde {\phi }}$, $\langle U_\phi \rangle _{\tilde {\phi }}$). The purpose of this manipulation is to analyse the modulation problem from the perspective of upstream/downstream/cross-stream regions which, by virtue of their constantly changing locations in a quasi-zero-mean flow, are difficult to elucidate without resorting to the adjustment described above. The instantaneous fields provide useful qualitative information (there is no obvious recirculation behind any droplet in this study, for instance), but building an ensemble sheds quantitative light on the expected/average behaviour.

The above procedure can be executed on unladen fields as well to provide points of comparison. Figure 11(a) depicts the $Re_\lambda = 100$ unladen field, first subjected to the angle adjustment procedure and then ensemble averaged to yield $\langle U_r \rangle _{\tilde {\phi }}$. The spatial distribution of $\langle U_r \rangle _{\tilde {\phi }}$ is easily explained by recalling that the flow is, on average, approaching from the left – hence the negative, positive and negligible values of $\langle U_r \rangle _{\tilde {\phi }}$ along $\tilde {\phi } = 0$, ${\rm \pi}$ and ${\pm }{\rm \pi} /2$, respectively. The distribution of $\langle U_\phi \rangle _{\tilde {\phi }}$ is omitted for brevity since it is analogous to the $\langle U_r \rangle _{\tilde {\phi }}$ field rotated by $-{\rm \pi} /2$. The droplet in figure 11(b) enforces a wake region, characterized by low $\langle U_r \rangle _{\tilde {\phi }}$ along the downstream axis, with a smaller extent of attenuation observed in the upstream region as well. On the other hand, figure 11(c) suggests that the spatial extent of $\langle U_\phi \rangle _{\tilde {\phi }}$ attenuation is slight. In general, figure 11 helps explain the dramatic difference in the attenuation profiles of figure 7 – the wake structure ensures that a protracted region of low radial velocity always exists and, although the orientation and magnitude will vary with time, the wake attenuation effect penetrates further into the field than the reduction in azimuthal velocity, which is driven by shear in the cross-stream directions.

Figure 11. Angle-adjusted mean velocity fields, $\langle U_i \rangle _{\tilde {\phi }}$. The median values of $Re_\lambda$ and $\bar {d}$ (100 and $819\,\mathrm {\mu }{\rm m}$, respectively) are selected for this representative example. (a) Mean radial velocity in the unladen field. The mean azimuthal velocity field is qualitatively similar if the field in (a) is rotated through an angle of $-{\rm \pi} /2$. The (b) radial mean velocity field and (c) azimuthal mean velocity field in the presence of a droplet. The FOV is cropped to the same extent as figures 8–10, but the tilde notation ($\tilde {x},\tilde {y}$) emphasizes the adjustment aspect.

The analysis leading to figure 11 was repeated for all $Re_\lambda$ and droplet sizes, and select mean velocity profiles along the angle-adjusted axes are plotted in figures 12–14. By definition, $\langle U_r \rangle _{\tilde {\phi }}$ is negligible along $\tilde {\phi } = {\rm \pi}/2$ and $3{\rm \pi} /2$, and $\langle U_\phi \rangle _{\tilde {\phi }}$ is negligible along $\tilde {\phi } = 0$ and ${\rm \pi}$ – hence, no profiles are provided for those combinations. These figures help validate the initial observations surrounding figure 11, the most important being the extended spatial region of recovery for $\langle U_r \rangle _{\tilde {\phi }}$ in the upstream and, to a much greater extent, downstream directions (figures 12–14(a,c), respectively). In contrast, the rapid convergence of the $\langle U_\phi \rangle _{\tilde {\phi }}$ laden profiles to the unladen base case (figures 12–14(b,d)) is also significant. In every case, the larger drop lags the smaller one in recovery distance, although the discrepancy may not be as large as expected, given the large droplet is five times larger than the small one. Furthermore, if the laden cases in figures 12–14 were plotted against radial distance from the droplet surface, $r-R$, instead of $r$, the profiles would be brought closer together by $\sim$0.5 mm (the difference between the maximum and minimum values of droplet radius).

Figure 12. Radial profiles of angle-adjusted mean velocity magnitudes for $Re_\lambda = 50$. Profiles are provided for the unladen flow, the smallest droplet ($\bar {d} = 273\,\mathrm {\mu }{\rm m}$) and the largest droplet ($\bar {d} = 1365\,\mathrm {\mu }{\rm m}$). (a) Radial mean velocity in the upstream direction. (b) Azimuthal mean velocity in the ${\rm \pi} /2$ cross-stream direction. (c) Radial mean velocity in the downstream direction with the wake width (less than 90 % recovery) depicted by the light fill. (d) Azimuthal mean velocity in the $3{\rm \pi} /2$ cross-stream direction.

Figure 13. Radial profiles of angle-adjusted mean velocity magnitudes for $Re_\lambda = 100$. See figure 12 caption for details.

Figure 14. Radial profiles of angle-adjusted mean velocity magnitudes for $Re_\lambda = 140$. See figure 12 caption for details.

Perhaps the most striking feature in figures 12–14 is the overall size of the $\langle U_r \rangle _{\tilde {\phi }}$ wake region, depicted in figures 12–14(c) with a light fill. At any radial location, the vertical extent of the fill (which is kept dimensionally consistent across all three figures to facilitate comparisons) depicts the penetration distance of the wake into the cross-stream direction, where the wake is said to exist until $\langle U_r \rangle _{\tilde {\phi }} > 0.9\langle U_r \rangle _{\tilde {\phi },0}$ (this 90 % recovery criterion is revisited shortly). Judging from these figures, one might assume that droplet diameter plays a major role in determining the overall attenuation shell thickness, since the radial wake increases significantly with $\bar {d}$, as expected. It is important to note that the azimuthal component, by virtue of its equivalence to the polar component, carries twice the weight into the TKE equation (2.5), and figures 12–14(b,d) indicate that not only does $\langle U_\phi \rangle _{\tilde {\phi }}$ recover quickly in the cross-stream direction (as previously noted), but there is also far less diameter dependence than what is witnessed in the upstream and downstream radial component. In figure 13, where the large droplet wake dramatically eclipses that of the small droplet in both width and depth, the actual thicknesses of the attenuation regions are nearly equivalent. Indeed, the thickness of an attenuation shell is relatively constant with diameter at a given background turbulence level. This is why smaller droplets affect larger normalized regions – the attenuation shell does not scale with diameter.

The radial extent of attenuation for all droplet diameters in helium flow fields is quantified in figure 15, which plots the shell-averaged normalized TKE, $\overline {(k/k_0)_{\Delta r}}$, vs the normalized radius $r/R$. These plots make clear that faster recovery, in the normalized spatial sense, is expected for larger droplets in highly turbulent fields. As $Re_\lambda$ increases, the kinetic energy value in the surface shell, ${\overline {k_{\Delta r,s}}}$, also increases. This observation is attributable to both the averaging effect of the finite-sized IAs and the absence of a no-slip boundary condition at the liquid–gas interface. Since the TKE at the surface is not identically zero (as it would be for a rigid, solid surface), it is possible that it varies between conditions. We note, however, that the impermeability/kinematic constraint is still valid, and that the overall surface kinetic energy is likely to be very small.

Figure 15. Radial profiles of $k$ in normalized units for all droplet sizes in (a) $Re_\lambda = 50$, (b) $Re_\lambda = 100$ and(c) $Re_\lambda = 140$. The DNS data of Vreman (Reference Vreman2016, figure 6(a)), $Re_\lambda \approx 55$, are included in (a), where $Re_\lambda$ was updated from the stated value of 32 to reflect the present calculation method, and $d/\eta = 2$. Profiles in nitrogen ($Re_\lambda = 180$ and 310) are qualitatively similar.

To assess the spatial extent of kinetic energy recovery, the $r/R$ values corresponding to the first occurrence of ${\overline {(k/k_0)_{\Delta r}}}{} \geq 0.9$ are extracted from figure 15 and plotted in figure 16. Notationally, we introduce $r^*_k$ to designate this radial location, where

(3.4)\begin{equation} r^*_k = \min{\left\{r/R\left(\overline{(k/k_0)_{\Delta r}} \geq 0.9 \right)\right\}}. \end{equation}

At $r^*_k$, the kinetic energy field has recovered to within 10 % of the unladen value. The value of 10 % was selected arbitrarily, but it represents a reasonable combination of a nearly recovered field while avoiding some of the run-to-run variation that exists as ${\overline {(k/k_0)_{\Delta r}}}{} \to 1$. Although ${\overline {(k/k_0)_{\Delta r}}}$ should equal unity at high $r/R$, we note that the far-field value, taken as the average in the ten outermost radial shells, varies between 0.92 and 1.08 across the 21 helium test cases in figure 15, although the average far-field value of the 21 cases is indeed 1.00. The deviation from unity is more severe for nitrogen tests, with an average far-field TKE recovery value of 1.06 (or 6 % greater than the unladen base case). It is not expected that the far-field kinetic energy is physically increased by the presence of a droplet, which is what ${\overline {(k/k_0)_{\Delta r}}}> 1$ suggests. Both Tanaka & Eaton (Reference Tanaka and Eaton2010) and Vreman (Reference Vreman2016) depict ${\overline {(k/k_0)_{\Delta r}}}$ asymptotically approaching a sub-unity value, although the fields of view do not extend beyond $r/R \approx 7$. Hence, the present deviation is best explained by experimental variation, despite every effort to maintain equivalent forcing and ambient conditions between the laden and unladen cases.

Figure 16. Normalized radius required for $k$ to return to within 10 % of the unladen value vs (a) $\bar {d}/\eta _0$ ($C_1 = 6.6$, $C_2 = 0.46$, $R^2 = 0.76$) and (b) $\bar {d}/\lambda _{f,0}$ ($C_1 = 1.0$, $C_2 = 0.62$, $R^2 = 0.87$). The dataset covers $Re_\lambda$ from 50 to 310 (table 1). The lines of best fit include all plotted data.

To help mitigate the effect of this variation, the curves in figure 15 were first normalized by their individual far-field values before extracting and plotting the 90 % recovery radius in figure 16. This approach artificially forces each profile to unity in the far field, but it is justified if the far-field value is assumed free from any droplet effect. If this condition is met (and figure 15 suggests that the profiles have recovered), then the adjustment procedure is equivalent to normalizing by the theoretical $\overline {k_{0,\Delta r}}$ profile of the particular case at hand, rather than the reference unladen case. Some increased scatter in the nitrogen TKE data notwithstanding, the adjustment generally improved the collapse of data but does not affect the interpretation of overall trends.

3.2.2. Dissipation rate

Figures 17–19 illustrate the dissipation rate field – centrally cropped about the droplet to ${\pm }5\,{\rm mm}$ – for the minimum, median and maximum droplet diameters at the three major $Re_\lambda$ levels (50, 100, and 140). For consistency, the minimum and maximum colour levels are set to the approximate unladen value and the maximum expected dissipation, respectively. Many of the observations detailed in the previous section are applicable here, including the overall radial symmetry of the field. There is a minor ‘cloverleaf’ effect visible in figure 17(c) which indicates some angular dependence in the dissipation calculation. This observation is not physical but rather results from the application of Cartesian finite differences and isotropic assumptions in the spatial gradient calculation (§ 2.3.4). The cloverleaf pattern is either reduced or non-existent for the other presented cases. The dissipation rate increases dramatically near the droplet surface – this finding could be inferred from the kinetic energy profiles in the previous section along with the dissipation results in the most relevant studies (Tanaka & Eaton Reference Tanaka and Eaton2010; Hoque et al. Reference Hoque, Mitra, Sathe, Joshi and Evans2016; Vreman Reference Vreman2016). The reader is reminded that the surface shell, despite being relatively thin, is still much coarser than what would be required to resolve the gradients at the droplet surface. Therefore, the surface-shell dissipation, $\overline {\varepsilon _{\Delta r,s}}$, is properly conceptualized as the very-near-field dissipation average (the surface shell is located at $1.1 \leq r/R \leq 1.5$, depending on $\bar {d}$) as opposed to the true surface dissipation, $\varepsilon _s$. Accurate experimental predictions of $\varepsilon _s$, in conditions similar to those presently developed, have not been achieved, due in part to the extremely fine resolution required.

Figure 17. Dissipation rate field surrounding the (a) minimum, (b) median and (c) maximum droplet diameters at $Re_\lambda = 50$.

Figure 18. Dissipation rate field surrounding the (a) minimum, (b) median and (c) maximum droplet diameters at $Re_\lambda = 100$.

Figure 19. Dissipation rate field surrounding the (a) minimum, (b) median and (c) maximum droplet diameters at $Re_\lambda = 140$.

Figure 20 plots the variation of the shell-averaged normalized dissipation rate, ${\overline {(\varepsilon /\varepsilon _0)_{\Delta r}}}$, against $r/R$. Each panel includes an inset to provide enhanced detail in the near-droplet region $(r/R \leq 10)$. In all three turbulent fields, there is a clear separation between ${\bar {d}/\eta _0}$ groups, where the larger droplets recover quicker. In the $Re_\lambda = 50$ field, $\overline {\varepsilon _{\Delta r,s}}$ is 20 to 22 times greater than the unladen value. As $Re_\lambda$ increases, the normalized surface-shell dissipation falls – first to ${\sim }10$ at $Re_\lambda$ = 100 and then to ${\sim }6$ at $Re_\lambda = 140$. These surface-shell dissipation spikes are not as extreme as the hundredfold increase reported by Vreman (Reference Vreman2016), but are still far larger than the values reported in any relevant experimental studies. The inability to resolve the boundary layer gradients is one possible reason for the discrepancy. Furthermore, Vreman (Reference Vreman2016) used rigid, solid spheres. Nevertheless, the degree of overlap in dissipation profiles between Vreman (Reference Vreman2016) and our conditionally closest case (figure 20a) is excellent (keeping in mind that $d/\eta _0 = 2$ for Vreman (Reference Vreman2016), hence justifying the leftward offset from our $\bar {d}/\eta _0 = 1.5$ case), which indicates that the dissipation profile leading up to the surface is being correctly captured. This agreement suggests that dissipation differences near solid vs liquid surfaces may only appear in the immediate vicinity of the surface itself. Furthermore, we should not automatically assume that the surface-shell dissipations are dramatically underpredicted, since the magnitudes are in general agreement with the decaying DNS simulations of Dodd & Ferrante (Reference Dodd and Ferrante2016), which clearly depicted near-droplet regions of normalized dissipation rates above 8 at $Re_\lambda \approx 80$. A follow-up article (Dodd & Jofre Reference Dodd and Jofre2019) presented average values of the dissipation rate conditioned upon distance from the droplet interface, and surface dissipation spikes between 5 and 40 are noted at early simulation times, depending on the Weber number, the density ratio and the viscosity ratio.

Figure 20. Radial profiles of $\varepsilon$ in normalized units for all droplet sizes in (a) $Re_\lambda = 50$, (b) $Re_\lambda = 100$ and(c) $Re_\lambda = 140$. The DNS data of Vreman (Reference Vreman2016, figure 6(d)), $Re_\lambda \approx 55$, are included in (a), where $Re_\lambda$ was updated from the stated value of 32 to reflect the present calculation method, and $d/\eta = 2$. The near interface region $(r/R \leq 10)$ is expanded in an inset for each case to better illustrate the trend separation of the various droplet diameters. Plot legends denote only the minimum and maximum diameters – see the analogous figure 15 for a full listing.

The recovery of dissipation is quantified using the 10 % criterion once again, where figure 21(a,b) plot the first $r/R$ coordinate that satisfies ${\overline {(\varepsilon /\varepsilon _0)_{\Delta r}}}{} \leq 1.1$ against ${\bar {d}/\eta _0}$ and ${\bar {d}/\lambda _{f,0}}$, respectively. The far-field correction discussed in § 3.2.1 is once again implemented. Adopting the same notation style as (3.4),

(3.5)\begin{equation} r^*_\varepsilon = \min{\left\{r/R\left(\overline{(\varepsilon/\varepsilon_0)_{\Delta r}} \leq 1.1 \right)\right\}}. \end{equation}

At $r^*_\varepsilon$, the dissipation field has recovered to within 10 % of the unladen value. The recovery trends for dissipation are clearly similar to those for kinetic energy, although $r^*_\varepsilon \geq r^*_k$ in all 21 combinations of $Re_\lambda$ and $\bar {d}$ presented thus far (excluding the nitrogen cases, where $r^*_\varepsilon$ cannot be computed).

Figure 21. Normalized radius required for $\varepsilon$ to return to within 10 % of the unladen value vs (a) $\bar {d}/\eta _0$ (${C_1 = 7.7}$, $C_2 = 0.53$, $R^2 = 0.93$) and (b) $\bar {d}/\lambda _{f,0}$ ($C_1 = 1.2$, $C_2 = 0.65$, $R^2 = 0.99$). Dissipation is not calculated for nitrogen. (c) Normalized dissipation rate in the surface shell vs ${\bar {d}/\eta _0}$. There is no qualitative difference when plotting surface-shell dissipation against ${\bar {d}/\lambda _{f,0}}$ instead of ${\bar {d}/\eta _0}$.

Returning briefly to the surface-shell dissipation, figure 21(c) indicates that $\overline {\varepsilon _{\Delta r,s}}$ is primarily a function of $Re_\lambda$ as opposed to ${\bar {d}/\eta _0}$. As $Re_\lambda$ increases, $\overline {(\varepsilon /\varepsilon _0)_{\Delta r, s}}$ appears to peak at larger values of ${\bar {d}/\eta _0}$, but the overall separation into distinct groups based on $Re_\lambda$ stands in stark contrast to the recovery plots in figure 21(a,b). According to Vreman (Reference Vreman2016), the dissipation should increase as the particle diameter decreases – this behaviour is not witnessed presently. The maximum dissipation in a surface shell is approximately $2200\,{\rm m}^2\,{\rm s}^{-3}$. The Kolmogorov scale in this shell is ${\sim }170\,\mathrm {\mu }{\rm m}$ which results in a PIV spatial resolution, $\Delta x/\eta$, of ${\sim } 1.1$. Data from a recent study (Verwey & Birouk Reference Verwey and Birouk2022) suggest a dissipation underprediction of up to 20 % at that resolution – this is not nearly sufficient to explain the surface dissipation discrepancy. The observation is, therefore, believed to be a physical phenomenon.

Figures 16 and 21 plotted recovery radii vs ${\bar {d}/\eta _0}$ and ${\bar {d}/\lambda _{f,0}}$. In both figures, the data collapse is superior for ${\bar {d}/\lambda _{f,0}}$ based on the $R^2$ values. Manipulation of the $d/\eta$ and $d/\lambda$ ratios using (3.1) and (3.2) along with the definitions of $\eta$ and $Re_t$, where $\eta = (\nu ^3/\varepsilon )^{1/4}$ and $Re_t = dk^{1/2}/\nu$, leads to

(3.6)\begin{gather} \frac{d}{\eta} \sim \left(\frac{d}{L}\right)^{1/4}Re_t^{3/4}, \end{gather}

(3.7)\begin{gather} \frac{d}{\lambda} \sim \left(\frac{d}{L}\right)^{1/2}Re_t^{1/2}. \end{gather}

The superior scaling of $r^*$ vs ${\bar {d}/\lambda _{f,0}}$ implies that the $d/L$ ratio and $Re_t$ play roughly equivalent roles in determining the extent of attenuation, as both are raised to the same power of 1/2. Interestingly, these are the two parameters suggested by Gore & Crowe (Reference Gore and Crowe1989) and Hetsroni (Reference Hetsroni1989) as being most influential in the question of augmentation vs attenuation in particle-laden pipe and jet flows (it should be noted that Hetsroni (Reference Hetsroni1989) used an unusual Reynolds number formulation based on the density difference between phases). Furthermore, as disclosed in the captions of figures 16 and 21, the scaling factor $C_1$ in both $r^*$ vs ${\bar {d}/\lambda _{f,0}}$ trend lines is close to unity, hence

(3.8)\begin{equation} r^* \approx ({\bar{d}/\lambda_{f,0}})^{{-}C_2}. \end{equation}

Equation (3.8) suggests that as ${\bar {d}/\lambda _{f,0}}{} \to 1$ from smaller values, $r^* \to 1$. In other words, as the droplet diameter becomes comparable to the unladen Taylor length scale, the region of attenuation shrinks to a potentially negligible value. Additionally, the ${\bar {d}/\lambda _{f,0}} \sim 1$ regime may signify a transition between attenuation and augmentation behaviour. Consider the experimental results of Hoque et al. (Reference Hoque, Mitra, Sathe, Joshi and Evans2016), who used a ZMF/fixed sphere configuration – they found a negligible TKE modulation effect for a 1 mm sphere, which corresponded to $(0.4 < {\bar {d}/\lambda _{f,0}}{} < 1.0)$. The three larger spheres ($d = 3, 5\ {\rm and}\ 8\ {\rm mm}, 1.3 < {\bar {d}/\lambda _{f,0}} < 7.8$) augmented the TKE. Adhering to the classic interpretation of the Taylor scale as the demarcation between viscous and inviscid regimes, the data suggest that the relative strength of viscous phenomena is responsible for attenuating the turbulent field surrounding a sub-Taylor-scale droplet. On the other hand, when $d > \lambda$, the viscosity effect is insufficient to prevent wake formation/shedding, which ultimately enhances the turbulent energy of the flow.

3.3.1. Evaporation rates

For context, the evaporation rates of ethanol droplets were collected and are presented in figure 22. The droplets were imaged using a standard backlight approach with a Fastec IL5Q camera and a Questar QM-100 long distance microscope. Once the desired turbulence level was set, a large droplet of ethanol was deposited on the cross-fibres and recorded at 24 fps until the evaporation process was completed. The same diameter-binning scheme was used to extract the evaporation rate constant, $K$, where $K = -\mathrm {d}(d^2)/\mathrm {d}t$ and is determined via a linear least squares fit as depicted in figure 22(a). For instance, all images with diameters between 1274 and $1456\,\mathrm {\mu }{\rm m}$ were used to construct the linear fit that defines $K$ for the $\bar {d} = 1365\,\mathrm {\mu }{\rm m}$ bin. The evaporation test was repeated five times per turbulent field and the average $K$ value associated with each diameter bin, $\bar {K}$, is presented in figure 22(b). The evaporation progression was observed to slow down in a non-physical way when the droplet diameter had fallen into the smallest bin – this was likely due to nodal interference or a build up of rhodamine B powder, and data are not presented for $\bar {d} = 273\,\mathrm {\mu }{\rm m}$. Although exceeding the scope of the present investigation, it is interesting to note the apparent collapse of $\bar {K}$ in figure 22(b) in the ${\bar {d}/\eta _0}{} \approx 1$ region. This suggests that the energy of the unladen field becomes less impactful (or perhaps altogether irrelevant) as the droplet diameter approaches and falls below the Kolmogorov scale. The limited ${\bar {d}/\eta _0}$ range in figure 22(b) would need to be expanded in both directions to better analyse this possibility.

Figure 22. (a) Extracting the evaporation rate from an example case ($Re_\lambda = 50$). One out of every 40 data points are shown for clarity. (b) Average $K$ values (5 runs).

3.3.2. Comparisons with non-volatile droplets

The 90 % recovery radii of the turbulent fields surrounding an ethanol drop are presented, along with all available previous data, in figure 23(a,b). As per the discussion at the end of § 3.2.2, ${\bar {d}/\lambda _{f,0}}$ is the superior scaling parameter – this remains true for ethanol and, as a result, plots of $r^*$ vs ${\bar {d}/\eta _0}$ are omitted for brevity. The rapidly evaporating ethanol drop has no effect on the far-field recovery. This is not surprising for small length-scale ratios, but verifying the collapse of water and ethanol data for large ratios – where the ethanol vaporization rate, $\dot {m}_v$, is high $(\dot {m}_v \propto d$ in classic Maxwell/Stefan formulations, with increasing diameter dependence in turbulence) and the recovery radius low – is useful.

Figure 23. Normalized radius required for (a) the TKE ($C_1 = 1.0$, $C_2 = 0.61$, $R^2 = 0.83$) and (b) the dissipation ($C_1 = 1.4$, $C_2 = 0.55$, $R^2 = 0.89$) to return to within 10 % of their unladen values. Trend lines are fitted using all presented data. (c) Ratio of dissipation rate in the surface shell of water over ethanol.

On the other hand, figure 23(c) indicates that the dissipation in the ethanol surface shell is reduced by a factor of ${\sim }$1.4–3 as compared with the slowly evaporating water droplets. Evidently, the disparity in $\overline {\varepsilon _{\Delta r,s}}$ is lessened as the length-scale ratio increases. While the vapour mass flux, $\dot {m}_v''$, is proportional to $d^{-1}$ in convection-free environments, there is a strong diameter effect in turbulent droplet vaporization. However, unless the turbulent evaporation rate increases in a nonlinear fashion with diameter (and experimental evidence indicates a linear relationship is indeed most reasonable Verwey & Birouk Reference Verwey and Birouk2017), the mass flux will still increase as the diameter decreases, which seemingly helps damp the near-surface dissipation spike. Indeed, the mass-average radial gas velocity at the surface, $V_r$, is a monotonically decreasing function of diameter, with $V_r$ estimates falling between 20 and $70\,{\rm mm}\,{\rm s}^{-1}$ across the ethanol tests. It will be shown that these velocities are two to four orders of magnitude larger than the velocities associated with drop motion and surface regression. It is generally understood that turbulence is more effective for low-volatility liquids – hence, as ${\bar {d}/\eta _0}$ increases, the relative increase in water vapour mass flux likely outpaces that of ethanol, leading to a surface dissipation ratio that decreases with ${\bar {d}/\eta _0}$ as witnessed in figure 23(c). The reduction of dissipation due to evaporation is consistent with the limited available literature (Mashayek Reference Mashayek1998; Wang & Rutland Reference Wang and Rutland2006), although the near-interface region, as opposed to global trends (which could be affected by phenomena such as droplet disappearance), has not been the focus of any studies to date. Shao et al. (Reference Shao, Jin and Luo2022) mentioned that the Stefan flow (bulk outward mass flux) at the surface of an evaporating droplet reduces the velocity gradients/dissipation which, although evident in the gas phase, was far more prevalent in the liquid phase. The present finding that evaporation significantly reduces the surface-shell dissipation in the gas phase is, therefore, expected to be useful for benchmarking future droplet-resolved simulations that focus on the interface region.

The differences in surface-shell dissipation between water and ethanol could also be attributable to the retracting surface, as $\dot {m}_v'' \propto \mathrm {d}R/\mathrm {d}t$. It is not difficult to envision a set of experiments that discriminate between the effects of surface regression and mass addition/flux due to evaporation. Yet, as is often the case, the design and execution of such a campaign would most certainly be challenging. A non-volatile liquid slowly withdrawn into a capillary tube would simulate regression in the absence of evaporation, just as a porous sphere – once a very popular test configuration – would help gauge the effect of mass transfer without boundary movement. Aside from the tubing needed to retract or supply liquid, the primary difficulty would likely be thoroughly populating the important sub-Kolmogorov regime. For example, porous spheres typically had diameters of the order of 1–10 mm. Realistically, both effects are almost always guaranteed to occur in tandem, so the data presented here are expected to be useful, even if the specific mechanism cannot be elucidated without additional studies. We note that the surface regression associated with ethanol droplets is of order 0.01 mm s$^{-1}$, which is two orders of magnitude smaller than the characteristic velocities associated with droplet movement on the fibres (discussed below), and three to four orders of magnitude smaller than the characteristic radial velocity associated with evaporation (discussed above).

Discrepancies in various properties including liquid density, $\rho _l$, and viscosity, $\mu _l$, the near-interface gas mixture density, $\rho _m$, and viscosity, $\mu _m$, and the liquid surface tension, $\sigma$, may also influence the interface results. All properties require a reasonable estimate of surface temperature, $T_s$. The surface temperatures of water and ethanol droplets are estimated to be ${\sim }285\,{\rm K}$ and ${\sim }280\,{\rm K}$, respectively, using the iterative technique outlined in Verwey & Birouk (Reference Verwey and Birouk2021). These $T_s$ values are only strictly applicable in convection-free environments, however, the present cursory analysis assumes that $T_s$ does not change dramatically in turbulence. Table 2 summarizes the aforementioned properties. The source of the properties are disclosed in the table caption.

Table 2. Estimated properties at the liquid–helium interface. All properties evaluated at $T_s$. The turbulent gas-phase Weber number, $We = \rho k d/\sigma$, is calculated using the density of helium at $T_\infty = 300\,{\rm K}$($\rho = 0.16\,{\rm kg}\,{\rm m}^{-3}$), the maximum unladen value of TKE ($k = 6.45\,{\rm m}^2\,{\rm s}^{-2}$) and the maximum diameter of $1456\,\mathrm {\mu }{\rm m}$. Liquid density and viscosity for water are from Bergman et al. (Reference Bergman, Lavine, Incropera and Dewitt2011); water surface tension is from Vargaftik, Volkov & Voljak (Reference Vargaftik, Volkov and Voljak1983). The issue of air vs helium as the ambient gas seemingly has little effect on $\sigma$, at least for water (Claussen Reference Claussen1967). Liquid density, viscosity and surface tension for ethanol are from Gonçalves et al. (Reference Gonçalves, Trindade, Costa, Bernardo, Johnson, Fonseca and Ferreira2010). The gas mixture density was calculated assuming saturation and applying the Wagner formula for vapour pressure (Poling, Prausnitz & O'Connell Reference Poling, Prausnitz and O'Connell2001). The gas mixture viscosity was estimated using the Lucas method (Poling et al. Reference Poling, Prausnitz and O'Connell2001).

The liquid density affects the inertial response of the droplet to aerodynamic drag. To gauge how gas-phase velocities are translated to the droplet, the shift in droplet centroid between sequential images is divided by the inter-frame time (0.1 s), thus providing average planar velocities, $U_c$ and $V_c$, between successive PIV images. The out-of-plane velocity, $W_c$, is assumed equal to the horizontal in-plane velocity $U_c$, and the overall velocity magnitude is calculated from the three components as $|\boldsymbol {U}_c| = \sqrt {U_c^2 + V_c^2 + W_c^2}$. $|\boldsymbol {U}_c|$ is then conditionally averaged based on fan speed and diameter, and these averages are found to be of the order of 1 mm s$^{-1}$ (approximately three orders of magnitude smaller than the characteristic turbulent velocity $k^{1/2}$). Average ethanol velocities are generally higher than water, but in no instance does this difference exceed 0.8 mm s$^{-1}$. The small velocities associated with the droplet indicate that liquid density (and, by extension, the modulation of TKE by kinetic energy exchange from the carrier phase to the bulk motion of the droplet) can likely be discarded as a factor.

Likewise, the liquid viscosity cannot be responsible for the decreased dissipation near the ethanol droplet surface, since $\mu _l$ is actually higher for ethanol. In their numerical study of droplet-laden decaying HIT, Dodd & Ferrante (Reference Dodd and Ferrante2016) suggested that the carrier-phase surface velocity gradient increases with viscosity ratio, $\gamma$, where $\gamma = \mu _l/\mu _m$. From this perspective, ethanol maintains a 30 % advantage in $\gamma$ over water, which reiterates that the reduction in surface-shell dissipation is not due to viscosity effects. The maximum Weber number is less than 0.1 for both liquids, hence surface deformations and distortions (which are pathways of TKE exchange as per Dodd & Ferrante (Reference Dodd and Ferrante2016)) are minimal. This is further verified by checking the circularity values, $C$, of the high-magnification images used to produce figure 22, where $C = 1$ for a perfect circle. Circularity values rarely fall below 0.98, and if they do, it is generally due to the fibres being removed imperfectly from the image as the droplet becomes small (in other words, not due to flow-induced surface distortions). In both water and ethanol cases, the presence of the vapour increases the gas mixture density while decreasing the viscosity. Ultimately, the kinematic viscosity of the gas mixture near the surface, $\nu _m = \mu _m/\rho _m$, is 22 % smaller for the helium–ethanol system – this decrease is primary driven by $\rho _m$ as evident in table 2. Because a constant kinematic viscosity has been assumed thus far in all dissipation calculations (that of pure helium at $T_\infty = 300\,{\rm K}$), the smaller surface kinematic viscosity associated with the helium–ethanol system suggests that the surface-shell dissipation ratios presented in figure 23 are probably underpredicted. In summary, all evidence points to the mass ejection phenomenon as the primary driver in surface-shell dissipation reduction for an ethanol droplet.

Extensive spatial heat maps and individual radial profiles of $k$ and $\varepsilon$ are not presented for the ethanol fields, as they qualitatively mirror those for water and the important similarities/differences are compactly quantified in figure 23. There is, however, one final perspective that deserves mention, as it further distinguishes the ethanol case from water. Figure 24(a) presents a spatial heat map of the isotropy ratio for an example ethanol test. A noticeable characteristic is the narrow band surrounding the drop where $I_r$ reaches a minimum before rising to the final surface-shell value. Although the scale is different and the present figure cropped to a greater extent, it is apparent that this behaviour does not exist for the water droplet in figure 6(b). In fact, no water droplet is associated with a protracted minimum region. The universality of this phenomenon for the ethanol drop is confirmed in figure 24(b–d), which plot the isotropy ratio in the familiar radial-shell format. The near-surface increase in $I_r$ indicates that the evaporation process mitigates the strong damping of $\langle u_r^2 \rangle ^{1/2}$ which was established in § 3.2.1. Radial profiles of $k$ are still monotonically decreasing functions as $r \to R$ from the far field, but the attenuation is certainly lessened when compared with water. Once again, the individual contributions of surface regression vs mass addition is unknown, but it is interesting to note that $U_r$ is no longer universally zero at the receding droplet surface in the fixed lab frame. The faster return to isotropy at elevated $Re_\lambda$ helps explain why high Reynolds number studies such as Petersen et al. (Reference Petersen, Baker and Coletti2019) ($200 \leq Re_\lambda \leq 500$) report a more isotropic field than low Reynolds experiments (e.g. Poelma et al. (Reference Poelma, Westerweel and Ooms2007), $Re_\lambda \approx 27$).

Figure 24. (a) Large-scale isotropy ratio field for an example ethanol droplet ($Re_\lambda = 100, {\bar {d}/\eta _0}{} = 2.0$). Panels (b–d) plot the radial profiles of the isotropy ratio for all ethanol tests. The dashed green line in (c) corresponds to the example image in (a). Legend entries in (b–d) identify the minimum and maximum ${\bar {d}/\eta _0}$ trends – see figure 15 for a full listing.