3.1 Droplet shape distribution

In order to quantify the degree of deformation in the droplet shape, we make use of a non-dimensional deformation parameter defined as $D=(d_{3}-d_{1})/(d_{3}+d_{1})$ (Taylor Reference Taylor1932, Reference Taylor1934; Guido & Villone Reference Guido and Villone1998; Maffettone & Minale Reference Maffettone and Minale1998; Guido, Minale & Maffettone Reference Guido, Minale and Maffettone2000; Biferale et al. Reference Biferale, Meneveau and Verzicco2014). When $D\sim 0$ , the droplet is close to spherical, while $D\gg 0$ implies the droplet is highly deformed. In order to properly quantify the correlation between the spatial position of the droplets and the degree of deformation, we plot the radial profiles of azimuthally, axially and time-averaged deformation parameter for droplets with four different $Ca$ in figure 1. The radial positions of the drops are normalised using $\tilde{r}=(r-r_{i})/(r_{o}-r_{i})$ ; the drop is close to the inner cylinder when $\tilde{r}\sim 0$ and close to the outer cylinder when $\tilde{r}\sim 1$ . Along with the averaged profiles we show the inner boundary layer (IBL) and outer boundary layer (OBL) thickness using grey shaded areas which have been computed following the approach described in Ostilla-Mónico et al. (Reference Ostilla-Mónico, Stevens, Grossmann, Verzicco and Lohse2013, Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ), Spandan et al. (Reference Spandan, Ostilla-Mónico, Verzicco and Lohse2016).

Figure 2. Zoomed in plot of figure 1(a). The schematic on the right shows interaction of a highly deformed drop (top) and a relatively less deformed drop (bottom) with the wall. The dotted line around each droplet shows the hypothetical bounding box constructed around each droplet. The centre of mass of a less deformed drop can approach more closely to the wall as compared to a highly deformed drop.

We can clearly observe in figure 1 that the capillary number $Ca$ has a strong influence on the mean deformation parameter $D$ . As expected, $D$ increases with increasing $Ca$ due to weakening surface tension forces. Additionally, as noted from figure 1 a clear indication of the spatial dependence of the deformation parameter $D$ is observed. $D$ is higher near the boundary layers (marked by grey shaded areas) for both Reynolds numbers as compared to the bulk where it is almost three times lower. Additionally, we also find that when $Ca=0.2$ and $Ca=0.3$ , the peak of the deformation parameter $D$ lies away from the inner cylinder. To observe this deviation more clearly, in figure 2 we show a zoomed-in plot of figure 1(a) and it can be clearly seen that for $Ca=0.2$ and $Ca=0.3$ there is a slight increase in $D$ away from the wall and the occurrence of this peak moves away from the wall with increasing $Ca$ . This is just a consequence of the modelling of interaction of the droplet with the wall. Since we assume elastic collision between the bounding box of a deformed droplet and the wall, the centre of mass of a highly deformed droplet lies farther away from the wall as compared to a less deformed droplet as shown in the schematic of figure 2. Indeed, this also depends on the relative orientation of the droplets with the wall and we will analyse this in detail in later sections. Through several test simulations it has been ensured that the shift in peak of $D$ is indeed due to the collision and not due to any additional boundary layer dynamics (e.g. droplets modelled without any collision do not show any shift in the peak of $D$ ). This argument is further strengthened in § 3.3 where we study the deformation of more viscous drops.

Figure 3. Azimuthally, radially and time-averaged axial profiles of the deformation parameter ( $D$ ) for four different $Ca$ ; (a) $Re_{i}=2500$ (b) $Re_{i}=5000$ . The viscosity ratio $\unicode[STIX]{x1D707}=1$ . The direction of the arrow indicates increasing $Ca$ (refer to figure 1 for description of the markers). Instantaneous snapshots of the azimuthal velocity of the corresponding $Re_{i}$ are shown on the top of each plot to better visualise the plume ejections and the Taylor rolls along with a quiver plot of half the domain to show the direction of flow. The colour scale shows the instantaneous azimuthal velocity.

Figure 4. Azimuthally and time-averaged profiles of the viscous dissipation: (a) radial profiles (b) axial profiles. Net dissipation is computed as a volume and time average of local dissipation rate and is normalised using its corresponding maximum value $\hat{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}/\text{max}(\unicode[STIX]{x1D716})$ .

Similar to figure 1 we plot axial profiles of the azimuthally, radially and time-averaged deformation parameter $D$ as a function of the normalised axial position $\tilde{z}=z/d$ for $Re_{i}=2500$ and 5000 and for different $Ca$ in figure 3. When $Re_{i}=2500$ (cf. bottom panel of figure 3 a), we observe that regardless of the capillary number, $D$ is higher near the plume ejection regions of the inner cylinder i.e. $\tilde{z}\sim 1$ and 3 (also refer to central and top panels in figure 3 a) which show an instantaneous snapshot of the carrier phase azimuthal velocity along with a quiver plot of a zoomed-in section). This is a result of relatively high strain rates stretching the ellipsoidal drops in the plume ejection region near the inner cylinder; an effect likely to be produced by the Taylor rolls which locally make the boundary layer produced by the cylinder rotation thinner or thicker. A similar observation is made even for $Re_{i}=5000$ in figure 3(b) where the plume ejection regions near the inner cylinder are at approximately $\tilde{z}=1.75$ and 3.75. Although relatively high strain rates are observed in the plume ejection regions, the plumes themselves which extend from the inner cylinder into the bulk have a relatively lower strain rate owing to a strong azimuthal velocity component. This causes a sudden drop in $D$ at the exact axial position where the plumes extend into the bulk, an effect which is seen very clearly for $Re_{i}=2500$ due to the more coherent nature of the plumes (figure 3 a). However, when the Reynolds number is increased the plumes are more erratic and turbulent and the sudden drop in $D$ gets averaged out and is not observed any more. The differences in plume structure in the bulk of the flow for both Reynolds number is clearly visible in the instantaneous snapshots of the azimuthal velocity in figure 3.

Deformation is directly related to the strain rates experienced by the droplets. In order the further understand the correlation between the strain rates in the flow and the net deformation observed we plot the averaged profiles of normalised viscous dissipation (square of the magnitude of the strain rate) against the radial and axial position. The total dissipation per unit mass is computed as a volume and time average of the local dissipation rate (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007b ). After spatial averaging the net dissipation is normalised using its corresponding maximum value $\hat{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}/\text{max}(\unicode[STIX]{x1D716})$ and is plotted against the normalised radial position in figure 4(a) and the normalised axial position in figure 4(b). On comparing the averaged radial and axial profiles of the deformation parameter (figures 1 and 3) along with the normalised dissipation (figures 4(a) and (b)) a very strong correlation is observed between both quantities, thus confirming the hypothesis that high local strain rates are crucial for strong drop deformation.

Until now we have looked at the variation of the deformation parameter $D$ with the radial and axial distance in the TC domain. Although $D$ can be used to study whether the droplet is close to spherical or highly deformed, it cannot be used to distinguish between oblate (disk-like) and prolate (cigar-like) droplets. In order to make this distinction, Biferale et al. (Reference Biferale, Meneveau and Verzicco2014) used the $s^{\ast }$ parameter which was introduced by Lund & Rogers (Reference Lund and Rogers1994) to quantify the probability distribution of strain rate tensor eigenvalues in turbulent flows. The shapes of the droplets can also be categorised based on simple ratios of the semi-axes i.e. $d_{2}/d_{1}$ and $d_{3}/d_{1}$ (where $d_{1}<d_{2}<d_{3}$ ). When the droplet is close to spherical $d_{2}/d_{1}\sim d_{3}/d_{1}\sim 1$ , while for deformed droplets both $d_{2}/d_{1}>1$ and $d_{3}/d_{1}>1$ . When the droplets are deformed they can additionally be categorised into oblate (disk-like) ellipsoids when $d_{2}/d_{1}\sim d_{3}/d_{1}$ and prolate (cigar-like) ellipsoids when $d_{2}/d_{1}\ll d_{3}/d_{1}$ .

Figure 5. Probability distribution function (p.d.f.) of the ratio of semi-axes ( $d_{1}<d_{2}<d_{3}$ ) for $Re_{i}=2500$ and $\hat{\unicode[STIX]{x1D707}}=1$ : (a–c) $Ca=0.05$ , (d–f) $Ca=0.2$ . The three panels for each $Ca$ correspond to droplets in the IBL, bulk and OBL.

In figure 5, we plot the p.d.f.s of the ratio of semi-axes ( $d_{2}/d_{1}$ and $d_{3}/d_{1}$ ) of droplets for two different capillary numbers $Ca=0.05$ and 0.2 while $Re_{i}=2500$ and $\hat{\unicode[STIX]{x1D707}}=1$ . The three different panels for each capillary number correspond to the position of the droplet in the IBL, bulk region (BULK) and the OBL. When $Ca=0.05$ , the droplets are almost spheroidal or in other words axis-symmetric (i.e. $d_{2}/d_{1}\sim 1$ ). In the bulk, where the strain rates are relatively weaker as compared to the boundary layer region, the droplet is close to being spherical with minimum deformation (i.e. $d_{2}/d_{1}\sim d_{3}/d_{1}\sim 1$ ). Additionally, the p.d.f.s of $d_{3}/d_{1}$ in the IBL and OBL have much wider tails indicating that the droplets are more prolate in comparison to the bulk region. Biferale et al. (Reference Biferale, Meneveau and Verzicco2014) found that with increasing capillary number ellipsoidal drops in homogenous isotropic turbulence tend to be more cigar-like or prolate where the major axis ( $d_{3}$ ) is relatively much larger than the other two axes ( $d_{1}$ , $d_{2}$ ). A similar behaviour can be observed in figure 5(d–f) when the capillary number is increased to $Ca=0.2$ . The tails of $d_{3}/d_{1}$ become much wider in the IBL and OBL regions as compared to $Ca=0.05$ which is a consequence of the weaker surface tension forces for higher capillary number drops. An additional consequence is that the droplets now tend to become less spheroidal and more triaxial.

Figure 6. Probability distribution functions of the streamwise orientation of the major axis of droplets for $Ca=0.05$ and $Ca=0.2$ : (a–c) $Re_{i}=2500$ , (b–d) $Re_{i}=5000$ . The three panels for each $Re_{i}$ correspond to droplets in the IBL, bulk and the OBL.

3.2 Orientational behaviour of the drops

In addition to the shape and size of the ellipsoidal droplets, it would also be interesting to understand their orientational preference. A number of studies (Mortensen et al. Reference Mortensen, Andersson, Gillissen and Boersma2008; Marchioli, Fantoni & Soldati Reference Marchioli, Fantoni and Soldati2010; Njobuenwu & Fairweather Reference Njobuenwu and Fairweather2015) investigating the orientation behaviour of axis-symmetric solid ellipsoidal particles in wall-bounded turbulent flows found that the major axis of the particles tend to align with the streamwise direction near the wall. Rigid rod-/fibre-like particles immersed in a isotropic turbulent flow align with the vorticity and the strain rate eigenvectors and their orientational behaviour has a strong dependence on the shape (Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012; Byron et al. Reference Byron, Einarsson, Gustavsson, Voth, Mehlig and Variano2015; Ni et al. Reference Ni, Kramel, Ouellette and Voth2015). The shape and size of solid ellipsoidal particles does not depend on their spatial position and underlying flow conditions. However, for deforming droplets the orientation dynamics can be strongly coupled to the shape evolution which would in turn depend on the local flow conditions. Biferale et al. (Reference Biferale, Meneveau and Verzicco2014) found that neutrally buoyant ellipsoidal droplets dispersed into a homogenous isotropic turbulent flow aligned with the most extensional strain rate eigendirection and the vorticity. With an increase in the capillary number $Ca$ the alignment is lost due to an increase in the interfacial relaxation time.

Figure 7. Probability distribution functions of the cosine of the angle between the eigenvector of the major axis ( $\boldsymbol{e}^{\mathbf{3}}$ ) of the ellipsoid and three strain rate eigendirections ( $\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{1}},\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{2}},\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{3}}$ )  $Re_{i}=2500~\hat{\unicode[STIX]{x1D707}}=1$ for (a–c) $Ca=0.05$ and (d–f) $Ca=0.2$ .

Figure 8. Probability distribution functions of the cosine of the angle between the eigenvector of the semi-major axis ( $\boldsymbol{e}^{\mathbf{3}}$ ) of the ellipsoid and the vorticity ( $\unicode[STIX]{x1D734}$ )  $\hat{\unicode[STIX]{x1D707}}=1$ for (a–c) $Re_{i}=2500$ and (d–f) $Re_{i}=5000$ .

Here we describe the relative orientation of the major axis of the ellipsoidal droplets with respect to the coordinate system in the form of direction cosines of the angle between them i.e. $l=\cos \unicode[STIX]{x1D703}_{\hat{\unicode[STIX]{x1D703}}}$ , $m=\cos \unicode[STIX]{x1D703}_{\hat{r}}$ , $n=\cos \unicode[STIX]{x1D703}_{\hat{z}}$ , where $l$ , $m$ , $n$ are the direction cosine of the orientation with $\hat{\unicode[STIX]{x1D703}}$ (streamwise), $\hat{r}$ (wall-normal) and $\hat{z}$ (spanwise) directions, respectively. In order to study the statistics of droplet orientation in detail, we plot the p.d.f.s of the cosine of the angle between the droplet major axis and the streamwise direction in figure 6. We observe that the droplets tend to be strongly aligned with the streamwise direction (i.e. $l\sim 1.0$ ) in the IBL and OBL. As can be seen from figure 6(a,d), the alignment is more pronounced in the case of $Re_{i}=2500$ as compared to $Re_{i}=5000$ . This may be possible due to the more erratic and time-dependent nature of the boundary layer region and plume ejections from the inner cylinder in the case of $Re_{i}=5000$ as was also seen in the contour plots of figure 3(b). A similar behaviour is also seen in the case of solid ellipsoidal particles in the vicinity of a wall in a turbulent channel flow (Mortensen et al. Reference Mortensen, Andersson, Gillissen and Boersma2008; Marchioli et al. Reference Marchioli, Fantoni and Soldati2010; Njobuenwu & Fairweather Reference Njobuenwu and Fairweather2015). The capillary number $Ca$ seems to have minimal influence on the orientational behaviour of drops suggesting that it is only dominant in the deformation process and not in the effective orientation of the drops. Since the drops are sub-Kolmogorov the drops experience uniform local flow conditions or in other words sample only the linear velocity gradients and thus the $Ca$ number plays an important role only in deforming the droplet but not in its relative orientation especially given the fact that $Ca$ is computed based solely on the ratios of the stretching forces and the retracting surface tension forces. Thus, no matter how much the droplet deforms the major axis is always oriented with the streamwise direction in the boundary layers due to the strong deformation of the underlying fluid material element in the boundary layers. In the bulk, the orientation of the droplets is highly isotropic showing no specific preference in alignment with the streamwise direction which is a consequence of the efficient mixing provided by the Taylor rolls.

Now we look at the orientation statistics of the major axis with respect to the local strain rate eigendirections and vorticity. In figures 7 and 8 we plot the p.d.f.s of the cosine of the angle between the eigenvector of the major axis ( $\boldsymbol{e}^{\mathbf{3}}$ ), the three strain rate eigendirections ( $\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{1}},\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{2}},\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{3}}$ , where $\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{1}}$ and $\boldsymbol{e}_{\boldsymbol{s}}^{3}$ are the compressive and extensional strain rates, respectively) and vorticity ( $\unicode[STIX]{x1D74E}$ ), respectively. In homogenous isotropic turbulence, Biferale et al. (Reference Biferale, Meneveau and Verzicco2014) found that the major axis of neutrally buoyant ellipsoidal droplets align with both the most extensional strain rate eigendirection and vorticity. In figures 7 and 8 we find that the preference of alignment with the strain rate eigendirections and vorticity depends again strongly on its relative spatial location.

In figure 7, we consider two different configurations with capillary numbers $Ca=0.05$ and $Ca=0.2$ for a fixed Reynolds number $Re_{i}=2500$ . In the boundary layer regions the ellipsoid major axis has a strong preference to align with the extensional strain rate eigendirection (here the extensional strain rate direction in the boundary layers is the azimuthal direction). In the bulk, the preference is relatively more uniform while some traces of droplets aligning with $\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{3}}$ can still be found, an indication that there seems to be a competition between the extensional strain rates trying to align itself with the droplet major axis and the homogenising nature of the Taylor rolls. This effect is clearly seen when $Ca=0.2$ given the prolate nature of the droplets (cf. figure 5). While the ellipsoidal droplets tend to align with the most extensional strain rate direction in the IBL region (figure 7 a,d) they prefer to be aligned orthogonally with the vorticity (figure 8 a,d). This is different from the behaviour of solid axis-symmetric ellipsoidal particles in an isotropic turbulent flow where rod-like particles tend to align with the vorticity (Shin & Koch Reference Shin and Koch2005; Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012; Chevillard & Meneveau Reference Chevillard and Meneveau2013). The difference in behaviour is a consequence of the inhomogeneity and anisotropy of the carrier flow along with the ellipsoidal droplets continuously deforming based to the underlying flow conditions unlike solid ellipsoidal particles which are fixed in shape. With increase in Reynolds number $Re_{i}$ , the preference of alignment with the vorticity stays qualitatively similar. Overall, we find that the spatial position of the droplet is crucial in determining the orientational preference of the immersed droplets owing to the strong inhomogeneity and anisotropy in the field.

Until now we have looked at the effect of capillary number $Ca$ and Reynolds number $Re_{i}$ on the droplet deformation and orientation statistics and in the next subsection we will study the effect of viscosity ratio $\hat{\unicode[STIX]{x1D707}}$ .

Figure 9. Azimuthally, axially and time-averaged radial profiles of deformation parameter ( $D$ ) for four different $Ca$ ; (a) $Re_{i}=2500$ , (b) $Re_{i}=5000$ . The viscosity ratio is $\hat{\unicode[STIX]{x1D707}}=100$ . The direction of the arrow indicates increasing $Ca$ . The grey shaded areas near the left and right extremes of the $x$ -axis represent the inner and outer cylinder boundary layer thickness, respectively.

3.3 Effect of viscosity ratio

In figure 9 we plot the averaged radial profiles of the deformation parameter $D$ for different $Ca$ in a high viscosity ratio system $\hat{\unicode[STIX]{x1D707}}=100$ . On comparison with $\hat{\unicode[STIX]{x1D707}}=1$ (cf. figure 1) the degree of deformation is at least 5 times smaller for all capillary numbers. This behaviour can be intuitively expected as for the same strain rates the stretching is lower for more viscous drops given the additional viscous dissipation due to the internal flow in the drops. Additionally, due to the relatively smaller deformations the peaks in the deformation parameter $D$ lie close to the wall for all $Ca$ . While it is clear from figure 9 that there is a reduction in the overall deformation of the droplets, it would be interesting to look at the shape of the droplets as we did in figure 5; i.e. in the case of $\hat{\unicode[STIX]{x1D707}}=1$ the droplets were more prolate (cigar-like). In figure 10 we plot the p.d.f.s of the ratio of semi-axes in the IBL, bulk region and OBL (similar to figure 5). It can be seen that the semi-axes ratios $d_{2}/d_{1}\sim d_{3}/d_{1}>1$ which is a clear indication that the droplets tend to be more oblate (disk-like) when $\hat{\unicode[STIX]{x1D707}}=100$ . A similar behaviour is observed when the capillary number of the drops is increased. While we found previously that the capillary number plays an important role in the degree of deformation of the droplets, here we find that the droplet shape also depends strongly on the viscosity ratio $\hat{\unicode[STIX]{x1D707}}$ .

Figure 10. Probability distribution function (p.d.f.) of the ratio of semi-axes ( $d_{1}<d_{2}<d_{3}$ ) for $Re_{i}=2500$ and $\hat{\unicode[STIX]{x1D707}}=100$ : (a–c) $Ca=0.05$ , (d–f) $Ca=0.2$ .

Figure 11. Probability distribution functions of the streamwise orientation of the major axis of droplets for $Ca=0.05$ and $Ca=0.2$ and $\hat{\unicode[STIX]{x1D707}}=100$ : (a)  $Re_{i}=2500$ , (b)  $Re_{i}=5000$ .

Figure 12. Probability distribution functions of the cosine of the angle between the eigenvector of the semi-major axis ( $\boldsymbol{e}^{\mathbf{3}}$ ) of the ellipsoid and three strain rate eigendirections ( $\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{1}},\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{2}},\boldsymbol{e}_{\boldsymbol{s}}^{\mathbf{3}}$ )  $Re_{i}=2500~\hat{\unicode[STIX]{x1D707}}=100$ for (a) $Ca=0.05$ and (b) $Ca=0.2$ .

In order to understand these dependencies we need to go back to the equation governing the droplet shape (i.e. (2.3)). By changing the viscosity ratio we influence the values of the parameters $f_{1}$ and $f_{2}$ which are calculated according to (2.4). $f_{1}$ and $f_{2}$ are monotonically decreasing functions of $\hat{\unicode[STIX]{x1D707}}$ and while the ratio $f_{1}/f_{2}$ is almost constant (see also figure 11 of Biferale et al. (Reference Biferale, Meneveau and Verzicco2014)), $f_{1}$ and $f_{2}$ decrease by up to a factor of almost 50 by increasing the viscosity ratio from $\hat{\unicode[STIX]{x1D707}}=1$ to $\hat{\unicode[STIX]{x1D707}}=100$ . This has a strong influence on the effective strengths of droplet stretching, relaxation and rotation of the ellipsoidal droplets. More importantly, in comparison to low viscosity ratio systems the influence of rotation becomes more dominant when the viscosity ratio is high (Biferale et al. Reference Biferale, Meneveau and Verzicco2014). Additionally, reducing $f_{1}$ results in an increase in the effective interfacial relaxation time (i.e. $\unicode[STIX]{x1D70F}/f_{1}$ ) or in other words the surface tension force acting on the ellipsoidal droplet has lesser time to adjust to the local flow conditions resulting in some highly stretched drops. The increased influence of rotation can also result in the loss of a preferential stretching axis for the ellipsoid thus resulting in more oblate or disk-like droplets.

Here it is important to note that the decorrelating effect of straining with rotation has been observed previously in multiple studies. Girimaji & Pope (Reference Girimaji and Pope1990) studied the evolution of infinitesimal material elements in homogeneous isotropic turbulence using DNS where they concluded that the material line elements align poorly with the straining direction due to the effect of rotation. This effect is attributed to the vorticity sweeping the material elements (in our case the axes of the ellipsoidal drop) away from the maximum strain rate direction along with the rotation of the principal strain rate axes relative to the material lines. Given that we are studying a highly inhomogeneous and anisotropic wall sheared flow, the former effect is more pronounced and is the cause of observed droplet shapes. More recently Johnson & Meneveau (Reference Johnson and Meneveau2015) calculated the Cramèr function for finite time Lyapunov exponents (FTLE) of Lagrangian particles in isotropic turbulence and by isolating the effect of the strain rate tensor they concluded that the deformation of a fluid volume is higher when the immersed particle does not rotate which resonates very well with the deformation statistics of highly viscous drops we observe in our study.

To understand the influence of rotation on the relative orientation of the drops we plot p.d.f.s of the direction cosine of the major axis with the streamwise direction in figure 11 for two different Reynolds numbers and $\hat{\unicode[STIX]{x1D707}}=100$ . It can be immediately observed that similar to $\hat{\unicode[STIX]{x1D707}}=1$ the orientation is highly isotropic in the bulk region even for $\hat{\unicode[STIX]{x1D707}}=100$ . In the IBL and OBL, while there is a tendency for the major axis to align with the streamwise direction which is similar to what we observed for $\hat{\unicode[STIX]{x1D707}}=1$ , the distribution of the orientation for $\hat{\unicode[STIX]{x1D707}}=100$ is more uniform than $\hat{\unicode[STIX]{x1D707}}=1$ which can be explained from earlier discussions about the increased influence of rotation. A similar behaviour is also observed in the p.d.f.s of orientation of the major axis with the eigendirections of the strain rate tensor which are shown in figure 12. In the IBL and OBL the droplets tend to align with the most extensional strain rate eigendirection which is similar to $\hat{\unicode[STIX]{x1D707}}=1$ while the distribution is relatively more uniform. This is in contrast to the behaviour of oblate disk-like solid ellipsoids in an isotropic turbulent flow (Chevillard & Meneveau Reference Chevillard and Meneveau2013) the droplets do not tend to align orthogonally to the most contracting eigendirection. These effects are connected to the anisotropy and continuously deforming nature of the drops as discussed previously.

From figures 7, 8 and 12 it is clear that the orientational behaviour of continuously deforming ellipsoidal droplets is completely different from that of solid axis-symmetric ellipsoids. For both viscosity ratios, the capillary number has negligible influence on the rotational behaviour of the droplets, suggesting that the droplets adjust to the underlying flow conditions regardless of the degree of deformation.