3.1. Asymmetric behaviours of the droplet size and the effective viscosity

The size of the dispersed droplets in a turbulent emulsion characterizes the microscopic structure of the emulsion, which affects the macroscopic stability and rheology of the emulsion. Also, the volume fraction of the dispersed phase, $\phi _{d}$, determines both the micro-scale structure (droplet size) and, consequently, the macro-scale rheological behaviours (effective viscosity).

First, we focus on the effect of the oil volume fraction, $\phi _{o}$, on the droplet size, for a given Reynolds number $Re = 5.2\times 10^{3}$. The volume fraction of oil, $\phi _{o}$, is varied from 0 (ethanol–water mixture) to $100\,\%$ (pure oil) by fixing the volume of each phase initially put into the TC system. After emulsification, the final state of the emulsion is observed to be O/W for $\phi _{o} \leqslant 50\,\%$, and W/O for $\phi _{o} \geqslant 60\,\%$. A phase inversion, in which the continuous phase and the dispersed phase are exchanged (Salager et al. Reference Salager, Márquez, Peña, Rondón, Silva and Tyrode2000; Bakhuis et al. Reference Bakhuis, Ezeta, Bullee, Marin, Lohse, Sun and Huisman2021), is thus expected to occur in between these two volume fractions. Note that the transient process of phase inversion, which consists of distinct runs at various volume fractions, is not observed in the present experiments. In this range, the behaviour of the system is too complex to allow for an accurate determination of the inversion point. In the following, we will consider only values of $\phi _{o}$ either below $50\,\%$ or larger than $60\,\%$, for which the natures of the dispersed and continuous phases are determined unambiguously. In this study, we focus on the global and local properties of O/W and W/O emulsions in a turbulent flow. Note that all emulsions that we obtained are of a simple type, and we did not observe multiple emulsions, such as O/W/O or W/O/W (Perazzo et al. Reference Perazzo, Preziosi and Guido2015). A typical O/W emulsion for $\phi _{o} = 5\,\%$ and W/O emulsion for $\phi _{o} = 95\,\%$ are shown in inset images in figure 2(a). Under steady stirring conditions, the droplet size in the turbulent emulsion eventually shows a statistically stationary distribution, giving an average of the droplet diameter $\langle D\rangle$ as an indicator for the droplet size. The average droplet diameters for various oil fractions are shown in figure 2(a). For both O/W emulsions (left branch, $\phi _{o} \leqslant 50\,\%$) and W/O emulsions (right branch, $\phi _{o} \geqslant 60\,\%$), it is found that the droplet size is almost independent of the oil fraction, at fixed $Re$. Remarkably, we find that the ethanol–water droplets in the right branch are about $50\,\%$ larger than the oil droplets in the left branch, indicating an obvious asymmetry of the droplet size between O/W and W/O emulsions. One may think that this asymmetric behaviour is due to the slight difference in physical properties of the two liquids used in experiments. However, we note that the densities of these two liquids are too close to account for the observed asymmetry. The interfacial tension between the two immiscible liquids is also identical for O/W and W/O emulsions. What about the viscosity? The viscosity of the silicone oil, $\nu _{o}=2.1 \times 10^{-6}\ {\rm m}^{2}\ {\rm s}^{-1}$, is slightly lower than that of the ethanol–water, $\nu _{w}=2.4 \times 10^{-6} {\rm m}^{2}\ {\rm s}^{-1}$, at the experimental temperature $\theta =22\,^\circ {\rm C}$, while it has been found that at least an order of magnitude difference between the viscosities of the two phases could change the droplet size by a measurable amount (Pacek et al. Reference Pacek, Nienow and Moore1994). Moreover, additional experiments that we have performed show that the asymmetry of the droplet size remains even when we eliminate viscosity difference by adjusting the working temperature (see Appendix D for more details). Thus the small viscosity difference cannot account for the observed asymmetry of the droplet size in these experiments. The asymmetry of the droplet size must therefore have another origin.

Figure 2.

(a) Average droplet diameter as a function of the oil volume fraction. The droplet diameter hardly changes with the oil volume fraction for both O/W emulsions ($\phi _{o} \leqslant 50\,\%$, hollow points) and W/O emulsions ($\phi _{o} \geqslant 60\,\%$, solid points). The ethanol–water droplets in W/O are found to be about $50\,\%$ larger than the oil droplets in O/W. The bluish band spans one standard deviation around the average diameter. The dashed lines denote the average values of the droplet diameters for O/W (or W/O). A typical O/W emulsion ($\phi _{o} = 5\,\%$) and a typical W/O emulsion ($\phi _{o} = 95\,\%$) are shown in the insets. The scale bar represents $200\ \mathrm {\mu }{\rm m}$. (b) The normalized effective viscosity of the emulsion as a function of the oil volume fraction $\phi _{o}$ for various Reynolds numbers $Re$. The inset shows the asymmetry factor $S$ as a function of $\phi _{o}$ for various $\omega _{i}$. The asymmetric trend of the effective viscosity between O/W (left branch) and W/O (right branch) emulsions is more pronounced at low Reynolds numbers.

Apart from the droplet size, also the effective viscosity of the emulsion shows an asymmetric behaviour. For various oil volume fractions ($0\,\% \leqslant \phi _{o} \leqslant 100\,\%$), we measured the effective viscosity of the emulsion, $\nu _{eff}$, which is calculated using a method that has been proposed recently for viscosity measurements in a turbulent TC flow (Bakhuis et al. Reference Bakhuis, Ezeta, Bullee, Marin, Lohse, Sun and Huisman2021; Yi et al. Reference Yi, Toschi and Sun2021). As is well known, an effective power-law dependence $G \propto Re^{\alpha }$ holds in the single-phase TC turbulent flow, where the power-law exponent $\alpha$ depends on the regime of the Reynolds number (Grossmann et al. Reference Grossmann, Lohse and Sun2016). Indeed, if the modified Reynolds number and dimensionless torque are used, then the current two-phase emulsion flow still follows the effective power-law dependence $G_{m} \propto Re_{m}^{\alpha }$, where $\alpha = 1.58$ in this study (see Appendix B). Based on this relation, we can calculate the effective viscosity of the emulsion, $\nu _{eff}$. The detailed calculation of the effective viscosity is documented in Appendix B. The results of the effective viscosity are shown in figure 2(b), which can also be divided into two parts: O/W emulsions for $\phi _{o} \leqslant 50\,\%$ (left branch), and W/O emulsions for $\phi _{o} \geqslant 60\,\%$ (right branch). Here, the effective viscosity is characterized by its normalized value, $\nu _{eff}/\nu _{n}$, where $\nu _{n} = \nu _{w}$ for O/W ($\phi _{o} \leqslant 50\,\%$), and $\nu _{n} = \nu _{o}$ for W/O ($\phi _{o} \geqslant 60\,\%$). For each branch, the effective viscosity increases with the increasing dispersed phase volume fraction $\phi _{d}$ for all Reynolds numbers. Note that the dispersed phase refers to oil for O/W or ethanol–water for W/O. The effective viscosity has only a weak dependence on the dispersed phase volume fraction in the dilute regime (i.e. for $\phi _{d} < 5\,\%$), while it displays a stronger dependence at larger dispersed phase volume fractions. The increase of the effective viscosity with increasing $\phi _{d}$ originates from the hydrodynamic or contact interactions between dispersed droplets, as it is observed in similar turbulent droplet dispersions (Pouplin et al. Reference Pouplin, Masbernat, Décarre and Liné2011) and in solid particle suspensions (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). Furthermore, the effective viscosity is found to decrease with the increasing Reynolds number for a given $\phi _{d}$, indicating that the turbulent emulsion somehow shows a shear-thinning behaviour (Yi et al. Reference Yi, Toschi and Sun2021). Though the qualitative trend of the effective viscosity versus the dispersed phase volume fraction is similar for both the left and right branches, an asymmetry of the effective viscosity between O/W and W/O emulsions is measured. The effective viscosity of O/W (left branch) is found to be higher than that of W/O (right branch) for a given Reynolds number, particularly for the case of the high dispersed phase volume fraction (see figure 2b). To represent quantitatively the degree of asymmetry, we define an asymmetry factor as

(3.1)\begin{equation} S = \frac{\left(\dfrac{\nu_{eff}}{\nu_{n}}\right)_{\phi_{o}} - \left(\dfrac{\nu_{eff}}{\nu_{n}}\right)_{1-\phi_{o}}}{\left(\dfrac{\nu_{eff}}{\nu_{n}}\right)_{\phi_{o}} + \left(\dfrac{\nu_{eff}}{\nu_{n}}\right)_{1-\phi_{o}}}, \end{equation}

where the subscripts $\phi _{o}$ and $1 - \phi _{o}$ denote the emulsion at $\phi _{o}$ and at $1 - \phi _{o}$, respectively. An asymmetry factor $S$ deviating from $0$ indicates asymmetry. The asymmetry factor, as a function of the oil volume fraction, is shown as the inset of figure 2(b). It is found that the asymmetry decreases with the increasing Reynolds number. In addition, the asymmetric trend between O/W and W/O is more pronounced for high dispersed phase volume fractions. Since it was already found that the droplet size has a dramatic influence on the emulsion rheology (Pal Reference Pal1996), the macroscopic asymmetry of the effective viscosity between O/W and W/O could be connected to the microscopic asymmetric behaviour of the droplet size.

3.2. Recovering the asymmetry between O/W and W/O emulsions using surfactant

We hypothesize that the possible reason for the asymmetry of the droplet size between O/W and W/O emulsions is the presence of surface-active contaminants. These surface-active contaminants are found widely on the liquid–liquid interface in practical environments, which is focused on by various studies related to interfacial phenomena (de Gennes Reference de Gennes2001; De Malmazet et al. Reference De Malmazet, Risso, Masbernat and Pauchard2015; Calvo et al. Reference Calvo, de Malmazet, Risso and Masbernat2019). On the one hand, these surface-active contaminants can modify the interfacial properties, yielding the change of the droplet size in the emulsion (Bazazi & Hejazi Reference Bazazi and Hejazi2020; Manikantan & Squires Reference Manikantan and Squires2020). On the other hand, the solubility of these contaminants is usually different in the oil phase and the aqueous phase (Kawaguchi Reference Kawaguchi2016). The preferential solubility can induce a different distribution of contaminants and different interfacial properties in O/W and W/O, which could be the source of the asymmetric behaviours.

To investigate the effect of surfactants on the asymmetric behaviour of turbulent emulsions, an effective way is to add a controlled amount of a selected surfactant into the system. First, we perform experiments using a kind of oil-soluble non-ionic surfactant: dimethylsiloxane block copolymer (30–35 % ethylene oxide). For convenience, we use its abbreviation (DBE) from the manufacturer. For the purpose of the present study, two contrasted concentrations of DBE in oil are selected. One is $12.5\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$, which is comparable to the critical micelle concentration (CMC) of DBE in water (around $13\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$) (Rheingans et al. Reference Rheingans, Hugenberg, Harris, Fischer and Maskos2000), and the second, 20 times larger, is $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$. The DBE is well mixed with the oil before each experiment. Two microscopic images of O/W and W/O emulsions with added $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ DBE are shown in figure 3(b). Figure 3(c) shows the results of the droplet size in emulsions using DBE, for various oil volume fractions. Here, the Reynolds number is fixed at $Re = 5.2\times 10^{3}$. The measured droplet sizes for emulsions using $12.5\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ and $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ DBE are denoted by yellow marks and red marks in figure 3(c), respectively. It is found that DBE only slightly reduces the droplet size in O/W emulsions (left branch) when compared to what has been found in emulsions without adding surfactant (blue marks). But for W/O emulsions (right branch), the droplet size decreases with the increasing concentration of DBE. When $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ DBE is used, remarkably, the droplet size difference between O/W and W/O emulsions is eliminated (see red marks in figure 3c). Consequently, we nearly recover the symmetry of the droplet size between O/W and W/O by adding oil-soluble surfactant DBE at high concentration into the emulsion system.

Figure 3.

(a,b) Schematic diagrams and microscopic images of the effects of surfactants (or surface-active contaminants) on the emulsions. Panel (a) depicts the emulsions without adding surfactant. The asymmetric behaviours of the droplet size between O/W emulsions and W/O emulsions are observed due to the difference in the concentrations of contaminants on the interfaces. The deformation of the droplet is observed clearly in W/O emulsions. Panel (b) represents emulsions with added DBE at $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$. The surfactant is adsorbed into the interface at saturation for both O/W and W/O emulsions. The symmetry of droplet size is almost recovered. Snapshots of the emulsion at an oil volume fraction $\phi _{o}=5\,\%$ are used to represent O/W emulsions, while W/O emulsions are denoted by snapshots of the emulsion at $\phi _{o}=95\,\%$. For all snapshots here, $Re = 5.2\times 10^{3}$. The scale bar represents $200\ \mathrm {\mu }{\rm m}$. (c) The droplets size in emulsions for various oil volume fractions, with and without adding the surfactant, for $Re = 5.2\times 10^{3}$. The droplet size almost does not change with the oil volume fraction for both O/W ($\phi _{o} \leqslant 50\,\%$) and W/O ($\phi _{o} \geqslant 60\,\%$) emulsions. The asymmetric trends of the droplet size between O/W and W/O are found to disappear when adding surfactant DBE into the emulsion systems. Note that $12.5\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ and $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ here refer to the concentrations of DBE in oil. The dashed lines represent the average values of the droplet diameters for O/W (or W/O). The coloured bands span the average diameter $\pm$ one standard deviation. For the sake of clarity, the coloured band for the case using $12.5\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ DBE is not present here. (d) The normalized effective viscosity of the emulsion for the oil volume fraction from $\phi _{o}=0\,\%$ to $\phi _{o}=100\,\%$, with added surfactant DBE at $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$, at various Reynolds numbers. The inset shows the asymmetry factor $S$ as a function of the oil volume fraction. The asymmetric behaviour of the effective viscosity between O/W and W/O emulsions becomes considerably smaller due to the presence of surfactant DBE at saturation.

The above results can be explained using the schematic diagrams in figures 3(a,b). We first focus on the cases without adding surfactant. In the practical environment, even in careful laboratory experiments, the emulsion inevitably contains some surface-active contaminants (Duineveld Reference Duineveld1995; Lalanne, Masbernat & Risso Reference Lalanne, Masbernat and Risso2020; Soligo et al. Reference Soligo, Roccon and Soldati2020) (black indicators in figure 3a), which could originate from the wall of the container in this study. These surface-active contaminants are expected to be preferentially soluble in the aqueous ethanol–water, a quite good solvent, and act as a surfactant. As illustrated in the left-hand part of figure 3(a) (O/W), the surface-active contaminants from the wall dissolve into the continuous phase of ethanol–water. These contaminants are then adsorbed onto the liquid–liquid interface and modify the oil droplet size for two reasons. On the one hand, contaminants on the surface suppress droplet coalescence, which is known to be a common effect of surfactants on emulsion systems (Cristini, Bławzdziewicz & Loewenberg Reference Cristini, Bławzdziewicz and Loewenberg1998; Ha, Yoon & Leal Reference Ha, Yoon and Leal2003; Dai & Leal Reference Dai and Leal2008; Baret Reference Baret2012). On the other hand, these surface-active contaminants cover the surface of oil droplets, inducing a reduction of the interfacial tension (Manikantan & Squires Reference Manikantan and Squires2020). Thus the breakup of droplets could be promoted. The effect of contaminants on the breakup and coalescence of the droplets finally reflects on the smaller size of oil droplets in O/W. As the interfacial tension is found to decrease only slightly with the concentration of DBE (see Appendix A), it is reasonable to assume that the inhibition of the droplet coalescence is the dominant factor affecting the droplet size here. However, when oil is the continuous phase (W/O), as shown in the right-hand part of figure 3(a), ethanol–water droplets embedded within the oil are not in contact with walls. Therefore, only a limited amount of surface-active contaminants are adsorbed onto the interface. The cleaner liquid–liquid interface brings less inhibition to the coalescence, yielding the larger droplet size for the given turbulent strength (see figure 3a). It should be noted that the difference between the O/W and W/O cases is not that there are surfactants in one case and no surfactant in the other case. The different behaviours arise from the fact that in the O/W case, the interfaces are fully covered by surfactants, which inhibit coalescence, and in the W/O case, there is not enough surfactant to significantly cover interfaces and inhibit coalescence.

The effective viscosity difference between O/W and W/O could be related to the deformability of the dispersed phase, which is closely connected to the droplet size (Verschoof et al. Reference Verschoof, Van Der Veen, Sun and Lohse2016; Bakhuis et al. Reference Bakhuis, Ezeta, Bullee, Marin, Lohse, Sun and Huisman2021). When compared to the larger ethanol–water droplets, the small and non-deformable oil droplets could behave as solid-like particles and bring an extra contribution to the effective viscosity of the emulsion in a turbulent flow (Bakhuis et al. Reference Bakhuis, Verschoof, Mathai, Huisman, Lohse and Sun2018; Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). In addition, as the droplet size is found to be comparable with the viscous sublayer thickness $\delta _{vis}$ (see table 1 below, and Appendix C), the exclusion of the droplets from the viscous sublayer could also be a reason for the reduced effectiveness of larger drops on the effective viscosity in W/O emulsions. Furthermore, surface-active contaminants in W/O could modify the dynamics of the emulsion through the hydrodynamic coupling interaction between the oil-droplet surface and the surrounding flow (Baret Reference Baret2012). As a result, the surface-active contaminants account for the measured higher effective viscosity of O/W (left branch) compared to that of W/O (right branch) (see figure 3b).

Table 1.

Some comparisons of length scales and capillary number. The O/W case at $\phi _{o} = 1\,\%$ is used here. Detailed calculations can be found in Appendix C.

Next, we consider the experimental results using the oil-soluble surfactant DBE. For the case of O/W emulsion shown in the left-hand part of figure 3(b), some DBE added in the system is competitively adsorbed into the interface. As the interface is already saturated due to the contaminants, the interfacial properties show no significant change when adding DBE, which is consistent with the previous result that the droplet sizes for O/W emulsions decrease only slightly when DBE is added. When the oil is the continuous phase (W/O) as shown in the right-hand part of figure 3(b), the surface of ethanol–water droplets is expected to be mostly covered by DBE adsorbed from oil, i.e. the surface is at saturation ($250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ case). Therefore, the coalescence of droplets is now inhibited. Consequently, the asymmetric trend of the droplet size between O/W and W/O is eliminated using DBE (figure 3c).

In this part, we focus on the effect of surfactant on another feature of the emulsion system: the effective viscosity. As shown in figure 3(d), it is found that the symmetry of the effective viscosity between O/W (left branch) and W/O (right branch) emulsions is partially recovered using $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ DBE. This is clearly indicated by the asymmetry factor $S$ close to $0$ (see the inset of figure 3d), which is expected to be mainly attributed to the recovery of the symmetry of the droplet size using DBE (see figure 3c). It should be noted that the symmetry of the effective viscosity is not fully recovered. In general, there are always some differences between O/W and W/O emulsions, such as the distribution of the surfactant in the flow. The effective viscosity for each case is found to be slightly larger than that for its corresponding case without adding surfactant (compare figures 3(d) and 3(b)). The reason could be that the polymeric surfactant (DBE in this study) with high molecular weight enhances the interfacial rigidification effect of the droplet surface when compared to the case with only contaminants (Erni Reference Erni2011). Indeed, the copolymer molecules of surfactant could form shell-like structures around the drops (Sundararaj & Macosko Reference Sundararaj and Macosko1995). This increases the resistance of the droplet to the surrounding flow, yielding the extra contribution of the viscous dissipation of the flow. Note that although there is still some uncertainty about the presence of the contaminant due to limitation of the experiments, the recovery of the symmetry using oil-soluble surfactant DBE supports the hypothesis that surface-active contaminants are the cause of the asymmetry.

3.3. The dependence of the droplet size on the Reynolds number

As the droplet size and the associated asymmetry have been discussed for various oil fractions, the next question is what sets the droplet size in turbulent emulsions. In this part, we study the dependence of the droplet size on the Reynolds number, at the low volume fraction of the dispersed phase (i.e. $\phi _{o}=1\,\%$ and $99\,\%$). Since the volume fraction of the dispersed phase is very low in these cases, the viscosity of the emulsion is approximately equal to that of the continuous phase, giving $Re_{m} \approx Re$ and $G_{m} \approx G$. We first consider the cases at $\phi _{o}=1\,\%$ without using surfactant. Note that the coalescence of droplets is inhibited due to the surface-active contaminants on the droplet surface. Therefore, the droplet size is determined mainly by the turbulent breakup mechanism. As shown in figure 4(a) by the black circles, the droplet size normalized by the gap, $\langle D\rangle /d$, is found to have a scaling dependence on the Reynolds number $Re$ with effective exponent $-1.18\pm 0.05$ obtained by a direct fit for $\phi _{o}=1\,\%$.

Figure 4.

(a) The average droplet diameter normalized by the gap width as a function of the Reynolds number for oil volume fractions $\phi _{o}=1\,\%$ and $99\,\%$. The black and red symbols denote droplet sizes in emulsions using no surfactant and using $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ DBE, respectively. The error bars are based on the errors of the edge detection. The black lines denote the scaling prediction of the Kolmogorov–Hinze (K–H) theory, $\langle D\rangle /d \sim Re^{-6/5}$. The asymmetry of the droplet size in O/W ($\phi _{o}=1\,\%$) and W/O ($\phi _{o}=99\,\%$) emulsions is removed using DBE at saturation. (b) The Weber number as a function of the Reynolds number in a log-log plot for the two theoretical approaches. The yellow, green and blue diamonds denote the results using K–H theory based on the energy dissipation rate in the bulk area, that using K–H theory based on energy dissipation rate in the boundary layer (BL), and that using Levich's theory, respectively. The O/W emulsions at $\phi _{o}=1\,\%$ without surfactant are considered here.

We now explore the physical mechanism behind the scaling dependence of the droplet size on the Reynolds number. According to the K–H theory, the droplet formation in a turbulent flow is determined by the competition between the deforming external dynamic pressure force (turbulent fluctuations) and the resisting interfacial tension over the droplet surface (Kolmogorov Reference Kolmogorov1949; Hinze Reference Hinze1955), of which the ratio is usually indicated by the droplet turbulent Weber number $We = \rho \, \overline {\delta u^2}\,D/\gamma$, where $\rho$ is the density of the continuous phase, $\overline {\delta u^2}$ is the mean-square velocity difference over a distance equal to the droplet diameter $D$, and $\gamma$ is the interfacial tension between the two phases (Risso & Fabre Reference Risso and Fabre1998). If the droplet diameter $D$ belongs to the inertial turbulent sub-range, then $\overline {\delta u^2}$ could be expressed as a function of the local energy dissipation rate: $\overline {\delta u^2} = C_{1}(\varepsilon D)^{2/3}$, where the constant is $C_{1}\approx 2$ according to Batchelor (Reference Batchelor1953). This yields the Weber number as $We = 2\rho \varepsilon ^{2/3}D^{5/3}/\gamma$. The force balance implies the existence of a critical value of the Weber number beyond which breakup occurs (Hinze Reference Hinze1955), and this value is found to be of the order of unity (i.e. $We\sim {O}(1)$) in various studies (Hesketh, Etchells & Russell Reference Hesketh, Etchells and Russell1991a; Risso & Fabre Reference Risso and Fabre1998; Lemenand et al. Reference Lemenand, Della Valle, Dupont and Peerhossaini2017). Thus the prediction of the maximum stable droplet size in a homogeneous and isotropic turbulent flow can be given by $D_{max} = C(\rho /\gamma )^{-3/5}\varepsilon ^{-2/5}$ (where $C$ is a constant coefficient), which is the main result of the work by Hinze (Reference Hinze1955). Moreover, various studies have shown that the average droplet diameter $\langle D\rangle$ can be used as the indicator of the droplet size in the K–H prediction (Lemenand et al. Reference Lemenand, Della Valle, Zellouf and Peerhossaini2003; Boxall et al. Reference Boxall, Koh, Sloan, Sum and Wu2012; Perlekar et al. Reference Perlekar, Biferale, Sbragaglia, Srivastava and Toschi2012).

First, we speculate that the droplet size could be dominated by the turbulent fluctuations in the bulk flow of the system, where most droplets distribute. The local energy dissipation rate in the bulk can be estimated as $\varepsilon _b \sim u_{T}^{3}/\ell$, where $u_{T}$ and $\ell$ are the typical velocity fluctuation and the characteristic length scale of the flow (Ezeta et al. Reference Ezeta, Huisman, Sun and Lohse2018). As the typical velocity fluctuation can be expressed as $u_{T} \sim \omega _{i}r_{i} \sim Re\,\nu /d$ (Van Gils et al. Reference Van Gils, Huisman, Grossmann, Sun and Lohse2012), we obtain $\varepsilon _b \sim Re^3\nu ^3/d^4$ by assuming $\ell \sim d$. Inserting $\varepsilon _b$ into the K–H prediction, the scaling dependence of the droplet size on the Reynolds number is obtained as $\langle D\rangle /d \sim Re^{-6/5}$, which agrees well with the experimental results for $\phi _{o} = 1\,\%$ without surfactant (see black circles in figure 4a). However, the discussion above is only a simple analysis based on the scaling law. A further quantitative study on the droplet formation in a turbulent flow needs to consider the Weber number, which can be calculated as $We = 2\rho _{w}\varepsilon _b^{2/3}D^{5/3}/\gamma$, where $\varepsilon _b$ can be estimated as $\varepsilon _b \approx 0.1T\omega _{i}/[{\rm \pi} (r_{o}^{2}-r_{i}^{2})L\rho _{w}]$ in a TC turbulent flow (Ezeta et al. Reference Ezeta, Huisman, Sun and Lohse2018). As shown in figure 4(b) (yellow diamonds, approach 1), the Weber number ranges between $0.013$ and $0.018$, two orders of magnitude smaller than the critical value obtained in previous studies, suggesting that the bulk of the system, where most droplets flow around, is not the place where the droplet size is determined.

Indeed, the droplet breakup is most often observed close to the area where the most intense stress participates in the deformation (Hesketh, Etchells & Russell Reference Hesketh, Etchells and Russell1991b; Afshar Ghotli et al. Reference Afshar Ghotli, Raman, Ibrahim and Baroutian2013). Considering that the coalescence is almost inhibited in the current system, the droplet size is dominated mainly by the place where small droplets are generated. Thus the droplet size is expected to be dominated by the boundary layer area close to the wall, where the K–H theory has some limitations. First, as shown in table 1, the droplet size is found to be smaller than the turbulent boundary layer thickness ($\langle D\rangle /\lambda _{u} < 1$), which supports the occurrence of breakup in the boundary layer. The droplet size is larger than the Kolmogorov length scale in the turbulent boundary layer ($\langle D\rangle /\eta _{bl} >1$). Additionally, the results for the capillary number are of the order of $10^{-2}$, indicating the negligible effect of the viscous deforming stress on the determination of the droplet size. Next, we check the validity of the K–H theory in the boundary layer. The energy dissipation rate in the boundary layer can be estimated as $\varepsilon _{bl} = u_{\ast }^{3}/\lambda _{u}$, where we use the shear velocity $u_{\ast } = \sqrt {\tau _w/\rho } = \sqrt {T/(2{\rm \pi} \rho r_{i}^{2}L)}$ and the boundary layer thickness $\lambda _{u} = d/(2Nu_{\omega }) \sim Re\,G^{-1}$ in a TC turbulent flow (Eckhardt, Grossmann & Lohse Reference Eckhardt, Grossmann and Lohse2007). Using the effective scaling $G \sim Re^{1.58}$ obtained in the current system, the energy dissipation rate is found to scale as $\varepsilon _{bl}/(\nu ^3 d^{-4}) \sim Re^{2.95}$. Inserting $\varepsilon _{bl}$ into the K–H prediction, one obtains $\langle D\rangle /d \sim Re^{-1.2}$, in agreement with the experimental data as well. The scaling exponent $-1.2$ suggests that the energy dissipation at the boundary layer ($\varepsilon _{bl}$) is just proportional to the local energy dissipation in the bulk of the system ($\varepsilon _{b}$), which is similar to what is observed for the case of the liquid–liquid dispersion in an agitated vessel (Wichterle Reference Wichterle1995). As shown in figure 4(b) (green diamonds, approach 2), Weber numbers calculated using $\varepsilon _{bl}$ are about $0.08$, which is an order of magnitude smaller than the critical value (${O}(1)$). This indicates that the energy dissipation rate $\varepsilon _{bl}$ near the wall is not large enough to cause the breakup of such small droplets. See Appendix C for more detailed calculations in this part. These results show that the K–H theory is not appropriate for modelling the droplet size in the present system.

A prediction of the droplet size in the non-homogeneous turbulent flow past a solid wall was proposed by Levich (Reference Levich1962), who gave the dynamic pressure force exerted on the two sides of the droplet using the logarithmic distribution of the mean velocity in the boundary layer. Note that the Reynolds number in the current study is in the interval where the logarithmic mean velocity distribution in the boundary layer can exist (Huisman et al. Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013). According to Levich (Reference Levich1962), the droplet diameter can be written as $\langle D\rangle = 2\sqrt {\gamma \nu _{w} /(25\rho _{w} u_{\ast }^{3})}$, thus the scaling dependence of the droplet diameter on the Reynolds number is derived as $\langle D\rangle /d \sim Re^{-1.19}$, where the exponent $-1.19$ is very close to the $-6/5$ in the K–H prediction and agrees again with the scaling dependence observed in experiments for $\phi _{o} = 1\,\%$. Based on the Levich theory, we also calculate the Weber number as the ratio of the dynamic pressure force induced by the mean flow to the interfacial tension: $We = 25\rho _{w} u_{\ast }^{3}\langle D\rangle ^{2}/(2\nu _{w}\gamma )$. As shown in figure 4(b) (blue diamonds, approach 3), the Weber number for the Levich prediction is about $5$, which is consistent with the critical value for the droplet breakup in a turbulent flow (Risso & Fabre Reference Risso and Fabre1998; Lemenand et al. Reference Lemenand, Della Valle, Dupont and Peerhossaini2017). The comparison of the Weber numbers based on the energy dissipation rate and that based on Levich's theory lead to the conclusion that the droplet fragmentation, which determines the droplet size, occurs within the boundary layer and is controlled by the dynamic pressure caused by the gradient of mean flow, in agreement with the mechanism originally proposed by Levich. Note that this conclusion also requires that the boundary layer thickness is larger than the droplets diameter, which is supported by the fact that boundary layer thickness is estimated as at least $5$ times the droplet size in this study (see table 1).

The discussion of the two approaches above is based on the droplet size at $\phi _{o} = 1\,\%$, whereas the droplet size at $\phi _{o}=99\,\%$ is found to follow the same $-6/5$ scaling dependence (see black triangles in figure 4a), indicating the robustness of the scaling law. Furthermore, this figure also shows the existence of the asymmetry of the droplet size between O/W and W/O at high $Re$, at least for low dispersed phase volume fractions (i.e. $\phi _{o}=1\,\%$ and $99\,\%$). Considering that the droplet size for W/O is about $50\,\%$ larger than that for O/W, only a slight variation of the Weber number is expected at $\phi _{o} = 99\,\%$ as compared to the case at $\phi _{o} = 1\,\%$. Therefore, all the qualitative conclusions given above are valid for emulsions at $\phi _{o} = 99\,\%$ as well. In this case, the droplet coalescence needs to be considered. The fact that the breakup theory, without accounting for coalescence, describes well the experimental result is particularly interesting. Indeed, for a steady state to be reached finally, the coalescence rate of droplets has to be equal to the breakup rate of coalesced droplets. We therefore observed an average size that is larger than that predicted by the breakup theory, but which remains proportional to it. Note that the interpretation given here is still not complete, and the results remain open for discussion. Additionally, the finding that the droplet formation is controlled by the boundary layer also provides a reasonable explanation for the observations that the droplet size hardly depends on the dispersed phase volume fraction $\phi _{d}$ (see figure 4a). The droplets that are generated close to the wall, where similar mean velocity gradient could distribute for various $\phi _{d}$, are expected to have similar size. Therefore, the similar droplet size is observed at various $\phi _{d}$ in the entire system for O/W (or W/O) emulsions.

Next, we turn to the results using surfactant. Experiments using $250\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$ DBE are performed, and the results are shown in figure 4(a). We note that the scaling dependence of the droplet size on the Reynolds number remains, suggesting the robustness of the scaling law for turbulent emulsions containing DBE. For the case $\phi _{o}=1\,\%$, it is found that the droplet size decreases only slightly due to DBE. However, the droplet size at $\phi _{o}=99\,\%$ shows a dramatic reduction to a value close to that at $\phi _{o}=1\,\%$, yielding the elimination of the asymmetry of the droplet size, for various Reynolds numbers. Since we have found that the recovery of the symmetry using DBE is due to the inhibition of droplet coalescence for the lowest $Re$ case (i.e. $Re = 5.2\times 10^{3}$), it is reasonable to conclude that the similar behaviours of the droplet size observed here at high $Re$ have the same physical interpretations. Moreover, similar results have been observed in additional experiments using a lower concentration of DBE ($12.5\ \mathrm {\mu } \text {l}\ \text {l}^{-1}$) (see Appendix D).