4.1. Effect of the liquid–gas density ratio in the drop regime

The scale distribution of kinetic energy $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ for different density ratios $R_\rho$ in the drop regime ( $\alpha = 25\,\%$ ) is plotted in figure 1. The scale is normalised by the width of the computational domain $L$ while $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ is normalised by twice the scale-integrated turbulent kinetic energy $\langle |\boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ . For the total fluctuating field $\mathbb{C} \equiv \mathbb{T}$ (figure 1 a), we observe that the small-scale content of kinetic energy is larger in multiphase turbulence than in single-phase turbulence. We further note that increasing the density ratio yields an increase of the scale-by-scale kinetic energy at small scales. The activity of turbulence at small scales is linked to the activity of the dissipative scales. Increasing $R_\rho$ , and/or moving from single-phase to multiphase flows, thus leads to an increase of the energy dissipation. This is confirmed by the values of $\varepsilon _f$ in the first five lines of table 1.

Figure 1.

Effect of the density ratio $R_\rho$ on the scale-by-scale kinetic energy for $\alpha = 25\,\%$ . The colours from blue to red correspond to $R_\rho = 1, \,5, \,25, \,125$ while the black curve is for single-phase turbulence: (a) $\mathbb{C} \equiv \mathbb{T}$ , (b) $\mathbb{C} \equiv \mathbb{L}$ and (c) $\mathbb{C} \equiv \mathbb{G}$ . The dashed line represents the $r^2$ scaling.

Looking at the curves for the liquid and gas conditional statistics (figures 1 b and 1 c for $\mathbb{C} \equiv \mathbb{L}$ and $\mathbb{C} \equiv \mathbb{G}$ , respectively), we note that the scale distribution of kinetic energy in the liquid is mildly changing contrary to the gas phase where kinetic energy substantially increases with $R_{\rho }$ when the separation $r$ decreases. Hence, the increase of $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{T}}$ at small scales is mostly associated with an increase of $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{G}}$ .

Appendix A.5 discusses the small-scale behaviour of the two-point kinetic energy. It is shown that the mixed structure function $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ scales as $r^2$ and is proportional to the enstrophy $\langle |\boldsymbol{\omega }|^2\rangle _{\mathbb{C}}$ averaged over $\mathbb{C} \in \{\mathbb{T},\mathbb{L},\mathbb{G}\}$ . This quadratic evolution with respect to $r$ is shown in figure 1(a–c) as grey dashed lines. As $r$ decreases, $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ approaches the expected scaling, although some small deviations are observed, particularly in the gas phase (figure 1 c). Previous studies (Ling et al. Reference Ling, Fuster, Tryggvason and Zaleski2019; Estivalezes et al. Reference Estivalezes2022) have noted that accurately resolving the enstrophy field in multiphase flow simulations is demanding. Therefore, while mass and kinetic energy are well resolved in our data (see Appendix B.3), the observed deviations from the $r^2$ scaling might reflect the need for an even finer mesh to fully capture the enstrophy field. Consequently, in the remainder of this paper, the small-scale scaling of $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ is not further addressed, and the vorticity field is discussed only qualitatively.

Figure 2.

Two-dimensional slices of the vorticity magnitude (in colour, units of $u'/L$ ) together with the liquid–gas interface (black curves) for $\alpha =25\,\%$ and (a–d) increasing $R_\rho$ .

Keeping this consideration in mind, our data indicate that the increase of the small-scale kinetic energy in the gas phase observed in figure 1(c) is related to an increase of the enstrophy field in the gas phase. From $R_\rho =1$ to $R_\rho =125$ , the averaged enstrophy in the gas phase is doubled, as shown in figure 1(c). To confirm this, one can scrutinise the visualisations of the flow portrayed in figure 2. We observe an increased amplitude of the vorticity magnitude, hence the small-scale activity, when one approaches the interface from the gas phase. This behaviour was already reported by Dodd & Ferrante (Reference Dodd and Ferrante2016). The growing production of kinetic energy and vorticity is occurring in the turbulent boundary layers and wakes surrounding the liquid structures. As per Dodd & Ferrante (Reference Dodd and Ferrante2016), this results from an increase of the liquid structures Stokes number which increases with $R_\rho$ . The observed increased vorticity magnitude is further consistent with the theory developed by Terrington, Hourigan & Thompson (Reference Terrington, Hourigan and Thompson2022).

Figure 3.

Effect of the density ratio $R_\rho$ on the scale-by-scale kinetic energy budgets for $\alpha = 25\,\%$ . The colours from blue to red correspond to $R_\rho = 1, \,5, \,25, \,125$ as in figure 1. The black curves are for the single-phase case at the same viscosity. The different lines correspond to $\langle \mathcal{T}\rangle _{\mathbb{C}}$ , $\langle \mathcal{P}\rangle _{\mathbb{C}}$ , $\langle \mathcal{S}\rangle _{\mathbb{C}}$ , $\langle \mathcal{V}\rangle _{\mathbb{C}}$ , $\langle \mathcal{F}\rangle _{\mathbb{C}}$ , with (a) $\mathbb{C} = \mathbb{T}$ , (c) $\mathbb{C} = \mathbb{L}$ and (d) $\mathbb{C} = \mathbb{G}$ . The horizontal dashed line indicates $4\langle F\rangle _{\mathbb{C}}$ , the limit at large separations of the forcing term. (b) The density correction terms $\langle \mathcal{C}(\boldsymbol{a}) \rangle _{\mathbb{T}}$ . Also represented in (a) are the scales $r_H$ with vertical lines and $r_{\mathcal{S}}$ with filled circles that are described later on.

The different terms of the scale-by-scale kinetic energy budget are displayed in figure 3. Each contribution is presented in units of $\langle F \rangle _{\mathbb{C}}$ which is the contribution of the forcing term in the one-point kinetic energy budget. By doing this, the scale distribution of the forcing term to the scale-by-scale budget appears to be independent of the investigated physical parameters and of the averaging volume $\mathbb{C} \in \{ \mathbb{T}, \mathbb{L}, \mathbb{G}\}$ , as shown in figure 3(a–c). The forcing term also appears to be the same as in single-phase flows. Consequently, all the other terms of the scale-by-scale budget can be studied in proportion of an invariant forcing contribution. This significantly eases the interpretation of the results.

Let us start by analysing the unconditionally averaged budget, i.e. $\mathbb{C} \equiv \mathbb{T}$ , portrayed in figure 3(a). We observe that the forcing term $\langle \mathcal{F} \rangle _{\mathbb{T}}$ contributes positively to the budget and acts at rather large scales. At intermediate scales, the kinetic energy is transferred from large to small scales through the nonlinear energy transport term $\langle \mathcal{T} \rangle _{\mathbb{T}}$ and the surface tension term $\langle \mathcal{S} \rangle _{\mathbb{T}}$ . The kinetic energy budget is finally equilibrated by the viscous term $\langle \mathcal{V} \rangle _{\mathbb{T}}$ which is negative. Therefore, our analysis based on the KHM framework confirms previous conclusions using other approaches such as, for example, spectral energy budgets (see e.g. Crialesi-Esposito et al. Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022; Cannon et al. Reference Cannon, Soligo and Rosti2024).

We further note that the nonlinear transfer term $\langle \mathcal{T} \rangle _{\mathbb{T}}$ is markedly reduced compared with its single-phase counterpart. The kinetic energy exchange in multiphase turbulence thus appears to comply with the following scenario. (i) Part of the kinetic energy injected at large scales is transferred into kinetic energy at smaller scales by the classical nonlinear transport process $\langle \mathcal{T} \rangle _{\mathbb{T}}$ . This transfer is, however, less important than in single-phase flows because (ii) part of the kinetic energy injected at large scales is also transferred into surface energy and then released back into kinetic energy, but at smaller scales. Finally, (iii) the increase of the overall kinetic energy transfer is compensated by a shift of the viscous term $\langle \mathcal{V} \rangle _{\mathbb{T}}$ in the direction of smaller scales. The pressure term also contributes as a loss of energy at small scales as soon as $R_\rho \neq 1$ . Its contribution is though rather marginal.

Although not shown in figure 3(a), summing up the contribution of the surface tension term and the nonlinear transport term indicates that the overall kinetic energy transfer (i.e. $\langle \mathcal{T} \rangle _{\mathbb{T}} + \langle \mathcal{S} \rangle _{\mathbb{T}}$ ) is larger than that pertaining to single-phase flows and peaks at smaller scales. It suggests that the overall transfer of kinetic energy in multiphase flows is quite likely to compare with that observed in single-phase flows but at a larger Reynolds number. This explains that numerical simulations of multiphase turbulence require a much finer resolution compared with single-phase turbulence, even when $R_\rho = 1$ .

When the density ratio $R_\rho$ increases, the contribution of the surface tension term $\langle \mathcal{S} \rangle _{\mathbb{T}}$ gets larger while that due to the nonlinear transport term $\langle \mathcal{T} \rangle _{\mathbb{T}}$ goes in the opposite direction. We note also that $\langle \mathcal{S} \rangle _{\mathbb{T}}$ peaks at smaller scales compared with $\langle \mathcal{T} \rangle _{\mathbb{T}}$ . The scale at which $\langle \mathcal{S} \rangle _{\mathbb{T}}$ is maximum is represented by the filled circles in figure 3(a). More insights into the behaviour of this scale are provided later in a dedicated section.

Consistently with our previous conclusions suggesting an increased activity of the smaller scales when $R_\rho$ becomes larger, the contribution of the viscous term starts being visible at smaller scales when $R_\rho$ increases. Finally, the contribution due to pressure $\langle \mathcal{P} \rangle _{\mathbb{T}} = -\langle \mathcal{C}(-\boldsymbol{\nabla }P) \rangle _{\mathbb{T}}$ is rather small and appears only in the small-scale range where it contributes negatively to the budget. It is zero for $R_\rho = 1$ , and increases in amplitude with $R_\rho$ .

In figure 3(b), we represent the contribution of $-\langle \mathcal{C}(\boldsymbol{a}) \rangle _{\mathbb{T}}$ for the viscous, forcing and surface tension terms. That corresponding to the pressure term is already plotted and discussed in figure 3(a). As expected, $\langle \mathcal{C}(\boldsymbol{a}) \rangle _{\mathbb{T}}$ is zero for $R_\rho = 1$ and its contribution increases when $R_\rho$ grows. We also have $\langle \mathcal{C}(\boldsymbol{a}) \rangle _{\mathbb{T}} \to 0$ when $r \to \infty$ (see Appendix A.4). The sign of the density correction for each term is the same as the term itself which means that variations of density tend to amplify the contribution of each process. We note that the density correction term contributes to about 1/3 of the surface tension at $R_\rho = 125$ , and about 10 % to the viscous term.

The different terms of the conditionally averaged KHM equation are presented in figures 3(c) and 3(d) for the liquid and gas phases, respectively. The first striking observation is for the gas phase (figure 3 d), where we note that the pressure term $\langle \mathcal{P} \rangle _{\mathbb{G}}$ is positive and thus represents a gain for the scale-by-scale kinetic energy budget. Recall that when conditionally averaged, only the pressure transport contributes to the budget while the term $\langle \mathcal{C} \rangle _{\mathbb{L,G}}$ is zero. Hence the significant contribution attributed to pressure arises solely from the scale-by-scale pressure transport. Its contribution increases greatly with $R_\rho$ and is even exceeding that of the forcing term $\langle \mathcal{F} \rangle _{\mathbb{T}}$ at $R_\rho = 125$ . This additional amount of kinetic energy injected into the gas phase yields an amplified contribution of the nonlinear transport term $\langle \mathcal{T} \rangle _{\mathbb{G}}$ . When the density ratio $R_\rho = 125$ , the latter is observed to be larger than that pertaining to single-phase flows. For $R_\rho = 1$ , the sum of the pressure and nonlinear transport, i.e. $\langle \mathcal{T} \rangle _{\mathbb{G}} + \langle \mathcal{P} \rangle _{\mathbb{G}}$ , is roughly similar to the nonlinear transport term of single-phase turbulence. The effect of the viscous stress $\langle \mathcal{V} \rangle _{\mathbb{G}}$ at each scale also becomes larger with $R_\rho$ . Here again, this is explained by an increased activity of the small scales in the gas phase.

In the liquid phase, our results show that the curves corresponding to $R_\rho = 1$ are distinguished substantially from the others. For instance, it appears that the pressure term $\langle \mathcal{P} \rangle _{\mathbb{L}}$ (the pressure transport) for $R_\rho = 1$ is very different from that for $R_\rho = 5$ . It seems, however, to vary only marginally for $R_\rho \geqslant 25$ . The same applies for the viscous term $\langle \mathcal{V} \rangle _{\mathbb{L}}$ but not for the nonlinear transfer term $\langle \mathcal{T} \rangle _{\mathbb{L}}$ which continues to grow as $R_\rho$ increases. That means that for the liquid phase, the effect of the density contrast between the two phases is not trivial. Some processes seem to saturate while some others continue to evolve. Therefore, the physics observed for $R_\rho = 1$ , which is a common simplification in the literature, cannot be readily extrapolated to flows with variations of density, even if the density contrast is moderate.

4.2. Effect of the liquid–gas density ratio in the bubble regime

The same analysis is performed for $\alpha = 75\,\%$ and $R_\rho = 1, 5, 25$ . The scale distribution of kinetic energy $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}/2 \langle | \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ is presented in figures 4(a), 4(b) and 4(c) for $\mathbb{C}=\mathbb{T},\ \mathbb{L}$ and $\mathbb{G}$ , respectively. Here, we note that $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{T}}/2\langle | \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ is mostly unchanged when the density ratio $R_\rho$ is increased. Hence, the total dissipation rate $\varepsilon _f$ is found to be constant with respect to $R_\rho$ (see table 1). At small scales, the scale-by-scale kinetic energy is slightly larger than that of single-phase turbulence. This is reflected by a larger dissipation rate $\varepsilon _f$ in multiphase turbulence compared with its single-phase counterpart. In the liquid phase, $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{L}}$ is almost independent of $R_\rho$ and is very comparable to single-phase turbulence. In the bubble regime, the effect of density ratio is only perceptible in the gas phase. We note again a systematic shift of $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{G}}$ towards smaller scales when the density ratio is increased. Here again, this trend is associated with an increased intensity of the small scales, which is reflected by a larger dissipation rate and a larger viscous transport in the gas phase.

Figure 4.

Effect of the density ratio $R_\rho$ on the scale-by-scale kinetic energy for $\alpha = 75\,\%$ . The colours from blue to red correspond to $R_\rho = 1, \,5, \,25$ while the black curve is for single-phase turbulence: (a) $\mathbb{C} \equiv \mathbb{T}$ , (Total) (b) $\mathbb{C} \equiv \mathbb{L}$ , (Liquid) and (c) $\mathbb{C} \equiv \mathbb{G}$ , (Gas).

Figure 5.

Two-dimensional slices of the vorticity magnitude (in colour, units of $u'/L$ ) together with the liquid–gas interface (black curve) at $\alpha = 75\,\%$ and for (a–c) increasing $R_\rho$ .

Figure 6.

Effect of the density ratio $R_\rho$ on the scale-by-scale kinetic energy budgets for $\alpha = 75\,\%$ . The colours from blue to red correspond to $R_\rho = 1, \,5, \,25, \,125$ as in figure 1. The black curves are for the single-phase case at the same viscosity. The different lines correspond to $\langle \mathcal{T}\rangle _{\mathbb{C}}$ , $\langle \mathcal{P}\rangle _{\mathbb{C}}$ , $\langle \mathcal{S}\rangle _{\mathbb{C}}$ , $\langle \mathcal{V}\rangle _{\mathbb{C}}$ , $\langle \mathcal{F}\rangle _{\mathbb{C}}$ , with (a) $\mathbb{C} = \mathbb{T}$ , (c) $\mathbb{C} = \mathbb{L}$ and (d) $\mathbb{C} = \mathbb{G}$ . The horizontal dashed line indicates $4\langle F\rangle _{\mathbb{C}}$ , the limit at large separations of the forcing term. (b) The density correction terms $\langle \mathcal{C}(\boldsymbol{a}) \rangle _{\mathbb{T}}$ .

This is indeed confirmed by looking at the visualisations of the vorticity magnitude presented in figure 5. Although less apparent than in figure 2 for the drop regime, we still observe larger amplitude of vorticity in the gas phase close to the interface. This is in agreement with the analytical considerations of Terrington et al. (Reference Terrington, Hourigan and Thompson2022).

The different contributions to the KHM equation are portrayed in figure 6. For the total fluctuating field (figure 6 a), the effect of the density contrast between the two phases is perceptible on the surface tension term $\langle \mathcal{S}\rangle _{\mathbb{T}}$ which appears to decrease in amplitude and slides towards smaller scales when $R_\rho$ increases. These variations are quite limited though. We further confirm that the transfer associated with the nonlinear transport $\langle \mathcal{T}\rangle _{\mathbb{T}}$ is much weaker than in single-phase turbulence. The overall transfer of kinetic energy $\langle \mathcal{T}\rangle _{\mathbb{T}}$ + $\langle \mathcal{S}\rangle _{\mathbb{T}}$ is substantially larger than in single-phase turbulence and is shifted towards smaller scales. This is compensated by a larger contribution of the viscous term in the small-scale region. The scale-by-scale contributions of the ’density correction’ terms $\langle \mathcal{C}(\boldsymbol{a}) \rangle _{\mathbb{T}}$ (figure 6 b) are similar to what was observed before. They have same sign as the term to which they apply and increase in amplitude with the density contrast $R_\rho$ . Hence, these results do not vary substantially from those related to the drop regime.

Scrutinising the conditionally averaged budget in the liquid or gas phase reveals that $R_\rho$ has a strong effect on the pressure and viscous term (see figure 6 d). In the gas phase, pressure transport appears to act as an additional gain in the budget which is compensated by the viscous term. These processes dominate at $R_\rho = 25$ as they reach values roughly five times larger than the forcing term. In the liquid phase (see figure 6 c), we note that the statistics are mostly unchanged between $R_\rho = 5$ and $R_\rho = 25$ . For $R_\rho \geqslant 5$ and $\alpha = 75\,\%$ , the mass of the liquid represents more than 90 % of the total mass. Hence, in the bubble regime, one can readily assume that the dynamics of the liquid phase dominates the whole flow, and thus increasing further $\rho _L$ has only a limited impact. Only the case with $R_\rho = 1$ differs from the two other sets of curves. The transfer of kinetic energy in the liquid phase due to the nonlinear transport is systematically smaller than in single-phase turbulence. This effect might be the consequence of the smaller $R_\lambda$ in multiphase turbulence compared with its single-phase counterpart (see table 1). The Reynolds number $R_\lambda$ does not vary much when $R_\rho$ varies.

4.3. Effect of the liquid volume fraction $\alpha$

Figure 7.

Effect of the liquid volume fraction $\alpha$ on the scale-by-scale kinetic energy. The colours from blue to red correspond to $\alpha = 12.5\,\%,\ 25\,\%,\ 50\,\%,\ 75\,\%,\ 87.5 \,\%$ : (a) $\mathbb{C} = \mathbb{T}$ , (b) $\mathbb{C} = \mathbb{L}$ and (c) $\mathbb{C} = \mathbb{G}$ .

The analysis is now repeated to emphasise the effect of the liquid volume fraction $\alpha$ . We start by investigating the scale-by-scale kinetic energy which is plotted in figure 7(a–c) for $\mathbb{C} \equiv \mathbb{T}, \ \mathbb{L} \ \textrm {and}\ \mathbb{G}$ , respectively. We note that the liquid volume fraction has a non-monotonic effect on the total kinetic energy distribution across scales. Indeed, scrutinising the small-scale region in figure 7(a) reveals that $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{T}}$ first increases when $\alpha$ grows from 12.5 % to 50 % before decreasing again for $\alpha$ from 50 % to 87.5 %. The scale distribution of kinetic energy in multiphase turbulence tends towards that of single-phase turbulence when $\alpha \to 1$ , i.e. when the liquid drives the whole flow dynamics.

The kinetic energy scale distribution in the liquid (figure 7 b) shows that the case with $\alpha = 12.5\,\%$ departs significantly from the different curves. At such a small liquid volume fraction, some confinement effects are likely to be at play which preclude turbulent eddies of size larger than the typical size of liquid structures from developing, thereby impeding the turbulent activity in the liquid phase. The scale distribution of kinetic energy in the liquid for the other values of $\alpha$ is roughly invariant. The scale-by-scale turbulent kinetic energy conditioned by the gas phase is presented in figure 7(c). The contribution of the small scales to the kinetic energy is observed to increase with $\alpha$ . Here again, the scenario proposed above stating that the liquid phase having much larger inertia yields an increased production of vorticity and thus small-scale kinetic energy in the gas phase is likely to hold true. Hence, the more liquid, the more kinetic energy production at small scales in the gas phase. Despite this increased activity of the small scales in the gas phase, table 1 indicates that the total kinetic dissipation rate $\varepsilon _f$ is maximum for $\alpha = 50\,\%$ . This is obviously due to the decreasing proportion in mass and volume of the gas phase when $\alpha$ increases.

The visualisations presented in figure 8 do not really show any substantial difference of the vorticity magnitude when $\alpha$ changes. However, when $\alpha \to 1$ , the volume of the gas phase which lies close to the interface increases relative to the total gas volume. Hence, the volume where the vorticity magnitude is larger increases in proportion. This is likely to explain why $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{G}}$ increases in the small-scale range when the liquid volume fraction increases.

Figure 8.

Two-dimensional slices of the vorticity magnitude (in colour, units of $u'/L$ ) together with the liquid–gas interface (black curves) for (a–e) increasing $\alpha$ .

Figure 9.

Effect of the liquid volume fraction $\alpha$ on the scale-by-scale kinetic energy budgets. The colours from blue to red correspond to $\alpha = 12.5\,\%,\ 25\,\%,\ 50\,\%,\ 75\,\%,\ 87.5 \,\%$ as in figure 7, while the black curves correspond to the single-phase case. The different lines correspond to $\langle \mathcal{T}\rangle _{\mathbb{C}}$ , $\langle \mathcal{P}\rangle _{\mathbb{C}}$ , $\langle \mathcal{S}\rangle _{\mathbb{C}}$ , $\langle \mathcal{V}\rangle _{\mathbb{C}}$ , $\langle \mathcal{F}\rangle _{\mathbb{C}}$ , with (a) $\mathbb{C} = \mathbb{T}$ , (c) $\mathbb{C} = \mathbb{L}$ and (d) $\mathbb{C} = \mathbb{G}$ . The horizontal dashed line indicates $4\langle F\rangle _{\mathbb{C}}$ , the limit at large separations of the forcing term. (b) Correction terms $\mathcal{C}(\boldsymbol{a})$ .

The different terms of the KHM equation for different volume fractions are presented in figure 9. Figure 9(a) indicates that the unconditionally averaged terms of the KHM equation corresponding to a given $\alpha$ (e.g. 12.5 % and 25 %) differ greatly from those corresponding to an equivalent $1-\alpha$ (i.e. 87.5 % and 75 %). For instance, the surface tension term $\langle \mathcal{S} \rangle _{\mathbb{T}}$ for $\alpha = 12.5\,\%$ appears to be much larger than that for $\alpha = 87.5\,\%$ , even though the surface area $A_\varGamma$ of the interface is similar (see table 1). The transfer due to the nonlinear transport $\langle \mathcal{T} \rangle _{\mathbb{T}}$ has also a non-monotonic evolution with respect to the liquid volume fraction. Nevertheless, it seems to tend towards its single-phase counterpart when $\alpha \to 1$ . The overall transfer of kinetic energy which is due to both the nonlinear transport and surface tension term is compensated by the viscous term $\langle \mathcal{V} \rangle _{\mathbb{T}}$ . The latter is shifted towards larger scales when $\alpha \geqslant 50 \,\%$ . Here again, the term $\langle \mathcal{V} \rangle _{\mathbb{T}}$ tends towards its single-phase counterpart when $\alpha \to 1$ .

The contribution of the density correction terms $\langle \mathcal{C}(\boldsymbol{a})\rangle _{\mathbb{T}}$ is presented in figure 9(b). Except for the forcing term, we note that the correction terms are maximum when $\alpha = 50\,\%$ . This suggests that $\langle \mathcal{C}(\boldsymbol{a})\rangle _{\mathbb{T}}$ should be related to some extent to the surface area of the liquid–gas interface which is minimum for $\alpha = 12.5\,\%\ {\textrm{and}}\ \,87.5\,\%$ and maximum at $\alpha = 50\,\%$ . The contribution of $\langle \mathcal{C}(\boldsymbol{S})\rangle _{\mathbb{T}}$ also peaks at very different scales. Its contribution is perceptible at small scales when $\alpha$ is small and systematically moves towards large scales when $\alpha$ becomes larger. For the viscous term, the correction appears to peak around the same scale, independent of $\alpha$ .

The terms of the KHM equation conditionally averaged in the liquid and gas phases are presented in figures 9(c) and 9(d), respectively. For the liquid phase, we note that when $\alpha \to 1$ , the contribution of the pressure transport term $\langle \mathcal{P} \rangle _{\mathbb{L}}$ tends to zero and the other terms (the viscous term $\langle \mathcal{V} \rangle _{\mathbb{L}}$ and nonlinear transport term $\langle \mathcal{T} \rangle _{\mathbb{L}}$ ) tend towards those obtained in single-phase flows. A careful examination of figure 9(d) reveals that the same holds true for the gas phase when $\alpha \to 0$ . For small values of $\alpha$ (i.e. the drop regime), the pressure term in the liquid phase $\langle \mathcal{P} \rangle _{\mathbb{L}}$ contributes negatively at large scales which means that it represents a loss for the liquid kinetic energy budget at those scales. On the other hand, the budget in the gas phase (figure 9 d) indicates that the pressure term $\langle \mathcal{P} \rangle _{\mathbb{G}}$ contributes positively and very substantially when $\alpha \to 1$ , i.e. in the bubble regime. Note the amplitude of the pressure term $\langle \mathcal{P} \rangle _{\mathbb{G}}$ for $\alpha = 87.5\,\%$ which is roughly six times larger than the forcing term $\langle \mathcal{F} \rangle _{\mathbb{G}}$ . This means that when the liquid volume fraction is large, most of the kinetic energy production in the gas phase comes from pressure transport. This process is compensated by an increased contribution from the viscous term $\langle \mathcal{V} \rangle _{\mathbb{G}}$ .

In summary, varying the liquid volume fraction reveals that the physics of the drop regime ( $\alpha \lt 50\,\%$ ) differs significantly from that of the bubble regime ( $\alpha \gt 50\,\%$ ). This stresses again that the results obtained by setting $R_\rho = 1$ can very hardly be representative of what occurs in ‘real’ multiphase flows characterised by fluctuations, even moderate, of the fluid density. Our results also reveal that the budget in the most abundant phase, when the minority phase represents less than say 5 %, could possibly correspond to that one observed in single-phase turbulence.

4.4. Effect of the Reynolds number $Re$

Figure 10.

Effect of the Reynolds number $Re \equiv 1/\nu$ on the scale-by-scale kinetic energy. The colours from blue to red correspond to $Re = 263, 714, 1887$ . The grey curves correspond to the single-phase (SP) data at the same Reynolds number: (a) $\mathbb{C} = \mathbb{T}$ , (b) $\mathbb{C} = \mathbb{L}$ and (c) $\mathbb{C} = \mathbb{G}$ .

Figure 11.

Two-dimensional slices of the vorticity magnitude (in colour, units of $u'/L$ ) together with the liquid–gas interface (black curves) for (a–c) increasing $Re$ .

The effect of Reynolds number $Re$ is studied independently of the other parameters by varying the kinematic viscosity (see table 1). The kinetic energy distribution among the different scales is plotted in figure 10. Decreasing the value of the kinematic viscosity shifts the viscous cutoff towards the small scales. The distribution of kinetic energy is thus spread over a wider range of scales and the small scales are more energetic when the Reynolds number increases. Here again, we further note that for $\mathbb{C}=\mathbb{T}$ (figure 10 a), the small-scale kinetic energy of multiphase turbulence is larger than its single-phase counterpart at the same Reynolds number. As with figure 1(a), the distribution conditioned in the liquid is very comparable to single-phase turbulence, only the distribution in the gas phase deviating significantly. Hence, the larger kinetic energy observed for $\mathbb{C}=\mathbb{T}$ at small scales is due to a larger kinetic energy in the gas phase.

The visualisations of the vorticity field presented in figure 11 indicate again that these small scales are produced in the gas phase in close vicinity of the interface. When the Reynolds number is increased, there are more vortical structures, they are of smaller size and their magnitude increases. This is reflected in the increase of $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{T}}$ and $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{G}}$ at small scales. Despite this, the amount of surface area of the two-fluid interface appears to be independent of the Reynolds number (see table 1).

Figure 12.

Effect of the Reynolds number $Re \equiv 1/\nu$ on the scale-by-scale kinetic energy budgets. The colours from blue to red correspond to $Re = 263, 714, 1887$ as in figure 10. The grey curves correspond to the single-phase data at the same Reynolds number. The different lines correspond to $\langle \mathcal{T}\rangle _{\mathbb{C}}$ , $\langle \mathcal{P}\rangle _{\mathbb{C}}$ , $\langle \mathcal{S}\rangle _{\mathbb{C}}$ , $\langle \mathcal{V}\rangle _{\mathbb{C}}$ , $\langle \mathcal{F}\rangle _{\mathbb{C}}$ , with (a) $\mathbb{C} = \mathbb{T}$ , (c) $\mathbb{C} = \mathbb{L}$ and (d) $\mathbb{C} = \mathbb{G}$ . The horizontal dashed line indicates $4\langle F\rangle _{\mathbb{C}}$ , the limit at large separations of the forcing term. (b) The density correction terms $\mathcal{C}(\boldsymbol{a})$ .

The terms of the KHM equation are displayed in figure 12 for the total fluctuating field and the liquid and gas phases. For the total fluctuating field (figure 12 a), decreasing the viscosity increases the separation between large and small scales and thus the intermediate scales develop. The viscous term is shifted towards smaller scales and the terms acting at the intermediate range, i.e. $\langle \mathcal{T}\rangle _{\mathbb{T}}$ and $\langle \mathcal{S}\rangle _{\mathbb{T}}$ , widen and increase in amplitude. This is qualitatively equivalent to what occurs in single-phase turbulence. The same observation stands true for the additional term due to density variations $\langle \mathcal{C}(\boldsymbol{a})\rangle _{\mathbb{T}}$ presented in figure 12(b). The overall transfer of kinetic energy, which is composed of both the nonlinear transport and the surface tension term, exceeds that of single-phase turbulence at the same Reynolds number. As observed before, this is compensated by the viscous term being active at smaller scales. The pressure term does not contribute much to the budget of $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{T}}$ .

Looking at the terms of the KHM equation in liquid and gas phases (figure 12 c,d), it is noted that $\langle \mathcal{T}\rangle _{\mathbb{L,G}}$ increases in amplitude and peaks at smaller scales when the Reynolds number increases. Its contribution is smaller than that of single-phase turbulence. Surprisingly, the pressure term appears to be independent of the Reynolds number. This applies to both the liquid and gas phases. Pressure transport is again found to contribute positively in the gas phase. For the liquid phase, it is a gain at small scales and a loss at large scales.

4.5. Effect of the Weber number $\textit{We}$

Figure 13.

Effect of the Weber number $\textit{We} \equiv \overline {\rho }/\sigma$ on the scale-by-scale kinetic energy. The colours from blue to red correspond to $\textit{We} = 12.5, \,25, \,50$ , while the black curves correspond to the single-phase case: (a) $\mathbb{C} = \mathbb{T}$ , (b) $\mathbb{C} = \mathbb{L}$ and (c) $\mathbb{C} = \mathbb{G}$ .

Figure 14.

Two-dimensional slices of the vorticity magnitude (in colour, units of $u'/L$ ) together with the liquid–gas interface (black curves) for (a–c) increasing $\textit{We}$ .

We now investigate the effect of Weber number $\textit{We}$ with $R_\rho = 25$ , $\alpha = 25\,\%$ and $Re = 714$ . The scale-by-scale kinetic energy $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{C}}$ is displayed in figures 13(a), 13(b) and 13(c) for $\mathbb{C} \equiv \mathbb{T},\ \mathbb{L} \ \textrm {and}\ \mathbb{G}$ , respectively.

As an overall observation, the scale-by-scale kinetic energy appears to vary only marginally when the surface tension is varied. One observes though some very slight modifications in the intermediate range of scales for the unconditionally averaged kinetic energy $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{T}}$ (figure 13 a). The small scales remain more or less unchanged consistently with the values of $\varepsilon _f$ reported in table 1. The same small variations in the intermediate range of scales are also visible in the liquid phase in figure 13(b). In the gas phase (figure 13 c), the distribution of kinetic energy remains independent of the Weber number in the large and intermediate range of scales. Some marginal differences appear when one travels towards smaller scales where $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{G}}$ appears to increase very slightly with $\textit{We}$ . Looking at $\langle |\delta \boldsymbol{u}|^2_\rho \rangle _{\mathbb{T}}$ in figure 13(a), the small-scale kinetic energy of multiphase turbulence is again observed to be larger than that pertaining to single-phase turbulence. This results from an increased small-scale activity in the gas phase (see figure 13 c).

The direct visualisations of the flow presented in figure 14 indicate that the magnitude of the vorticity field does not vary substantially when $\textit{We}$ is varied. The increase of the Weber number yields more pronounced corrugations of the interface and hence a larger surface area (see also table 1). The theory of Terrington et al. (Reference Terrington, Hourigan and Thompson2022) indicates that one particular term of the vorticity production is proportional to the surface tension $\sigma$ and the gradient along the interface of the mean curvature. Hence, when the Weber number increases, the decrease of surface tension is likely compensated by larger gradients of curvature (more corrugations), leading to roughly the same vorticity magnitude, and hence a comparable small-scale kinetic energy distribution in the gas phase.

Figure 15.

Effect of the Weber number $\textit{We} = \overline {\rho }/\sigma$ on the scale-by-scale kinetic energy budgets. The colours from blue to red correspond to $\textit{We} = 12.5, \,25, \,50$ as in figure 13, while the black curves correspond to the single-phase case. The different lines correspond to $\langle \mathcal{T}\rangle _{\mathbb{C}}$ , $\langle \mathcal{P}\rangle _{\mathbb{C}}$ , $\langle \mathcal{S}\rangle _{\mathbb{C}}$ , $\langle \mathcal{V}\rangle _{\mathbb{C}}$ , $\langle \mathcal{F}\rangle _{\mathbb{C}}$ , with (a) $\mathbb{C} = \mathbb{T}$ , (c) $\mathbb{C} = \mathbb{L}$ and (d) $\mathbb{C} = \mathbb{G}$ . The horizontal dashed line indicates $4\langle F\rangle _{\mathbb{C}}$ , the limit at large separations of the forcing term. (b) The density correction terms $\mathcal{C}(\boldsymbol{a})$ .

The effect of surface tension on the scale-by-scale kinetic energy budget is presented in figure 15. Scrutinising the unconditionally averaged terms (figure 15 a), we note that the surface tension term $\langle \mathcal{S}\rangle _{\mathbb{T}}$ drops when surface tension decreases. It is thus plausible that the scale-by-scale contribution of the surface tension term will tend to zero at infinite Weber number. The Weber number further appears to have an influence on the pressure term $\langle \mathcal{P}\rangle _{\mathbb{T}}$ , whose contribution changes sign when the Weber number changes. This behaviour is rather surprising. The two other terms of (2.2) due to the nonlinear transport $\langle \mathcal{T}\rangle _{\mathbb{T}}$ and the viscous term $\langle \mathcal{V}\rangle _{\mathbb{T}}$ are only marginally influenced by variations of the surface tension. However, they are substantially different from those obtained in single-phase flows. For instance, the viscous term starts to contribute at much smaller scales to compensate an increased overall transfer of kinetic energy, which is again much larger in multiphase than in single-phase flows.

The density correction terms are portrayed in figure 15(b). As observed previously, we note that $-\langle \mathcal{C}(\boldsymbol{S})\rangle _{\mathbb{T}}$ , $-\langle \mathcal{C}(\boldsymbol{V})\rangle _{\mathbb{T}}$ and $-\langle \mathcal{C}(\boldsymbol{F})\rangle _{\mathbb{T}}$ have the same sign as $\langle \mathcal{S}\rangle _{\mathbb{T}}$ , $\langle \mathcal{V}\rangle _{\mathbb{T}}$ and $\langle \mathcal{F}\rangle _{\mathbb{T}}$ , respectively. The contribution of these terms seem to decrease when the Weber number increases.

The terms of (2.2) conditionally averaged budget in the liquid and gas phases are displayed in figures 15(c) and 15(d), respectively. Noticeable is the evolution of the pressure transport term in the liquid phase $\langle \mathcal{P}\rangle _{\mathbb{L}}$ when the surface tension is varied. The latter is positive at small scales (a gain) and negative (a loss) at large scales. The scale at which the change of sign occurs decreases with surface tension. The effect of surface tension is also clearly visible on the pressure term in the gas phase, $\langle \mathcal{P}\rangle _{\mathbb{G}}$ , which appears to increase when the Weber number increases. Its contribution to the scale-by-scale budget is positive (a gain) and is compensated mainly by an increased contribution of the viscous term $\langle \mathcal{V}\rangle _{\mathbb{G}}$ .

4.6. Phenomenological predictions for the pivotal scale

So far, we have shown that surface tension yields an additional transfer term of kinetic energy between scales. Of particular interest is the scale denoted $r_{\mathcal{S}}$ at which the term $\langle \mathcal{S} \rangle _{\mathbb{T}}$ is maximum. Indeed, for scales larger than $r_{\mathcal{S}}$ , the interface pumps kinetic energy, stores it as surface energy before releasing it back into kinetic energy but at scales smaller than $r_{\mathcal{S}}$ .

As already mentioned in § 1, Crialesi-Esposito et al. (Reference Crialesi-Esposito, Chibbaro and Brandt2023b ) interpreted $r_{\mathcal{S}}$ as the pivotal scale between, at large scales, a range dominated by breakup and, at small scales, a range dominated by coalescence. They thus suggested that $r_{\mathcal{S}}$ should coincide with the Kolmogorov–Hinze scale, denoted $r_H$ , which is known as the largest fluid parcel which would resist breakup.

Going back to the original idea of Hinze (Reference Hinze1955), Mukherjee et al. (Reference Mukherjee, Safdari, Shardt, Kenjereš and Van den Akker2019) extracted the Kolmogorov–Hinze scale from a local, i.e. scale-dependent, Weber number (see also Perlekar et al. Reference Perlekar, Biferale, Sbragaglia, Srivastava and Toschi2012). We proceed similarly here, and the local Weber number is defined from the scale-by-scale kinetic energy:

(4.1) \begin{eqnarray} We(r) := \frac {1}{2} \frac {\overline {\rho }}{\langle \rho \rangle _{\mathbb{T}}} \frac { \langle |\delta \boldsymbol{u}|_\rho ^2\rangle _{\mathbb{T}} r}{\sigma }. \end{eqnarray}

Since $\langle |\delta \boldsymbol{u}|_\rho ^2\rangle _{\mathbb{T}} / \langle \rho \rangle _{\mathbb{T}} \to 2u'^2$ when $r \to L$ , this definition of the local Weber number is such that $\textit{We}(r) \to We$ as $r \to L$ . The Kolmogorov–Hinze scale is then simply given as $\textit{We}(r=r_H) =1$ .

Note that the classical definition for the Kolmogorov–Hinze scale can be recovered if the Reynolds number is sufficiently large for an inertial range to exist and for $r_H$ to fall in such an inertial range. In this situation, Kolmogorov scaling might apply for the Favre-averaged structure functions:

(4.2) \begin{eqnarray} \frac {\langle |\delta \boldsymbol{u}|_\rho ^2 \rangle _{\mathbb{T}}}{ \langle \rho \rangle _{\mathbb{T}}} = C_K(\epsilon _f r )^{{2}/{3}}, \end{eqnarray}

where $C_K$ is known as the Kolmogorov constant. This yields the following relation for the Kolmogorov–Hinze scale:

(4.3) \begin{eqnarray} r_H = \left (\frac {2}{C_K}\right )^{3/5} \left (\frac {\sigma }{\overline {\rho }}\right )^{3/5} \varepsilon _f^{-2/5} , \end{eqnarray}

which is very similar to the classical definition of the Kolmogorov–Hinze scale (see e.g. Hinze Reference Hinze1955; Perlekar et al. Reference Perlekar, Biferale, Sbragaglia, Srivastava and Toschi2012; Crialesi-Esposito et al. Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022, Reference Crialesi-Esposito, Chibbaro and Brandt2023b ; Cannon et al. Reference Cannon, Soligo and Rosti2024). Defining $r_H$ from (4.2) instead of (4.3) applies to cases where the Reynolds number is not sufficiently large for an inertial range scaling to be invoked. Note that Hinze (Reference Hinze1955) introduced an ad hoc critical Weber number, $\textit{We}_c$ , which initially was aimed at accounting for the variations of viscosity. This can readily be done here as well by defining $r_H$ as $\textit{We}(r=r_H) = We_c$ . Note also that (4.3) differs from the classical definition of the Kolmogorov–Hinze scale through the appearance of the mean density $\overline {\rho }$ instead of the density of the most abundant fluid. One can thus think of (4.3) as a likely more general expression for $r_H$ that applies to the dense regime where the notion of carrier versus dispersed phase is irrelevant.

The values for $r_{\mathcal{S}}$ and $r_H$ are reported in table 1. They are also represented in figures 3(a), 6(a), 9(a), 12(a) and 15(a) as filled circles and vertical lines, respectively. Comparing the values of $r_{\mathcal{S}}$ and $r_H$ reveals that the Kolmogorov–Hinze scale $r_H$ does not systematically scale with $r_{\mathcal{S}}$ . This is particularly visible when the density ratio between the two phases is varied. In this situation, $r_H$ is roughly constant while $r_{\mathcal{S}}$ decreases when $R_\rho$ is varied from 1 to 125. It is also visible when the Weber number is varied. In this situation, both $r_H$ and $r_{\mathcal{S}}$ decrease when $\textit{We}$ increases. However, $r_H$ does not vary in proportion to $r_{\mathcal{S}}$ .

The discrepancy between $r_H$ and $r_{\mathcal{S}}$ was already noted by (Crialesi-Esposito et al. Reference Crialesi-Esposito, Chibbaro and Brandt2023b ) at $R_\rho = 1$ but for varying viscosity ratio $R_\mu$ . This difference could possibly be taken into account by using an ad hoc critical Weber number function of $R_\rho$ . One could otherwise seek for another characteristic scale as done below.

Using some different forms of the scale-by-scale kinetic energy budget equations in gravity-driven bubbly flows, Pandey et al. (Reference Pandey, Ramadugu and Perlekar2020, Reference Pandey, Mitra and Perlekar2023) and Ramirez et al. (Reference Ramirez, Burlot, Zamansky, Bois and Risso2024) have shown that the surface tension term has qualitatively the same scale distribution as that one reported here or in Crialesi-Esposito et al. (Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022, Reference Crialesi-Esposito, Chibbaro and Brandt2023b ) and Cannon et al. (Reference Cannon, Soligo and Rosti2024). However, the regime of bubbly flows investigated by Pandey et al. (Reference Pandey, Ramadugu and Perlekar2020, Reference Pandey, Mitra and Perlekar2023) and Ramirez et al. (Reference Ramirez, Burlot, Zamansky, Bois and Risso2024) does not reveal any breakup or coalescence, and thus does not fall into the Kolmogorov–Hinze framework. This means that breakup/coalescence may not be required to explain the behaviour of the surface tension term.

Here, we extend somewhat the phenomenology proposed by Crialesi-Esposito et al. (Reference Crialesi-Esposito, Chibbaro and Brandt2023b ) and replace breakup/coalescence events by the more general mechanisms of interface deformation/restoration. We hypothesise that scales larger than $r_{\mathcal{S}}$ contribute to interface deformation, thereby loosing kinetic energy which is converted into surface energy. The mechanism of interface deformation can, but not necessarily, include breakup. On the other hand, for scales smaller than $r_{\mathcal{S}}$ , surface energy is released back into kinetic energy through interface restoration. By restoration, we mean that any corrugations of the interface smaller than $r_{\mathcal{S}}$ are flattened through the action of surface tension. Coalescence can also contribute to the disappearance of scales smaller than $r_{\mathcal{S}}$ , thereby releasing kinetic energy at those scales. Breakup can also contribute to restoration. This is true for instance during the Rayleigh–Plateau instability where the deformed ligaments have larger surface area than the formed droplets after breakup. For simplicity, the range of scales $r\gt r_{\mathcal{S}}$ is now referred to as the deformation range, and the range of scales $r\lt r_{\mathcal{S}}$ as the restoration range.

In recent work (Thiesset, Ménard & Dumouchel Reference Thiesset, Ménard and Dumouchel2021; Gauding et al. Reference Gauding, Thiesset, Varea and Danaila2022; Thiesset & Federrath Reference Thiesset and Federrath2023), we introduced a general morphological analysis of corrugated interfaces. It was shown in particular in Gauding et al. (Reference Gauding, Thiesset, Varea and Danaila2022) and Thiesset & Federrath (Reference Thiesset and Federrath2023) that when observed at a sufficiently small scale, an interface appears flat. If observed at larger scales, the interface is seen to be curved though smooth, and can eventually be observed to be fractal at even larger scales. Building on the exact analytical results of Kirste & Porod (Reference Kirste and Porod1962) and Frisch & Stillinger (Reference Frisch and Stillinger1963) (see also Gauding et al. Reference Gauding, Thiesset, Varea and Danaila2022; Thiesset & Federrath Reference Thiesset and Federrath2023) a typical scale below which an interface is observed to be flat can be obtained. The latter is written

(4.4) \begin{eqnarray} r_c = \sqrt {\frac {8}{3\langle H^2 \rangle _s - \langle G \rangle _s}}, \end{eqnarray}

where $H$ and $G$ represent mean curvature and Gaussian curvature, respectively, while $\langle \bullet \rangle _s$ stands for a surface area weighted average. For iso-surfaces formed by passive scalar, Gauding et al. (Reference Gauding, Thiesset, Varea and Danaila2022) proved that $r_c$ scales with the Batchelor length scale.

To say it differently, an observer zooming on the interface down to scale $r_c$ will see a flat surface. The curvature of the interface is visible when it is observed with a field of view larger than $r_c$ . In other words, the smallest interface corrugation is related to $r_c$ . This scale is thus likely to be a good candidate to represent $r_{\mathcal{S}}$ , the pivotal scale between the deformation and the restoration range. The values for $r_c$ computed from (4.4) are reported in table 1. Unfortunately, although the above reasoning is appealing on paper, we find that the variations of $r_c$ do not coincide with the variations $r_{\mathcal{S}}$ . The scale $r_{\mathcal{S}}$ is systematically smaller and does not vary in proportion to $r_c$ . Nevertheless, it is found that $r_c$ coincides rather satisfactorily with $r_H$ so that one can conclude that the Kolmogorov–Hinze framework provides a rather suitable prediction for the smallest corrugation scales of the interface.