3.1. Local void fraction and identification of meso-scale structures

Paralleling what we did for turbulent particle-laden flows (Monchaux, Bourgoin & Cartellier Reference Monchaux, Bourgoin and Cartellier2010; Sumbekova et al. Reference Sumbekova, Cartellier, Aliseda and Bourgoin2017; Mora et al. Reference Mora, Aliseda, Cartellier and Obligado2018), we exploit 1-D Voronoï tessellations built from the gas phase indicator function $X_G(t)$ (Raimundo Reference Raimundo2015; Mezui, Cartellier & Obligado Reference Mezui, Cartellier and Obligado2018; Raimundo et al. Reference Raimundo, Cloupet, Cartellier, Beneventi and Augier2019). Here, $X_G(t)$ is deduced from the signal delivered by an optical probe. For the gas phase indicator function measurements presented here, the probe orientation was held fixed (the probe was directed downwards). As shown in figure 2, Voronoï cells are then built as successive time intervals, each containing a single bubble. For that, the centres $T_k$ of successive gas residence times $t_{gk}$ are identified. The mid-distance between successive centres $T_k$ and $T_{k+1}$ defines a Voronoï cell boundary. This process is repeated for all detected bubbles, and the width of the $k$th Voronoï cell that contains the $k$th bubble is given by $\Delta T_k = (T_{k+1} - T_{k-1})/2$.

Figure 2.

Construction of a 1-D Voronoï tessellation from the gas phase indicator function.

Probability density functions (p.d.f.s) of the Voronoï cell width $\Delta T_k$ normalised by the average $\langle \Delta T_k\rangle$ are presented in figure 3(a) for various gas superficial velocities: all these data have been collected on the bubble column axis at $H/D=3.625$. Care was taken to ensure a correct convergence of these distributions. The latter comprises between 8000 and 13 000 bubbles: these samples correspond to measuring durations from 95 to 950 s depending on flow conditions.

Figure 3.

(a) Centred p.d.f.s of 1-D Voronoï cells width $\Delta T_k/\langle \Delta T_k\rangle$ built from Doppler probe signals at various gas superficial velocities. The dash line represents the 1-D Voronoï distribution for an RPP, i.e. a random Poisson process. The vertical dashed lines indicate the thresholds for the definitions of clusters and voids (as defined by Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010). (b) Evolution of the standard deviation of 1-D Voronoï distributions with the gas superficial velocity. The horizontal dash line indicates the standard deviation for an RPP while the vertical dashed line delineate the homogeneous–heterogeneous transition. Measurements performed in a $D=0.4$ m column, on the column axis at $H/D=3.625$. P.d.f.s of 1-D Voronoï cells width (c) $\Delta T_k$ and (d) $\Delta T_k / (D/V_{sg})$ . We remark that in the last two panels the binning used to compute the p.d.f.s calculation is different from panel (a).

Qualitatively, the width $\Delta T_k$ of the time interval containing the $k$th bubble is an indication of the local concentration. A short duration $\Delta T_k$ means the presence of a close-by bubble while a large duration indicates that the $k$th bubble is somewhat isolated. We will come back later to the connection between normalised cell durations $\Delta T_k/\langle \Delta T_k\rangle$ and concentration. For the time being, let us focus on the allure of these p.d.f.s. The dash line in figure 3(a) represents the p.d.f. of normalised cell durations $\Delta T_k/\langle \Delta T_k\rangle$ for a random Poisson process – RPP in short (Ferenc & Néda Reference Ferenc and Néda2007) – that has no correlation at any scale. Clearly, and as noted by Raimundo et al. (Reference Raimundo, Cloupet, Cartellier, Beneventi and Augier2019), measured distributions at large enough $V_{sg}$ differ from the RPP case. In particular, both very large cell durations (corresponding to dilute conditions) and very small cell durations (corresponding to dense conditions) are more probable than for an RPP. Figure 3(c,d) show the p.d.f. with non-normalised cell durations ($\Delta T_k$) and with the alternative normalisation $\Delta T_k / (D/V_{sg})$, respectively. The latter represents a large-scale time scale for the gas phase, and gives similar results in the heterogeneous regime than the normalisation $\Delta T_k/\langle \Delta T_k\rangle$.

Following Monchaux et al. (Reference Monchaux, Bourgoin and Cartellier2010), the distance to an RPP is commonly appreciated by examining the standard deviation $\sigma _{voronoi}$ of the p.d.f. of Voronoï cells widths. As shown in figure 3(b), such standard deviation drastically increases from a low value, comparable to that of an RPP, to a much higher value (close to unity) when the system shifts from the homogeneous to the heterogeneous regime. In the homogeneous regime, the measured standard deviation of Voronoï cells p.d.f.s evolves between 0.8 and 0.85. This is slightly larger than the 0.71 limit for an RPP of point particles as determined by Ferenc & Néda (Reference Ferenc and Néda2007) (according to Uhlmann (Reference Uhlmann2020), the standard deviation for an RPP with finite-size particles is even lower). The origin of that small difference is unclear. This could be the mark of an inhomogeneous spatial repartition of bubbles in the homogeneous regime because of some gas maldistribution at injection (Nedeltchev Reference Nedeltchev2020): such a scenario is supported by the analysis of liquid velocity profiles (Lefebvre et al. Reference Lefebvre, Mezui, Obligado, Gluck and Cartellier2022). Alternately, that small difference could be due to the measuring method itself because the optical probe allows detecting the centres of gas chords and not the centres of bubbles, and because most bubbles are not spherical. Hence, the value of the standard deviation measured in the homogeneous regime could be interpreted at the reference RPP level as detected with the probe technique. The key points in figure 3(b) are the very sharp increase in $\sigma _{voronoi}$ observed at the homogeneous–heterogeneous transition, and the large value, well above that of an RPP, that $\sigma _{voronoi}$ reaches at high $V_{sg}$. The shortcomings of 1-D Voronoï analysis in complex flows have been discussed elsewhere (Mora et al. Reference Mora, Aliseda, Cartellier and Obligado2018, Reference Mora, Aliseda, Cartellier and Obligado2019): one key result is that the clear difference observed with the standard deviation of an RPP unambiguously demonstrates that clustering does occur in the present flow conditions. Furthermore, for all heterogeneous conditions investigated (that is, for $V_{sg}$ up to $24\,{\rm cm}\,{\rm s}^{-1}$), the standard deviation $\sigma _{voronoi}$ remains nearly the same: that feature also indicates that clustering is a central characteristic of the heterogeneous regime. Finally, let us underline that, as for turbulent flows laden with inert particles (Sumbekova et al. Reference Sumbekova, Cartellier, Aliseda and Bourgoin2017), the main contribution to the standard deviation comes from cells at large $\Delta T_k/\langle \Delta T_k\rangle$ corresponding to low void fractions, compared with the contribution from cells with intermediate $\Delta T_k/\langle \Delta T_k\rangle$ (void fractions close to the mean value) or with low $\Delta T_k/\langle \Delta T_k\rangle$ (high void fractions).

To quantify the connection between cell width $\Delta T_k/\langle \Delta T_k\rangle$ and concentration, we consider two approaches. First, we follow what we did for a turbulent flow laden with droplets (Sumbekova et al. Reference Sumbekova, Cartellier, Aliseda and Bourgoin2017; Mora et al. Reference Mora, Aliseda, Cartellier and Obligado2018), by connecting the ratio $\Delta T_k/\langle \Delta T_k\rangle$ to linear number densities, i.e. with the number of inclusions detected per unit length. The length corresponds to the measuring duration multiplied by the axial velocity $V_{axial}$ of inclusions. The local number density $\gamma _k$ (number of inclusions per meter) in the $k$th cell equals $1/[\Delta T_k V_{axial}]$, while $1/[\langle \Delta T_k\rangle V_{axial}]$ is the mean number density $\gamma$. Therefore, the normalised cell width $\Delta T_k/\langle \Delta T_k\rangle =\gamma / \gamma _k$ represents the inverse of the instantaneous (i.e. at the scale of the Voronoï cell) number density divided by the mean number density. When applied to bubble columns (Raimundo Reference Raimundo2015; Mezui et al. Reference Mezui, Cartellier and Obligado2018; Raimundo et al. Reference Raimundo, Cloupet, Cartellier, Beneventi and Augier2019), we considered $V_{axial}$ as the mean bubble velocity, and $\gamma$ was assumed to be proportional to the mean dispersed phase concentration. Under these assumptions, the inverse of $\Delta T_k/\langle \Delta T_k\rangle$, i.e. $\gamma _k/\gamma$, provides the magnitude of the local gas concentration (local at the scale of the Voronoï cell) with respect to the mean gas fraction at the measuring location. In figure 3(a), the abscissa $\Delta T_k/\langle \Delta T_k\rangle$ varies from 0.07 to 10 so that $\gamma _k/\gamma$ covers more than two decades as it evolves between 0.1 and approximately 14.

However, a second approach is required because, for the heterogeneous conditions considered here, $\gamma _k/\gamma$ does not coincide with the ratio $\varepsilon _k / \varepsilon$ of the void fraction $\varepsilon _k$ relative to the $k$th cell to the mean gas hold-up $\varepsilon$ at the measuring location. Indeed, in the turbulent particle-laden flows we have previously analysed, all inclusions travelled with almost the same axial velocity. This is no longer the case for bubbles in the heterogeneous regime as their velocities experience strong variations (see figure 2 in Mezui et al. Reference Mezui, Obligado and Cartellier2022), leading to a standard deviation as large as 60 % of the mean. Hence, the selection of a mean bubble velocity to transform time into space induces very large distortions on the concentration estimate by way of $\gamma _k$. To correct for these distortions and to evaluate reliable local void fractions, it is appropriate to rely on gas residence times as the latter naturally account for the actual velocity of each bubble. The void fraction relative to the $k$th Voronoï cell equals the sum of gas residence times included in that cell divided by the cell duration $\Delta T_k$. As shown in Appendix A, the ratio $\Delta T_k/\langle \Delta T_k\rangle$ is indeed related to $\varepsilon /\varepsilon _k$, but it does not coincide with $\varepsilon /\varepsilon _k$ as the prefactor between these two quantities varies with the gas residence time (see (A1) in Appendix A). In consequence, in the following, we will use the ratio $\gamma _k/\gamma$ as a crude, qualitative characterisation of meso-scale structures in terms of concentration, while exact measurements of the gas fraction $\varepsilon _k / \varepsilon$ will be considered in § 5 for discussing modelling issues.

Going back to figure 3(a), and whatever the flow conditions, the measured p.d.f.s of Voronoï cells cross the RPP at two fixed abscissa represented by vertical dashed lines. A third intersection sometimes occurs in the very dense limit (at $\Delta T_k/\langle \Delta T_k\rangle$ approximately 0.1, that is, for $\gamma _k/\gamma$ approximately 10), but it will not be considered here because its occurrence is far too sensitive to the sample size. As for turbulent particle-laden flows (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010), we define three populations out of the two stable thresholds. A Voronoï cell (and the bubble it contains) belongs to a ‘dense’ region when $\Delta T_k/\langle \Delta T_k\rangle$ is below 0.51, or equivalently when $\gamma _k/\gamma$ is higher than 1.96. A Voronoï cell (and the bubble it contains) belongs to an ‘empty’ or ‘void’ region when $\Delta T_k/\langle \Delta T_k\rangle$ is above 2.89, or equivalently when $\gamma _k/\gamma$ is lower than 0.34. In between, the cell (and its bubble) pertains to an ‘intermediate’ region. Let us underline that these thresholds are stable with respect to $V_{sg}$ (see figure 3a). They seem also independent of the column diameter as the same thresholds were observed for a $D=1$ m column at $V_{sg} =0.25\,{\rm m}\,{\rm s}^{-1}$ (see figure 11 in Raimundo et al. Reference Raimundo, Cloupet, Cartellier, Beneventi and Augier2019).

Owing to figure 3(a), the probability for bubbles to belong to ‘empty’ or to ‘dense’ regions is larger than in RPP. This is confirmed by the data presented in table 1. On average, 38 % of the bubbles belong to dense regions, while 5 % are within empty regions and 57 % are in intermediate regions: these figures remain stable within approximately 5 % over the whole heterogeneous regime, that is, for $V_{sg}$ from $6\,{\rm cm}\,{\rm s}^{-1}$ to $25\,{\rm cm}\,{\rm s}^{-1}$. As expected, the figures relative to dense and to empty regions are significantly larger than those for an RPP. Table 1 also confirms that, in the homogeneous regime, the repartition of bubbles in number between dense, empty and intermediate regions is very close to the repartition in number for an RPP.

Table 1.

Typical distributions of the dispersed phase between void regions, intermediate regions and dense regions in the heterogeneous and homogeneous regimes and comparison with an RPP. From measurements on the axis of a $D=0.4$ m bubble column at $H/D=3.625$.

The difference between homogeneous and heterogeneous conditions is also manifest in terms of void fractions. In the heterogeneous regime, the contributions to the local void fraction are typically 17 % for the dense regions, 10 % for the empty regions and 70 % for intermediate regions (table 1). These values correspond to average values for $V_{sg} \geq 9 \,{\rm cm}\,{\rm s}^{-1}$: they change by less than 1 % when considering data over the interval $V_{sg} \geq 6 \,{\rm cm}\,{\rm s}^{-1}$. Note that there is a slight decrease of the contribution of dense regions to the local void fraction as $V_{sg}$ increases, which is compensated by a slight increase with $V_{sg}$ of the contributions of empty and intermediate regions.

3.2. Characterisation of meso-scale structures

Once all bubbles have been distributed within the three populations, meso-scale structures are then formed using the following procedure. Bubbles belonging to a ‘dense’ region and successive in time are assembled to form a ‘cluster’. Similarly, successive bubbles belonging to an ‘empty’ region are assembled to form a ‘void’. The same process was used for intermediate regions. The characteristics of the resulting meso-scale structures in terms of size and concentration are then extracted.

1. (i) The void fraction (in absolute value) in a given meso-scale structure is evaluated as the sum of gas residence times for all bubbles pertaining to that structure divided by the duration of that structure, the later being the sum of all involved $\Delta T_k$. The distributions of void fraction in clusters and in voids are exemplified in figure 4 for various $V_{sg}$.

2. (ii) The size of a given meso-scale structure is estimated as the duration of the structure multiplied by the average bubble velocity, the latter being evaluated for the bubbles belonging to the structure considered: these conditional velocities are analysed in the next section. Length distributions for clusters and for voids are provided in figure 5 for various $V_{sg}$.

Figure 4.

P.d.f.s of void fraction (in absolute value) (a) in clusters and (b) in void regions for different superficial velocities. For these statistics, we considered clusters comprising at least two bubbles. Measurements performed in a $D=0.4$ m column, on the column axis at $H/D=3.625$.

Figure 5.

P.d.f.s of lengths (a) of clusters and (b) of void regions for different superficial velocities. For these statistics, we considered clusters that comprise at least two bubbles. Measurements performed in a $D=0.4$ m column, on the column axis at $H/D=3.625$.

We considered two options for clusters: either the minimum number of bubbles in a cluster is set to one so that all Voronoï cells with a $\Delta T_k/\langle \Delta T_k\rangle$ below the threshold are considered as clusters, or the minimum number of bubbles is set to two so that clusters involving a single bubble are excluded. The second option has been suggested to help distinguishing between ‘coherent’ and ‘random’ clusters in turbulent particle-laden flows (Mora et al. Reference Mora, Aliseda, Cartellier and Obligado2019). Here, and for all the flow conditions pertaining to the heterogeneous regime, it happens that 37 % to 40 % of clusters involve a single inclusion.

It should also be underlined that zones below or beyond the above-defined thresholds also exist for an RPP. Hence, one can still identify and statistically characterise ‘dilute’ and ‘dense’ regions in homogeneous conditions even though the corresponding Voronoï distributions are very close to and/or almost collapse with an RPP. Using the same data processing routine to analyse homogeneous and heterogeneous conditions, the characteristics of clusters and of empty regions are presented over the whole range of $V_{sg}$ from homogeneous to heterogeneous regimes, bearing in mind that different physical origins are associated with meso-scale structures for these two regimes. In particular, the data in the homogeneous regime are not expected to bear any particular significance as they could be of random origin, or they could be related to some correlation induced by ‘defects’ in the system (due for example to gas injection, see Lefebvre et al. Reference Lefebvre, Mezui, Obligado, Gluck and Cartellier2022).

Figures 4 and 5 clearly demonstrate that, for void regions as well as for clusters, the distributions in the heterogeneous regime markedly differ from the distributions observed in the homogeneous regime. Moreover, in the heterogeneous regime, the distributions tend to collapse indicating that clusters and void regions reach an asymptotic state when the gas superficial velocity becomes large enough. As shown in figure 6, that limiting state is almost the same when considering clusters with a minimum of one bubble or with a minimum of two bubbles.

Figure 6.

Comparisons of the p.d.f.s (a) of void fraction in clusters and (b) of cluster lengths when the minimum number of bubbles is set to one or to two. Measurements performed in a $D=0.4$ m column, on the column axis at $H/D=3.625$ and $V_{sg}=24.7\,{\rm cm}\,{\rm s}^{-1}$.

The average characteristics of clusters and of void and intermediate regions are given in figure 7 as a function of the gas superficial velocity.

1. (i) The average number of bubbles is approximately 1.8 in void regions and approximately 4 in intermediate regions. In clusters, it is approximately 4.5 when $n \geq 2$, and it drops to 3.2 when accounting for clusters consisting of a single bubble. The decrease from 4.5 to 3.2 is consistent with the fact that, as seen above, 2/3 of the clusters comprise more than one bubble. These average numbers of bubbles are quite low: they indicate that the clusters are not organised as compact assemblies of bubbles, but are more like thin sheets. The fact that the probability to find a cluster comprising $N$ bubbles decays like $N^{-1.17}$, i.e. that it strongly drops with $N$, also supports the proposed picture. In particular, 1-D clusters comprising more than ten bubbles are very rare: they represent only 3.7 % of the clusters (with $n\geq 1$) present in the heterogeneous regime.

2. (ii) The size of void regions and of intermediate regions varies from 6–7 cm to 20 cm while the size of clusters ranges from a few millimetres up to 6–7 cm. In the heterogeneous regime, the mean size of clusters $\langle L_{cluster} \rangle$, that of void regions $\langle L_{void} \rangle$ and that of intermediate regions $\langle L_{int} \rangle$ all remain fairly stable. The mean cluster length asymptotes at $21\pm 3\,{\rm mm}$: it is marginally affected if one considers a minimum of one bubble instead of two to form clusters. The asymptotic mean length of void regions is significantly larger as $\langle L_{void} \rangle \rangle \sim 74\,{\rm mm}\pm 10\,{\rm mm}$, and similarly, for intermediate regions, $\langle L_{int} \rangle$ is approximately $62\,{\rm mm}\pm 4\,{\rm mm}$.

3. (iii) The average concentration (in absolute value) in voids steadily increases with the gas superficial velocity. A similar behaviour holds for intermediate regions. In clusters, the average concentration sharply increases at the homogeneous–heterogeneous transition, and for $V_{sg}$ above $\sim 0.15\,{\rm m}\,{\rm s}^{-1}$, it tends to stabilise at a large void fraction, say approximately 50 %. Interestingly, when scaled by the local void fraction $\varepsilon$ (here $\varepsilon$ equals the void fraction on the axis $\varepsilon _{axis}$), the mean concentrations in voids and in intermediate regions increase with the mean gas hold-up, while the concentration in clusters slightly decreases: additional data are needed to confirm if the asymptotic trend corresponds to a decrease or to a plateau. The same question holds concerning the asymptotic behaviour of the difference in concentration between dense and dilute regions.

Figure 7.

Mean characteristics of clusters, of void regions and of intermediate regions versus the gas superficial velocity: (a) average number of bubbles in meso-scale structures; (b) average size; (c) average absolute gas concentration in meso-scale structures; (d) average concentration scaled by the void fraction on the column axis. Measurements performed in a $D=0.4$ m column, on the column axis, at $H/D=3.625$. Vertical dashed lines delineate the homogeneous to heterogeneous transition.

5.1. Test of the relevance of the scaling proposed for the meso-scale structure relative velocity

Concerning the quantities $L_s [\varepsilon - \varepsilon ]$, and as all the data exploited here have been collected on the column axis, the local void fraction $\varepsilon$ for the mixture is equal to $\varepsilon _{axis}$. The mean values $\langle L_{void} [\varepsilon _{void}-\varepsilon ]\rangle$, $\langle L_{cluster} [\varepsilon _{cluster}-\varepsilon ]\rangle$ and $\langle L_{int} [\varepsilon _{int}-\varepsilon ]\rangle$ are shown versus $V_{sg}$ in figure 11. All these quantities happen to remain fairly stable at large $V_{sg}$, say for $V_{sg}$ above ${\approx }10\unicode{x2013}15\,{\rm cm}\,{\rm s}^{-1}$. Note also that setting the minimum number of bubbles in clusters to one or to two does not induce any significant difference on $\langle L_{cluster} [\varepsilon _{cluster} -\varepsilon ] \rangle$.

Figure 11.

Evolution of the products $\langle L_{cluster} [\varepsilon _{cluster}-\varepsilon ]\rangle$ for $n \geq 1$ and for $n \geq 2$, $\langle L_{int} [\varepsilon _{int}-\varepsilon ]\rangle$ and $\langle L_{void} [\varepsilon _{void}-\varepsilon ]\rangle$ with $V_{sg}$. Measurements performed in a $D=0.4$ m column, on the column axis at $H/D=3.625$.

The last ingredient needed to test (5.1)–(5.3) is a connection between the mixture velocity $U_m$ and the mean liquid velocity $V_L$. By definition, $U_{m}$ is the mixture volumetric flux, that is, the velocity of the centre of volume of both phases (Ishii Reference Ishii1975). Therefore, $U_{m}$ is related to unconditional phasic velocities by $U_{m} = (1-\varepsilon )U_L+\varepsilon V_G$, and $U_{m}-V_L$ writes

(5.4)\begin{equation} U_{m}-V_L= \varepsilon (V_G-V_L) =\varepsilon U_R. \end{equation}

The prefactors $C_{structure}$ in (5.1)–(5.3) can now be evaluated. Indeed, for each meso-scale structure, one has

(5.5)\begin{equation} U_{R\ structure\text{-}mixture}= U_{s}-U_{m} =U_{s}+V_L-V_L-U_{m} =V_{b \vert s}-V_L-\varepsilon U_R. \end{equation}

Note that, in the last equality of (5.5), $U_{s}$ has been identified with $V_{b \vert s}$. Strictly speaking, these two quantities are not the same as $V_{b \vert s}$ represents the mean velocity of bubbles within the structure considered while, as discussed in the introduction of § 5, $U_{s}$ corresponds to the velocity of the whole coherent region forming the structure meaning that $U_{s}$ includes information on both gas and liquid phases. In clusters, owing to their large void fraction in the heterogeneous regime (see figure 7c), it is reasonable to assume that both phases move at nearly the same velocity and hence that $U_{clusters} \sim V_{b \vert clusters}$. A similar argument could be put forward for intermediate regions when $V_{sg}$ is large. For void regions, experience shows that $V_{b \vert voids}$ and $V_L$ nearly coincide when in the heterogeneous regime (see figure 9). Hence, assuming $U_{s} \sim V_{b \vert s}$ seems reasonable (to confirm that, liquid velocity measurements conditioned by the local gas concentration would be useful), and (5.5) combined with (5.1), (5.2) or (5.3) provides an estimate of the prefactor $C_{s}$, namely,

(5.6)\begin{equation} C_{s}= U_{R\ structure\text{-}mixture} / (g L_{s} \vert \varepsilon_{s} - \varepsilon \vert)^{1/2}, \end{equation}

where the denominator is known from figure 11. The values $C_{s}$ deduced from (5.6) are given as a function of $V_{sg}$ in figure 12. They all tend to nearly constant values at large $V_{sg}$: the mean values of $C_{s}$ in the heterogeneous regime are provided in table 4; all these figures are almost insensitive to the range of $V_{sg}$ selected to compute the average.

Figure 12.

Evolution of the prefactors $C_{cluster}$, $C_{int}$ and $C_{void}$ deduced form (5.6) combined with (5.1)–(5.3) with $V_{sg}$.

Table 4.

Mean values of prefactors $C_{s}$ in the heterogeneous regime when assuming $U_{s} = V_{b \vert s}$.

Moreover, all prefactors are of order unity, with $C_{cluster} \sim 3$, $C_{int} \sim 1.2$ and $C_{void} \sim -0.36$. The scalings proposed in (5.1)–(5.3) are therefore consistent, and these models provide the correct magnitude of the relative velocity of meso-scale structures with respect to the mixture. These results also confirm that the dynamics of these meso-scale structures is indeed controlled by a buoyancy–inertia equilibrium applied at their respective scales.

Here, each conditional bubble relative velocity $V_{b \vert s} - V_L$ has been connected with the characteristics of the corresponding meso-scale structure. Hence, thanks to (4.1), the unconditional bubble relative velocity is related with the characteristics of all three meso-scale structures present in the flow, combined with the repartition of bubbles between these three populations. This result demonstrates that the enhancement of the bubble relative velocity observed in the heterogeneous regime is the direct consequence of the collective dynamics occurring in these buoyancy-driven bubbly flows when the gas fraction is large enough.

5.2. Scaling of the mean relative velocity: discussion

At this stage, it is worthwhile to come back to the scaling of the relative velocity of (2.1) that involves the reference velocity $(g D \varepsilon )^{1/2}$ identified by Mezui et al. (Reference Mezui, Obligado and Cartellier2022). Let us first recall that (2.1) arises from scaling laws for mean velocities of both phases that have been corroborated over a large range of column diameters and flow conditions. However, direct measurements of the relative velocity in the heterogeneous regime such as those presented in § 2 are absent from the literature, so that the dependency of the relative velocity with the column diameter predicted by (2.1) cannot be directly tested from the available data. Similarly, the analysis in terms of meso-scale structures developed above concerns but a single column diameter, and more experiments are required to further investigate meso-scale structure characteristics. Yet, as shown in Mezui et al. (Reference Mezui, Obligado and Cartellier2022), the existence of an asymptotic heterogeneous state at large $V_{sg}$ is supported by many experiments performed in various column diameters and flow conditions. The presence of such an asymptotic behaviour prompted us to rescale the relative velocities $U_{R\ structure\text {-}mixture}$ using $D$ as the relevant scale for the dimensions $L_s$ of meso-scale structures and using the gas hold-up $\varepsilon$ for scaling the differences in concentration $[ \varepsilon _{s} - \varepsilon ]$. Table 5 provides $L_s/D$, $[\varepsilon _{s} - \varepsilon ]/\varepsilon$ and the quantity $\langle g L_{s} [ \varepsilon _{s} - \varepsilon ] \rangle ^{1/2} / (g D \varepsilon )^{1/2}$ that enter (5.1)–(5.3). The last line of table 5 provides for each structure the relative velocities $U_{R\ structure\text -mixture}$ scaled by the velocity $(gD\varepsilon )^{1/2}$; the resulting coefficients are comprised between 0.1 and 0.6; such $O(1)$ values seem reasonable.

Table 5.

Average characteristics of meso-scale structures in the heterogeneous regime measured on the axis of a $D=0.4$ m bubble column and at $H/D=3.625$, pre-factors $C_{structure}$ and ratio $U_{R\ structure\text {-}mixture} / (gD\varepsilon )^{1/2}$.

The scaling for the unconditional relative velocity $U_R=V_G-V_L$ can be deduced from the above information. Starting from (4.2) and still assuming that $U_{s} =V_{b \vert s}$, we have $U_R = N_{clusters} (U_{clusters} - U_{m}) + N_{int} (U_{int} - U_{m}) + N_{voids} (U_{voids} - U_{m}) + \varepsilon U_R$ that transforms into

(5.7)\begin{align} &(1-\varepsilon) U_R = N_{clusters} ( U_{clusters} - U_{m}) + N_{int} (U_{int} - U_{m}) \nonumber\\ &\quad + N_{voids} (U_{voids} - U_{m}) = \large( N_{clusters} \left[(U_{clusters} - U_{m}) / (gD\varepsilon)^{1/2} \right] \nonumber\\ &\quad + N_{int} \left[(U_{int} - U_{m}) / (gD\varepsilon)^{1/2} \right] \nonumber\\ &\quad + N_{voids} \left[(U_{voids} - U_{m}) / (gD\varepsilon)^{1/2} \right] \large) (gD\varepsilon)^{1/2} = C_R (g D \varepsilon)^{1/2}. \end{align}

The prefactor $C_R$ has been evaluated over various ranges of $V_{sg}$ within the heterogeneous regime: it is given in table 6 where we have also considered the two options for clusters (namely $n \geq 1$ and $n \geq 2$). Overall, the dispersion is small, and one gets

(5.8)\begin{equation} (1-\varepsilon) U_R \sim 0.30 \pm 0.01 (g D \varepsilon)^{1/2}. \end{equation}

As the void fraction in the heterogeneous regime ranges from 20 to 37 % for the experimental conditions considered here, the ratio $U_R/(gD\varepsilon )^{1/2}$ evolves from 0.37 to 0.50, to be compared with the value $U_R \sim 0.41 (gD \varepsilon )^{1/2}$ deduced from direct velocity measurements (see (2.1)). The difference between these two results remains in the interval $[-11\,\%; +23\,\%]$. Such a difference is quite acceptable owing to the variety of independent measurements involved in that analysis (the latter include void fraction, unconditional and conditional relative velocities, statistics on size and on concentration for the three meso-scale structures, repartition of bubbles among these structures based on Voronoï tessellations) and possibly also owing to the assumption $U_{s} \approx V_{b \vert s}$ we made.

Table 6.

Prefactors for the relative velocities $U_{R\ structure\text {-}mixture}$ and for the unconditional relative velocity $U_R$ with respect to the velocity scale $(gD\varepsilon )^{1/2}$ in the heterogeneous regime (from measurements on the axis of a $D=0.4$ m bubble column at $H/D=3.625$).

The expression of the relative velocity from (2.1) derived from scaling considerations is recovered here using the partition of bubbles into three distinct populations. Again, and as seen in § 4, (5.7) and (5.8) show that the increase of the bubble relative velocity beyond the terminal velocity value originates from the dynamics of the meso-scale structures present in the heterogeneous regime. In this process, clusters bring the strongest contribution. Thanks to the significant proportion of bubbles they gather (table 1) and thanks to their high relative velocity with respect to the mixture (clusters correspond to the largest ratio $U_{R\ structure\text {-}mixture}/(gD\varepsilon )^{1/2}$ in table 5), they contribute 65–68 % to $U_R$. Intermediate regions host the majority of bubbles but their relative velocity with respect to the mixture is approximately three times smaller than that of clusters: they contribute 33–35 % to $U_R$. Last, void regions are sinking in the mixture: they carry few bubbles and their (negative) contribution to $U_R$ is almost negligible ($\sim$2 %).

Although data are still lacking to directly examine how the relative velocity evolves with the column diameter, (5.7) and (5.8) provide an indirect way to discuss the dependency of $U_R$ on $D$. According to experiments, the void fraction weakly varies with $D$ (in Mezui et al. (Reference Mezui, Obligado and Cartellier2022), the void fraction on the column axis is found to evolve as $D^{-0.2}$), so that (5.8) indicates that the relative velocity monotonously increases with $D$. At first sight, that prediction seems odd if one refers to a single bubble dynamic that is controlled by its interaction with the liquid at a scale commensurable with the bubble size, and not with the size of the domain. However, we have seen that collective dynamics in these buoyancy-driven bubbly flows plays a central role in the formation of meso-scale structures, and that the presence of both dense and dilute structures drives the momentum exchange between phases and leads to an enhancement of the relative velocity. Further, the various contributions $U_{R\ structure\text {-}mixture}$ appearing in (5.7) are directly related with the characteristics in terms of size and concentration that each meso-scale structure has in the asymptotic limit (i.e. in the limit of large $V_{sg}$). The question left is thus whether the scalings with $D$ and with $\varepsilon$ used to build table 5 are relevant or if they are artificial.

Concerning $\varepsilon$, an examination of the evolution of the coefficients $C_s$ of figure 12 versus $\varepsilon$ instead of $V_{sg}$ indicates that a scaling with the void fraction is indeed acceptable for gas hold-up above approximately 30 %.

Regarding the dimensions of meso-scale structures, it is unlikely that the typical width of clusters grows with $D$, but the void regions do have an extension of order $D$. Such a statement is supported by our experiments in the $D=0.4$ m column (figure 7). It is also sustained by the results presented in the next section where it is shown that void regions correspond to large-scale vorticity regions whose dimension is of order $D$.

5.3. Flow structure in the heterogeneous regime

Concerning the internal structure of the flow in the heterogeneous regime, the spatial organisation of the gas phase deduced from 1-D Voronoï tessellations has already demonstrated the presence of thin regions at high void fractions, and of large regions at low void fractions. When examining the flow through column walls using direct lightning, these void regions correspond to dark zones (figure 13) while bright regions indicate a significant presence of bubbles (whose interface reflects light back towards the observer). These dark regions seem to correspond to the large-scale vortical-like structures that have often been reported in the literature (and that are illustrated in the video enclosed as supplementary material associated with the reference Mezui et al. Reference Mezui, Obligado and Cartellier2022).

Figure 13.

Images of flow near column walls in the heterogeneous regime. Bright zones indicate the presence of many bubbles while dark zones correspond to structures comprising few bubbles. The vertical extent of the images corresponds to sections at 0.8 and at 2 metres above gas injection ($D=0.4$ m, static liquid height = 2 m). Movies are included in the supplementary material of Mezui et al. (Reference Mezui, Obligado and Cartellier2022).

To quantify such vortical structures, we exploited the local liquid velocity provided by a Pavlov tube. Spatial correlations were not accessible with a single sensor, and liquid velocity measurements conditioned by the local gas concentration were not attempted. Instead, we considered time series collected from a single Pavlov tube even though its temporal resolution was limited (approximately 20 Hz, see Mezui et al. (Reference Mezui, Cartellier and Obligado2023) for further details).

The zero crossings of the signal $v_L(t) - V_L$ were detected and the density of zero crossing $n_s$ per unit length was evaluated using the mean liquid velocity to transform time into space. The resulting characteristic length scale $n_s^{-1}$ provides the mean vertical size of vortical structures. Let us mention that, for a turbulent single-phase flow, Liepmann & Robinson (Reference Liepmann and Robinson1953) related the Taylor microscale $\lambda$ to the average distance $n_s^{-1}$ between zero crossings of a streamwise velocity signal of a turbulent flow, and they show that $n_s^{-1}=B\lambda$, where $B$ is a constant that accounts for intermittency ($B= {\rm \pi}$ for a Gaussian time series with also a Gaussian derivative). In our case, it is not clear if the vertical spatial scale we construct is related to any turbulence scale, as we are far from the conditions of homogeneity and isotropy required by the model from Liepmann & Robinson. From measurements in the $D=0.4$ m column, $n_s^{-1}$ was found equal to $D$ (within 10 %) for $V_{sg}$ larger than ${\approx }10\,{\rm cm}\,{\rm s}^{-1}$. Although acquired in a single column, this result supports the idea that the size of vortical structures does scale as the bubble column diameter. A few other results on the integral length scale of turbulence (Mezui et al. Reference Mezui, Cartellier and Obligado2023) and on the size of vortical structures in various bubble columns (Cartellier Reference Cartellier2019) also indicate that $D$ is the relevant scale.

In parallel, let us show that the regions containing bubbles are not in the form of ‘compact’ clusters of bubbles. Indeed, the high particle Reynolds number bubbles considered here (see § 2) are in a constant drag coefficient regime (when isolated). If one considers a compact, close to spherical assembly of $N$ such bubbles, the dynamics of that ensemble would be also governed by a constant drag coefficient, and its relative velocity would be equal to $N^{1/6} U_T$. Therefore, the relative velocities between approximately $2 U_T$ and $2.5 U_T$ that we measured in heterogeneous conditions at large $V_{sg}$ (for $V_{sg}$ above $10\,{\rm cm}\,{\rm s}^{-1}$, as discussed in § 2), would be recovered with $N=64$ for $2 U_T$, or with $N=244$ for $2.5 U_T$: these figures are $10$ to $50$ times larger than the average number of bubbles detected in clusters (figure 7). Clearly, the existence of compact assemblies of bubbles does not correspond to observations. The question is now how thin bubbly ‘sheets’ could induce such an enhancement of the bubble relative velocity.

From these findings, a tentative cartoon of the spatial organisation of phases in the heterogeneous regime emerges: bubbles accumulate in narrow (a few bubbles in size) regions located in between large (typically $\approx 0.2 D$ according to figure 7b) vortical structures that are almost free of bubbles. This situation is sketched in figure 14. The dense regions containing most bubbles are like thin ‘sheets’ or ‘curtains’ (a few bubble diameter wide) located in between vortices whose extent is of order $D$. Thus, these arguments combined with (5.7) support a relative velocity controlled by the lateral dimension of the column.

Figure 14.

Tentative sketch of the flow organisation. As it can be appreciated in the movies included as supplementary material in Mezui et al. (Reference Mezui, Obligado and Cartellier2022) (see figure 13), large vortical liquid structures are seen as dark zones (as they comprise few bubbles), while clusters of bubbles correspond to bright zones. Such large dark vortices, that are almost free of bubbles, show bubble ‘curtains’ between them. Although collected near the walls, these structures can be seen as representative of the internal flow structures inside the column.

To conclude, in this section, the relative velocity observed in the heterogeneous regime has been connected with the characteristics in terms of size and concentration of dense, intermediate and dilute regions formed in these buoyancy-driven bubbly flows. These meso-scale structures and the dynamics they induce are believed to be at the origin of the swarm factor introduced in Eulerian two-fluid simulations to evaluate the momentum exchange between phases in the heterogeneous regime. Moreover, a number of experimental results support the existence of an asymptotic flow organisation at large $V_{sg}$, in particular, with the saturation of the gas concentration in dense regions, and with limiting values of the spatial extent of meso-scale structures. Additional investigations are required to fully determine how these asymptotic values evolve with parameters, and in particular with the column diameter. Let us finally underline that, in their simulations of heterogeneous conditions, Panicker, Passalacqua & Fox (Reference Panicker, Passalacqua and Fox2020) captured the presence of bubble swarms with a characteristic length scale of order $V_G^2/g$ and predicted a significant increase of the mean gas velocity compared with homogeneous conditions: these findings are consistent with the experimental results presented here.