3. Results and discussion

We now confront the predictions obtained with this method to the air–water experimental data of Matas et al. (Reference Matas, Marty and Cartellier2011). We first consider their series of data obtained at a fixed $U_{G}=27~\text{m}~\text{s}^{-1}$ : the prediction of the viscous spatiotemporal analysis without confinement for this series overestimates the experimental data by a factor of 2 for most liquid velocities (see figure 31d of Otto et al. Reference Otto, Rossi and Boeck2013). Figure 2 shows what happens for $U_{L}=0.26~\text{m}~\text{s}^{-1}$ when a confinement similar to the experimental one is taken into account and when ${\it\omega}_{i}$ , the imaginary part of ${\it\omega}$ , is reduced: a confinement branch appears along the $k_{i}$ axis, and pinching between this branch and the shear instability branch occurs for positive ${\it\omega}_{0i}$ . While the shear instability branch involved in the pinching observed by Otto et al. (Reference Otto, Rossi and Boeck2013) at large $U_{L}$ is what they call the weaker mode (as in figure 25b of their paper), the branch involved in the pinching mechanism of figure 2 is the stronger mode, which extends to lower wavenumbers. The confinement branch involved in the pinching is of the type described in Healey (Reference Healey2007): it arises because of the oscillatory nature of the confined perturbation when $k$ is close to the imaginary axis. The frequency predicted without taking confinement into account would be the frequency corresponding to the minimum of the negative growth rate (there is no pinch point on the weaker mode for the conditions of figure 2), namely 66 Hz, while the pinching caused by confinement occurs for a much lower frequency of 28 Hz. The latter frequency is much closer to the experimental frequency $f_{exp}=28.8~\text{Hz}$ .

Figure 2. Pinching at positive ${\it\omega}_{0i}$ occurs between confinement and shear controlled branches when confinement is taken into account; $U_{G}=27~\text{m}~\text{s}^{-1}$ , $U_{L}=0.26~\text{m}~\text{s}^{-1}$ , ${\it\delta}_{d}=1$ , $H_{G}=1~\text{cm}$ and $H_{L}=1~\text{cm}$ . Symbols correspond to ${\it\omega}_{r}$ in the range $(80{-}680)~\text{s}^{-1}$ .

For the case of an absolute instability, nonlinearity is expected to affect the properties of the eigenmode at the pinch point, and comparison between growth rate at the pinch point and experimental spatial growth rate is therefore not attempted. It is however interesting to note the value of the wavenumber at the pinch point: figure 2 shows that the pinching induces a major reduction in the predicted wavenumber, from $k_{r}\approx 720~\text{m}^{-1}$ down to $k_{r}\approx 140~\text{m}^{-1}$ . The corresponding phase velocity ${\it\omega}_{r}/k_{r}$ (we denote ${\it\omega}_{r}$ the real part of ${\it\omega}$ ) is then about $1.28~\text{m}~\text{s}^{-1}$ , very close to the value given by (1.2), $U_{c}=1.2~\text{m}~\text{s}^{-1}$ . Wave velocity was not measured in the experiment of Matas et al. (Reference Matas, Marty and Cartellier2011), but previous experiments by Raynal (Reference Raynal1997) and Ben Rayana (Reference Ben Rayana2007) had both found wave velocity in close agreement with $U_{c}$ . In addition, expression (1.2) for velocity $U_{c}$ was derived by Dimotakis (Reference Dimotakis1986) through a balance of dynamic pressure, and a similar expression was found by Matas et al. (Reference Matas, Marty and Cartellier2011) in the frame of the inviscid linear theory via an asymptotic expansion in the limit of low density ratio. This expression therefore corresponds to a perturbation driven by an inviscid mechanism: the fact that the predicted phase velocity is close to (1.2) is a first indication that the dominant mechanism for these ${\it\omega}$ and $k$ is inviscid. Note that the velocity associated with the most unstable modes in Fuster et al. (Reference Fuster, Matas, Marty, Popinet, Hoepffner, Cartellier and Zaleski2013) or Otto et al. (Reference Otto, Rossi and Boeck2013), which are viscous modes, is closer to $U_{i}$ and significantly lower than this estimate.

Figure 3 next shows how the predicted frequency compares to the experimental one (filled squares) for the experimental data at $U_{G}=27~\text{m}~\text{s}^{-1}$ for each of the six liquid velocities investigated. The asterisks show the prediction of Otto et al. (Reference Otto, Rossi and Boeck2013) for ${\it\delta}_{d}=0.1$ , a prediction that works well for the two higher velocities, but clearly fails for the four lower velocities. For ${\it\delta}_{d}=1$ the predictions of Otto et al. (Reference Otto, Rossi and Boeck2013) for all $U_{L}$ far exceed experimental data (see figure 31d of their paper), and are not shown here. The circles show the frequency obtained at pinching when confinement is taken into account, for ${\it\delta}_{d}=1$ (no velocity deficit): this frequency is in relatively good agreement with experimental data. Note that for the four lower liquid velocities the mode predicted by Otto et al. (Reference Otto, Rossi and Boeck2013) when ${\it\delta}_{d}=1$ is associated with a convective mode, and the present absolute instability caused by confinement is therefore expected to dominate. For ${\it\delta}_{d}=1$ and the two largest liquid velocities in this series, however, the pinching occurs at negative ${\it\omega}_{0i}$ , and the confinement mechanism is consequently not relevant.

Figure 3. Comparison at $U_{G}=27~\text{m}~\text{s}^{-1}$ between experimental data of Matas et al. (Reference Matas, Marty and Cartellier2011) (▪) and available spatiotemporal predictions: *, prediction by Otto et al. (Reference Otto, Rossi and Boeck2013), for ${\it\delta}_{d}=0.1$ ; ○, prediction taking into account confinement without any velocity deficit ( ${\it\delta}_{d}=1$ ); ▫, prediction with confinement and a velocity deficit ( ${\it\delta}_{d}=0.5$ ).

If a velocity deficit is included in the base flow profile, the absolute instability is enhanced (i.e. it occurs at even larger ${\it\omega}_{0i}$ ), and the frequency slightly increases too; these data are shown by the open squares in figure 3, for ${\it\delta}_{d}=0.5$ . If a stronger deficit is included ( ${\it\delta}_{d}<0.4$ ), the shear instability branch is displaced and the pinch point due to confinement disappears. Figure 4 shows the same comparison for the series of experimental data of Matas et al. (Reference Matas, Marty and Cartellier2011) at $U_{G}=22~\text{m}~\text{s}^{-1}$ , and the same agreement is found when confinement is included. Figure 5 shows the variation of the absolute growth rate ${\it\omega}_{0i}$ at the pinch point: it decreases when $U_{L}$ is increased, and eventually becomes negative for $U_{L}>0.6~\text{m}~\text{s}^{-1}$ . The inclusion of a moderate velocity deficit ( ${\it\delta}_{d}=0.5$ , open squares) leads to a slight increase of ${\it\omega}_{0i}$ .

Figure 4. Comparison at $U_{G}=22~\text{m}~\text{s}^{-1}$ between experimental data of Matas et al. (Reference Matas, Marty and Cartellier2011) (▪) and available spatiotemporal predictions: *, prediction by Otto et al. (Reference Otto, Rossi and Boeck2013), for ${\it\delta}_{d}=0.1$ ; ○, prediction taking into account confinement without any velocity deficit ( ${\it\delta}_{d}=1$ ).

Figure 5. Variation of the absolute growth rate ${\it\omega}_{0i}$ at the pinch point caused by the confinement branch, as a function of liquid velocity, for fixed $U_{G}=27~\text{m}~\text{s}^{-1}$ , $H_{G}=1~\text{cm}$ and $H_{L}=1~\text{cm}$ : ●, ${\it\delta}_{d}=1$ ; ▫, ${\it\delta}_{d}=0.5$ . The growth rate decreases when liquid velocity is increased, and increases when a velocity deficit is included.

The fact that agreement between prediction and experiments occurs for ${\it\delta}_{d}=1$ at lower $U_{L}$ , while it occurs for smaller ${\it\delta}_{d}$ at larger $U_{L}$ , is consistent with the idea that the velocity deficit will be resorbed over a longer distance when the liquid velocity is larger. At any rate, only a global approach, in the sense introduced by Huerre & Monkewitz (Reference Huerre and Monkewitz1990), may clarify how mode selection occurs in the presence of strong spatial variations, when unstable modes predicted close to the splitter plate (where ${\it\delta}_{d}\ll 1$ ) differ from those farther downstream (where ${\it\delta}_{d}=1$ ).

In order to clarify the nature of the dominant mechanism for the new pinch point caused by confinement, we carry out an energy budget similar to the one introduced by Boomkamp & Miesen (Reference Boomkamp and Miesen1996) for temporal modes. The energy budget can be written as

Here $E_{kin}$ is the total kinetic energy of the eigenmode $(\text{gas}+\text{liquid})$ ; $\mathit{REY}_{L}$ (respectively $\mathit{REY}_{G}$ ) is the transfer of energy from the base flow to the perturbation via Reynolds stresses in the liquid (respectively gas) stream; $\mathit{TAN}$ is the work of tangential stresses; $\mathit{NOR}$ is the contribution of normal stresses (surface tension and gravity in the present case); and $\mathit{DIS}_{L}$ (respectively $\mathit{DIS}_{G}$ ) is the dissipation in the liquid phase (respectively gas phase). The analytical expressions for each of these terms are very similar to the expressions given in Otto et al. (Reference Otto, Rossi and Boeck2013), and are not repeated here. The only differences here are that the wavenumber is complex, and that the kinetic energy variation term must be generalized to include a contribution from the spatial growth:

We compute the energy budget for the six data points of figure 3 showing the best agreement with experimental results, more precisely the four data points obtained with confinement for lower $U_{L}$ and ${\it\delta}_{d}=1$ (no velocity deficit), and the two points obtained by Otto et al. (Reference Otto, Rossi and Boeck2013) for $U_{L}=0.74$ and $1~\text{m}~\text{s}^{-1}$ (with a velocity deficit, ${\it\delta}_{d}=0.1$ ). The energy contributions are normalized by the total positive kinetic energy rate. We omit the contribution of NOR, which is negligible for most of the present points. Table 1 shows that, for the four conditions for which confinement triggers the absolute instability, the mechanism is inviscid, and that the perturbation draws its energy from the gas side. This is consistent with phase velocity being close to $U_{c}$ (1.2). For the two data points at larger velocity, the mechanism is confirmed to be viscous, driven by the work of interfacial tangential stresses (Otto et al. Reference Otto, Rossi and Boeck2013). Dissipation occurs almost exclusively in the gas phase.

The idea behind the generalized budget of (3.1) is to identify among the right-hand contributions which are dominant and which are negligible when at the pinch point. It might be tempting to carry out the energy budget in a more classical form, in order to avoid the generalization to spatiotemporal modes. However, for a purely temporal mode, the results would depend very much on how the $\{\text{spatiotemporal}\rightarrow \text{temporal}\}$ transposition is made. For a purely temporal mode at the same ${\it\omega}_{r}$ as the pinch point, for example, and conditions of the first line of table 1, the wavenumber would become $k_{r}\approx 380~\text{m}^{-1}$ instead of $k_{r}\approx 170~\text{m}^{-1}$ for the spatiotemporal mode. For these conditions, the energy budget says the temporal mode is 85 % viscous (the same budget would be found for the spatiotemporal mode at this larger $k_{r}$ ). If a $\{\text{spatiotemporal}\rightarrow \text{spatial}\}$ comparison is carried out for the wavenumber of the pinch point, then the energy budget says the mode is 60 % viscous and 40 % viscous. Hence a transposition would be extremely sensitive to which ${\it\omega}$ and $k$ values are retained.

At any rate, the inviscid nature of the instability for the pinch point due to confinement is confirmed by figure 6, which shows the impact of a reduction of viscosity on spatial branches for the case $U_{G}=27~\text{m}~\text{s}^{-1}$ and $U_{L}=0.26~\text{m}~\text{s}^{-1}$ (conditions of figure 2). When viscosity is reduced, the shear branch is impacted, but the confinement branch is unchanged, and the pinch point remains at the same location. Frequency at the pinch point slightly decreases when viscosity is reduced, from $f=28~\text{Hz}$ for ${\it\nu}_{G}=1.5\times 10^{-5}~\text{m}~\text{s}^{-1}$ and ${\it\nu}_{L}=10^{-6}~\text{m}~\text{s}^{-1}$ down to $f=20~\text{Hz}$ for ${\it\nu}_{G}=1.5\times 10^{-7}~\text{m}~\text{s}^{-1}$ and ${\it\nu}_{L}=10^{-8}~\text{m}~\text{s}^{-1}$ .

Figure 6. Impact of a variation in viscosity on spatial branches for $U_{G}=27~\text{m}~\text{s}^{-1}$ , $U_{L}=0.26~\text{m}~\text{s}^{-1}$ and ${\it\delta}_{d}=1$ : *, ${\it\nu}_{G}=1.5\times 10^{-5}~\text{m}~\text{s}^{-1}$ , ${\it\nu}_{L}=10^{-6}~\text{m}~\text{s}^{-1}$ and ${\it\omega}_{i}=85~\text{s}^{-1}$ ; ▫, ${\it\nu}_{G}=1.5\times 10^{-6}~\text{m}~\text{s}^{-1}$ , ${\it\nu}_{L}=10^{-7}~\text{m}~\text{s}^{-1}$ and ${\it\omega}_{i}=65~\text{s}^{-1}$ ; ●, ${\it\nu}_{G}=1.5\times 10^{-7}~\text{m}~\text{s}^{-1}$ , ${\it\nu}_{L}=10^{-8}~\text{m}~\text{s}^{-1}$ and ${\it\omega}_{i}=50~\text{s}^{-1}$ ; $+$ , ${\it\nu}_{G}=1.5\times 10^{-7}~\text{m}~\text{s}^{-1}$ , ${\it\nu}_{L}=10^{-6}~\text{m}~\text{s}^{-1}$ and ${\it\omega}_{i}=45~\text{s}^{-1}$ . The shear branch is displaced when viscosity is reduced, but the location of the pinch point associated with confinement remains unchanged.

This result is satisfying, because it explains the paradox exposed in the introduction, namely why the simplified analysis proposed by Raynal (Reference Raynal1997) was successful, and why the inclusion of viscosity in the analysis had degraded the quality of the prediction. The inclusion of viscosity gives rise to a much stronger shear mode based on the viscous mechanism (Boeck & Zaleski Reference Boeck and Zaleski2005; Otto et al. Reference Otto, Rossi and Boeck2013), but resonance due to confinement triggers an absolute instability for the lower-wavenumber part of the shear branch, which is dominated by the inviscid mechanism. This occurs for all the lower liquid velocity points of Matas et al. (Reference Matas, Marty and Cartellier2011), as in figure 3. For the larger liquid velocity conditions, we find that the shear branch is displaced away from the confinement branch, and the mechanism is then the one predicted by Otto et al. (Reference Otto, Rossi and Boeck2013), an absolute instability between the shear branch controlled by viscosity and a lower branch controlled by surface tension.

Figure 7. Change in the confinement branches when $H_{G}$ and $H_{L}$ are reduced, for fixed $U_{G}=27~\text{m}~\text{s}^{-1}$ , $U_{L}=0.26~\text{m}~\text{s}^{-1}$ , ${\it\delta}_{d}=1$ , ${\it\omega}_{i}=120~\text{s}^{-1}$ and ${\it\omega}_{r}$ in the range $(60{-}430)~\text{s}^{-1}$ : ●, $H_{G}=H_{L}=2~\text{cm}$ ; $\times$ , $H_{G}=H_{L}=1~\text{cm}$ ; ○, $H_{G}=H_{L}=0.5~\text{cm}$ .

Figure 8. Pinching for $U_{G}=27~\text{m}~\text{s}^{-1}$ , $U_{L}=0.26~\text{m}~\text{s}^{-1}$ , ${\it\delta}_{d}=1$ , ${\it\omega}_{i}=60~\text{s}^{-1}$ and ${\it\omega}_{r}$ in the range $(60{-}280)~\text{s}^{-1}$ : ○, $H_{G}=3~\text{cm}$ and $H_{L}=1~\text{cm}$ ; *, $H_{G}=1~\text{cm}$ and $H_{L}=3~\text{cm}$ . Both thicknesses play a symmetric role.

The velocity profile considered in figure 1 includes both a finite $H_{G}$ and a finite $H_{L}$ . In order to clarify the influence of these thicknesses, we keep them equal and vary them, while keeping ${\it\delta}_{G}$ and ${\it\delta}_{L}$ constant and equal to their values for $H_{G}=H_{L}=H=1~\text{cm}$ . Figure 7 shows that the location of the confinement branch along the $k_{i}$ axis behaves as $1/H$ , and that the extension of the branches increases strongly with $1/H$ : these branches therefore set the order of magnitude of $k$ at the pinch point. If these thicknesses are distinct, we observe that their role is symmetric. Figure 8 shows two sets of branches obtained for ( $H_{G}=3~\text{cm}$ ; $H_{L}=1~\text{cm}$ ) and for ( $H_{G}=1~\text{cm}$ ; $H_{L}=3~\text{cm}$ ). Though experimentally a thick liquid and a thin gas stream may look quite different from a thin liquid stream and a thick gas stream, we find that the branches obtained in both situations are very close. They extend to the same $\mathit{real}(k)$ as the branch obtained for $H_{G}=H_{L}=1~\text{cm}$ , the smaller length, the only difference being that the absolute growth rate ${\it\omega}_{0i}$ is significantly smaller for $H_{G}\neq H_{L}$ than for $H_{G}=H_{L}$ . The fact that a symmetric confinement can enhance the absolute instability of a shear layer has already been pointed out in Healey (Reference Healey2009).

As shown in figure 3, the present mechanism dominates for the largest $U_{G}/U_{L}$ cases but is absent for the largest liquid velocities investigated. Figure 9 sheds light on this behaviour. When $U_{L}$ is increased, the maximum growth rate of the shear branch decreases. If the maximum growth rate of this branch, which is controlled by the viscous mechanism (Otto et al. Reference Otto, Rossi and Boeck2013), is shifted above the location of the confinement branch, then the two branches cannot collide any more when ${\it\omega}_{i}$ is reduced.

Figure 9. Influence of liquid velocity $U_{L}$ , for fixed $U_{G}=27~\text{m}~\text{s}^{-1}$ , $H_{G}=H_{L}=1~\text{cm}$ , ${\it\delta}_{d}=1$ and ${\it\omega}_{i}=120~\text{s}^{-1}$ : ●,  $U_{L}=0.26~\text{m}~\text{s}^{-1}$ ; ○, $U_{L}=0.5~\text{m}~\text{s}^{-1}$ ; *, $U_{L}=1~\text{m}~\text{s}^{-1}$ . The ranges of ${\it\omega}_{r}$ are respectively (120–370), (170–500) and $(260{-}820)~\text{s}^{-1}$ . When liquid velocity is increased, the shear branch is shifted away from the confinement branch.

Figure 10. Influence of vorticity thickness ${\it\delta}_{G}$ on ${\it\omega}_{r}$ (circles) and ${\it\omega}_{0i}$ (squares), for fixed $U_{G}=27~\text{m}~\text{s}^{-1}$ , $U_{G}=0.26~\text{m}~\text{s}^{-1}$ , ${\it\delta}_{d}=1$ and $H_{G}=H_{L}=1~\text{cm}$ . Open symbols correspond to ${\it\delta}_{L}=500~{\rm\mu}\text{m}$ and filled symbols to ${\it\delta}_{L}={\it\delta}_{G}$ .

All the above results have been obtained with ${\it\delta}_{G}$ set to its value in experiments, and with a fixed ${\it\delta}_{L}$ . We show in figure 10 the impact of ${\it\delta}_{G}$ for fixed $U_{G}$ on both the real and imaginary parts of ${\it\omega}$ . Increasing ${\it\delta}_{G}$ causes a strong reduction in the frequency and absolute growth rate, both for ${\it\delta}_{L}={\it\delta}_{G}$ (filled symbols) and for fixed ${\it\delta}_{L}=500~{\rm\mu}\text{m}$ (open symbols). The impact of ${\it\delta}_{G}$ on the wavenumber at the pinch point is weaker: $k$ remains mostly constant when ${\it\delta}_{G}$ is increased, except for a slight decrease, and the location of the pinch point remains essentially controlled by the confinement branch.

The scalings of (1.1)–(1.3), in agreement with experiments, can be derived from the partially successful inviscid analysis. Up to now they had no justification in the context of viscous analyses. The fact that the energy of the perturbation is fed by the gas Reynolds stress indicates that the argument behind the simplified derivation of (1.1) is still valid. This is a significant result, since with ${\it\omega}\sim U_{G}/{\it\delta}_{G}$ we recover the $f\sim U_{G}^{3/2}$ scaling observed in most experimental studies, where ${\it\delta}_{G}$ scales as $U_{G}^{-1/2}$ . The instability predicted here is absolute, and in the experiment nonlinearity is therefore expected to take over: the same arguments used by Dimotakis (Reference Dimotakis1986) to derive (1.2) are expected to hold. Provided this is valid for the waves generated by the present mechanism, the scaling law (1.3) for the wavelength can then be derived from the scalings for velocity and frequency.