Liquid inertia versus bubble cloud buoyancy in circular plunging jet experiments

Abstract

When a liquid jet plunges into a pool, it can generate a bubble-laden jet flow underneath the surface. This common and simple phenomenon is investigated experimentally for circular jets to illustrate and quantify the role played by the net gas/liquid void fraction on the maximum bubble penetration depth. It is first shown that an increase in either the impact diameter or the jet fall height to diameter ratio at constant impact momentum leads to a reduction in the bubble cloud size. By measuring systematically the local void fraction using optical probes in the biphasic jet, it is then demonstrated that this effect is a direct consequence of the increase in air content within the cloud. A simple momentum balance model, including only inertia and the buoyancy force, is shown to predict the bubble cloud depth without any fitting parameters. Finally, a Froude number based on the bubble terminal velocity, the cloud depth and also the net void fraction is introduced to propose a simple criterion for the threshold between the inertia-dominated and buoyancy-dominated regimes.

1. Introduction

The impact of a plunging jet on the free surface of a pool of the same or a different liquid above a critical velocity transports ambient gas into the continuous liquid phase, thus forming a cluster of bubbles, a bubble cloud (Lin & Donnelly 1966; Bonetto, Drew & Lahey 1994; Zhu, Oğuz & Prosperetti 2000; Lorenceau, Quéré & Eggers 2004). This phenomenon is encountered widely in industrial applications, such as the stirring of chemicals (McKeogh & Ervine 1981) or hydroelectric applications (Guyot 2019), and in nature, as in breaking waves and cascades (Chanson 2002; Kiger & Duncan 2012).

Previous studies discussed the inception of air bubbles below jet impact and proposed models for the air entrainment mechanism based on the morphology in circular (McKeogh & Ervine 1981; Sene 1988; El Hammoumi, Achard & Davoust 2002) and planar (Cummings & Chanson 1999; Bertola, Wang & Chanson 2018) jets, as well as multi-droplet streams (Speirs et al. 2018). Kiger & Duncan (2012) reviewed the air entrainment mechanism for laminar, turbulent and disintegrated jets of different viscosities. Once the bubble entrainment conditions are reached, the maximum penetration depth of the bubble cloud, denoted by $H$ in figure 1(a), is an essential parameter to model in various applications (Clanet & Lasheras 1997). Indeed, a change in $H$ will affect the volume of the biphasic region and alter the gas mixing rate in chemical industries. In hydroelectric power plants, $H$ is critical in designing dams to prevent erosion and structural weakening caused by the incoming biphasic jet hitting the river bed. As can be inferred from table 1, previous studies have addressed bubble cloud formation over a wide range of scales (Biń 1993; Clanet & Lasheras 1997; Chirichella et al. 2002; Roy, Maiti & Das 2013; Qu et al. 2013; Harby, Chiva & Muñoz-Cobo 2014; Kramer, Wieprecht & Terheiden 2016; Miwa et al. 2018; Guyot, Cartellier & Matas 2020). Evidence for strong scale effects exists not only regarding bubble cloud size but also for bubble count rate or void fraction (Chanson, Aoki & Hoque 2004). Whereas numerous empirical correlations for the cloud depth can be found in the literature depending on their size, from a few centimetres to about a few metres, the underlying physics has been elaborated only recently.

Figure 1. (a) Schematic of the control volume of the bubble cloud. (b) Schematic of the experimental set-up, illustrating flow lines to generate the bubble cloud, the backlighting imaging set-up and the void fraction measurement set-up with optical probes.

Table 1. Flow conditions in previous works on vertical ($\downarrow$), inclined ($\swarrow$) and oscillating ($\rightleftarrows$) plunging jets from circular ($\bigcirc$) and planar ($\unicode{x25AD}$) nozzles. All authors used water. Data in Lin & Donnelly (1966) refer also to oil and glycol jets.

In this context, Clanet & Lasheras (1997) developed a jet momentum conservation argument, similar to the work of Suciu & Smigelschi (1976), but by postulating that (i) the biphasic jet expands with a constant cone half-angle $\gamma =12.5^{\circ }$, as is well known in turbulent jets (Horn & Thring 1956; Ervine & Falvey 1987; L'vov et al. 2008), and (ii) bubbles would escape the biphasic zone as soon as the local advection speed, due to the liquid jet below the free surface, decreases down to the terminal velocity $U_t$. They thereby derive a simple prediction for the bubble cloud depth $H_i$:

(1.1)\begin{equation} H_i = \frac{1}{2\tan\gamma} \left( \dfrac{V_{i}D_{i}}{U_{t}} \right)\!, \end{equation}

where $V_i$ is the impact velocity and $D_i$ is the impact diameter. Note that there is no adjustable parameter nor any liquid or gas physical properties in (1.1), except for $U_t$, which is taken equal to $22\ {\rm cm}\ {\rm s}^{-1}$ for all air bubbles in water larger than $1$ mm in diameter (Maxworthy et al. 1996). Equation (1.1) shows that in this case, the cloud depth is directly proportional to $V_{i}D_{i}$, which, up to a factor of liquid density $\rho$, is the square root of the impact momentum. Clanet & Lasheras (1997) obtained very good agreement for their experiments with micro-jets, of at most $2.16$ mm in nozzle diameter. However, unlike single-phase jets in a neutrally buoyant environment, momentum conservation may not be valid for voluminous biphasic jets, for which buoyancy cannot be neglected. For example, in a $15$ cm deep conical bubble cloud that contains only a small void fraction of air, say $\phi = 10\,\%$, the outgoing momentum $\dot {Q}_o \sim (1 - \phi ) \rho {U_t}^2 {\rm \pi}H^2\tan ^2\gamma$ in figure 1(a) is already comparable to the net buoyancy force $F_b \sim \phi \rho g {{\rm \pi} }H^3\tan ^2\gamma /3$ acting on it. Indeed, the largest cloud in experiments by Clanet & Lasheras (1997) is about $16$ cm deep, and (1.1) overpredicts the cloud depth data in the previous literature (Biń 1993; see figure 29 and references within) for much deeper clouds.

More recently, Guyot et al. (2020) extended the above momentum balance model by accounting for buoyancy effects under the assumption that the void fraction within the bubble cloud was uniform. They applied the momentum balance across the truncated conical control volume in figure 1(a) to obtain the following equation for the modelled cloud depth $H_b$:

(1.2)\begin{align} \underbrace{\rho V_i V_n\,\dfrac{{\rm \pi} {D_n}^2}{4}}_\text{Incoming momentum flux} &= \underbrace{\left(1-\bar{\phi}\right) \rho {U_t}^2 {\rm \pi} \left(H_b \tan\gamma + \dfrac{D_i}{2} \right)^2}_\text{Outgoing momentum flux}\nonumber\\ &\quad +\underbrace{\bar{\phi} \rho g {\rm \pi}\left(\dfrac{1}{3}\,H_b^3 \tan^2 \gamma + \dfrac{D_i }{2}\,H_b^2 \tan \gamma + \dfrac{D_i^2}{4}\,H_b \right)}_\text{Buoyancy force}, \end{align}

where $D_n$ is the nozzle diameter, $V_n$ is the velocity at the nozzle and $\bar {\phi }$ is the constant void fraction. Note that the impact void fraction and diameter have been removed from the impact momentum term on the left-hand side using mass conservation, $V_nD_n^2 = (1-\bar {\phi })V_iD_i^2$. For the limiting case of micro-jets and bubble cloud depths of $10$ cm or less, the buoyancy force may become less important than the outgoing momentum flux in (1.2), which then reduces to (1.1). Conversely, in the limit of very large jets, or massive bubble clouds ($H > 20$ cm), the contribution of the outgoing momentum flux becomes negligible, and (1.2) reduces to the impact momentum being consumed entirely by buoyancy. The fact that a contribution proportional to a surface term is balanced by a contribution proportional to a volume term then leads to $H$ being proportional to $(V_i V_n)^{1/3} D_n^{2/3}$. This is close to the empirical correlation $H \propto (V_{n}D_{n})^{0.66}$ proposed by van de Donk (1981) and $H \propto (V_{n}D_{n})^{0.7}$ proposed by McKeogh & Ervine (1981). Hence, based on physical arguments, (1.2) elucidates the two most common empirical relations, namely, $H \propto V_n D_n$ and $H \propto ( V_n D_n )^{0.66}$, over a very wide range of jet diameters and all available data in the current literature (Biń 1993; Guyot et al. 2020).

Furthermore, all parameters in the cubic polynomial for $H_b$ are known a priori, except for the average gas/liquid void fraction $\bar {\phi }$. So Guyot et al. (2020) adopted $\bar {\phi } = 15\,\%$ based on results from past studies (van de Donk 1981; McKeogh & Ervine 1981) and obtained a relatively good agreement with large-scale experiments. However, they observed a significant dispersion for the cloud depth in both their data and previous investigations when the jet diameter is large. Guyot et al. (2020) attributed this dispersion to variations in the void fraction $\bar {\phi }$ between various experiments since it is expected to depend strongly on both the jet geometry and the dynamics (McKeogh & Ervine 1981; Sene 1988; Bonetto et al. 1994; Zhu et al. 2000). Therefore, even if the scaling law for the bubble cloud depth is understood to some extent, measurements of the actual values of the void fraction $\phi$ are needed to clarify the role of this quantity in producing the observed dispersion in the cloud depth data. The conditions for the transition from jet-momentum-dominated to buoyancy-dominated clouds remain to be established as well. In addition, such investigations can provide insights into how, when and where additional features of the bubble cloud – such as turbulent dissipation, bubble size distribution and bubble cloud shape – are necessary for dimensioning industrial applications.

Biń (1993) lists correlations based on both global gas/liquid entrainment ratio measurements, using gas holdup techniques, and local gas/liquid void fraction measurements, often using resistivity probes within a bubble cloud. More recent developments on various correlations, along with inception conditions and some mechanistic viewpoints on air entrainment rate, can be found in Kiger & Duncan (2012). Past studies evidence that in the developing region, $\phi$ radial profiles consist of two maxima located at the radius of the free jet and a minimum at the jet axis (van de Donk 1981; Bonetto & Lahey 1993; Cummings & Chanson 1997; Brattberg & Chanson 1998; Chanson & Manasseh 2003; Ma et al. 2010). McKeogh & Ervine (1981) and van de Donk (1981) measured $\phi$ in the fully developed region and reported that the peak void fraction $\phi _0$ occurs along the cloud axis, remains almost constant with depth and then sharply decreases. However, the current literature lacks measurements of the cloud depth $H$ along with measurements of the local void fraction $\phi$ in the developed zone of the plume. Such data over various values of the impact momentum and jet fall height are necessary for better understanding of the dynamics of bubble clouds in the buoyancy-dominated regime.

To address this gap, the objective of the present work is to report concurrent local void fraction ($\phi$) and bubble cloud depth ($H$) measurements, for three typical nozzle diameters and at several impact momentum values and fall heights $Z_f$. Thereby, our aim is twofold: we examine not only the robustness and the relevance of the momentum balance models (Suciu & Smigelschi 1976; Clanet & Lasheras 1997; Guyot et al. 2020) by incorporating in situ void fraction data into (1.2), but also the effect of the void fraction on the maximum penetration depth of a two-phase plume, by varying independently three major parameters, namely, nozzle diameter, impact momentum and fall height. Using such a case study, we also try to propose a simple criterion to determine when the bubble cloud size is controlled by inertia and when it is controlled by buoyancy.

2. Methodology

Lab-scale experiments are carried out using a $1.2\ {\rm m} \times 0.6\ {\rm m} \times 0.6\ {\rm m}$ glass tank that is slightly tilted on one side so that overflowing water is drained through a drainage pipe to the reservoir bucket to keep the water level constant, as shown in figure 1(b). Pumps supply water from the drainage bucket to the straight injector, from which it is issued as a plunging jet. Three nozzle diameters $D_n$ are used: $D_n = 2.7$, 8 and 10 mm. For nozzle diameter $D_n = 2.7$ mm, a series of two centrifugal pumps from Pan World (NH-100PX and NH-200PS) is used. A heavy-duty centrifugal pump (2KVC AD 45/80M) from DAB Pumps generates the jet from the larger 8 mm and 10 mm injectors. The injector length to diameter ratio $(L/D_n)$ is kept equal to 50, and the height of fall to diameter ratio $Z_f/D_n$ is varied from 20 up to 100. The nozzle velocity $V_n$ varies from 1.5 to 12 m s$^{-1}$ and is controlled using a flow meter and ball valve. The impact velocity $V_i$ is deduced using the equation $V_i= \sqrt {V_n^2 + 2gZ_f}$. The present work considers only cases where the jet is not broken into droplets before impact.

Backlight imaging is used to film the jet just before the impact, illuminated by an LED panel, using a CMOS fast camera from Ximea (CB262MG) at 300 fps and $20\ \mathrm {\mu }$s shutter speed at $2496 \times 2418$ resolution. Images of the biphasic bubble cloud are filmed using a digital camera (Sony a7 III) and a zoom lens (Sony FE 24–70 mm F/4 ZA OSS Carl Zeiss, with focal length kept at 35 mm) at 50 fps and 200 $\mathrm {\mu }$s exposure at resolution $1920 \times 1080$ pixels. Using a MATLAB-integrated calibration application, lens distortion was found to be negligible, and constant spatial resolutions 54 and 333 $\mathrm {\mu }$m pixel$^{-1}$ were used for jet and bubble cloud images, respectively. The cloud images are analysed using the open-source freeware ImageJ (Schindelin et al. 2012) and algorithms therein for brightness thresholding (Kapur, Sahoo & Wong 1985; Tsai 1985), along with in-house MATLAB code to detect and trace the outer boundaries of the cloud in order to measure $H$. The mean penetration depth $H$ is measured by averaging over 3000 images. A low frame rate was chosen in order to ensure that the images of the bubble cloud are decorrelated and so that the ensemble average value of the cloud depth $H$ converges faster for a given number of images.

The contours of the jet are extracted from images using machine-learning-based object detection models, namely, grounding DINO (Liu et al. 2023) and the Segment Anything model (Kirillov et al. 2023). The detected jet edges are then analysed using in-house MATLAB code to obtain the impact diameter ($D_i$) and the roughness ($\epsilon$). The latter is defined as the average of the root mean square values of the lateral departure from the mean edge position, for each side, and is obtained by averaging over $200$ images.

The local void fraction $\phi$ is measured using an optical fibre probe (A2 Photonics Sensors) of diameter 125 $\mathrm {\mu }{\rm m}$ mounted upright on a slender plate (thickness 4 mm), as shown in the magnified view in figure 1(b). This horizontal plate is fixed to a vertical cylindrical rod (diameter 1.5 cm). The probe tip is located 4.5 cm upstream of this arm. This ensures that obstruction to the incoming flow is minimized as much as possible at the probe tip, where the measurement is carried out. The probe assembly is translated in the vertical ($r$–$z$) plane using two motorized linear stages (Igus) having accuracy $0.01$ mm. The probe signals are sampled at $250$ kHz for $60$ s, a duration that is enough to ensure convergence of the void fraction for all conditions. The signals are analysed using A2 Photonics software (SO6 v4.7). The probe is centred at the jet axis, and the measurements are carried out in the radial direction of the cloud with step size $2$ mm to obtain radial void fraction profiles. This was repeated at various depths, such that $z/H = 0.4$, 0.6 and $0.8$. Note that when the bubble cloud size is measured, the probe assembly is pulled out of the tank with the help of the translation stage.

3. Results and discussion

3.1. Measurement of bubble cloud depth $H$

Figure 2 illustrates the variation of $H$ for the $D_n=2.7$ mm and $D_n=8$ mm nozzles, as a function of $\sqrt {V_iV_n}\,D_n$, which is proportional to the square root of the impact momentum flux, up to a factor of $\rho$, as in (1.2). Note that the largest bubble cloud obtained for our conditions is $45$ cm deep, which is significantly smaller than the tank depth ($1.25$ m). We therefore assume that there is no significant pressure gradient caused by confinement effects in our experiments. The bubble cloud size $H$ for jets issued from the $D_n =2.7$ mm nozzle (red and black discs) varies linearly with $\sqrt {V_iV_n}\,D_n$. This is consistent with the model proposed by Clanet & Lasheras (1997) (solid line; see (1.1)), since for this small-scale jet and for the experimental range considered here, $V_i\approx V_n$ and $D_i\approx D_n$. When $Z_f/D_n$ is increased for this $D_n =2.7$ mm nozzle, a negligible change in $H$ is observed. For the larger $D_n = 8$ mm nozzle (open symbols), (1.1) predicts accurately the behaviour of $H$ up to $\sqrt {V_iV_n}\,D_n \approx 0.02\ {\rm m}^2\ {\rm s}^{-1}$. Beyond this threshold, a transition occurs, and $H$ scales as $(V_iV_n)^{1/3}D_n^{2/3}$, in accordance with the model of Guyot et al. (2020) ($2/3$ power law shown by dashed line). This suggests that beyond a threshold, the buoyancy force becomes dominant over the outgoing momentum flux, leading to a decrease in cloud depth $H$ compared with what (1.1) would predict. Also, figure 2 shows the coexistence of two regimes in a narrow range, $\sqrt {V_iV_n}\,D_n= 0.02$–$0.03$ m$^2$ s$^{-1}$, wherein the inertia-dominated regime occurs for $D_n = 2.7$ mm while buoyancy forces become important for the bigger nozzle diameter. This is perhaps a signature of the dependence of the transition threshold on the void fraction at different jet diameters. Finally, in the buoyancy-dominated zone for $D_n = 8$ mm, some dispersion in the values of $H$ is observed when $Z_f/D_n$ is varied. As mentioned in the Introduction, this dispersion arises from the variations in the net air/water void fraction, as will be discussed in the next subsection.

Figure 2. Variation of $H$ with the square root of the impact momentum $\sqrt {V_iV_n}\,D_n$ for $D_n = 2.7$ and 8 mm at various $Z_f/D_n$. The dotted line shows the trend expected from (1.1). The dashed line shows the $2/3$ scaling law expected at large scales. The inset graph is a blow-up of the $D_n=8$ mm data showing that increasing $Z_f$ at constant impact momentum leads to a decrease in $H$. The data points I–VI correspond to the flow conditions that are compared in § 3.2.

3.2. Local void fraction measurements

The aim of this subsection is to present measurements of the local void fraction $\phi$ within the bubble cloud and discuss them in relation to the bubble cloud size. We discuss first $\phi$ measurements along the jet axis and then radial profiles of $\phi$ at a given depth within the cloud. Finally, general remarks on the void fraction profiles are presented.

3.2.1. Axial measurements

Figure 3(a) presents the air-to-water volume fraction at the jet axis, $\phi (r = 0, z)$, denoted here by $\phi _0$. These measurements correspond to the bubble cloud generated by the $D_n = 2.7$ mm nozzle for two distinct values of jet fall height to diameter ratio $Z_f/D_n$ while maintaining nearly identical impact momentum. The values of $\phi _{0}$ are nearly constant with depth $z$, until they decrease sharply at the end of the cloud, as reported previously by van de Donk (1981), McKeogh & Ervine (1981), Chanson & Manasseh (2003) and Hoque & Aoki (2008). When $Z_f/D_n$ is increased, a similar axial profile is observed, but $\phi _{0}$ is increased by almost $26\,\%$.

Figure 3. (a) Axial evolution of the void fraction $\phi _0 (z)$ for two different ratios $Z_f/D_n$ at a fixed jet diameter $D_n = 2.7$ mm and $\sqrt {V_iV_n}\,D_n = 0.016\pm 0.0002\ {\rm m}^2\ {\rm s}^{-1}$. Instantaneous images of the freely falling jet just before impact: (b) $Z_f /D_n = 20$; (c) $Z_f /D_n = 50$.

The jet morphology just before impact is illustrated in figure 3(b,c) for $Z_f/D_n = 20$ and $50$, and it shows a slightly larger amplitude of the jet undulations for the larger $Z_f/D_n$ case. Using the algorithm introduced in § 2, we measure that the undulations on the jet surface exhibit roughness $\epsilon = 170\ \mathrm {\mu }{\rm m}$ for figure 3(b) and $\epsilon = 225\ \mathrm {\mu }{\rm m}$ for figure 3(c). As discussed by McKeogh & Ervine (1981) and Sene (1988), we expect the air entrainment rate and therefore the void fraction to be proportional to the volume enclosed by the corrugations observed on the jet surface. The increase in the size of these undulations is therefore certainly what leads to the increase in void fraction observed in figure 3(a). Supplementary movies 1 and 2 (available at https://doi.org/10.1017/jfm.2023.1019) provide additional visualizations of the jets of figure 3(b,c), respectively (frame rate 300 images s$^{-1}$). These two conditions are indicated by labels I and II in figure 2. They are both situated in the inertia-dominated regime, where buoyancy effects are still negligible. As a result, the effect of the void fraction on $H$ is negligible, even though $\phi$ increases.

3.2.2. Radial measurements

Radial profiles of the void fraction are measured for three nozzle diameters, and two heights of fall, at a constant impact momentum. The radial profiles are taken at depths larger than $4D_i$, in order to ensure that they are in the developed region (Ervine & Falvey 1987). Figure 4(a) shows the bubble cloud generated by a jet of nozzle diameter $D_n=2.7$ mm. In comparison, the bubble cloud depth issued from the jet of larger diameter $D_n = 8$ mm, as shown in figure 4(c), is significantly smaller (see also labels III and IV in figure 2). The visualizations of figure 4(a,c) are illustrated more extensively in supplementary movies 3–6. The void fraction profiles for the 2.7 and 8 mm injectors are shown in figure 4(b,d), respectively. They are measured for three different depths. Dispersion in $\phi$ profiles is relatively small in the developed region. These profiles can be well approximated using a Gaussian distribution (solid lines) as previously reported by van de Donk (1981). This finding is similar to the case where an air jet is injected into the pool (Kobus 1968; Freire et al. 2002). When our data are compared with fits based on air/bubble diffusivity and air-to-water volume flux ratio, as in Cummings & Chanson (1997), the best fit was found for an air-to-water entrainment ratio of $\mathcal {O}$(10)! This unrealistic value results perhaps from their constant diffusivity assumption for the advection–diffusion process in the two-phase mixing layer. In our case, and as described by Clanet & Lasheras (1997), the bubbles are convected by the large-scale eddies dominating the evolution of the submerged jet.

Figure 4. (a,c) Instantaneous pictures of the falling jet just before impact for $D_n = 2.7$ and $8$ mm, respectively, at constant $\sqrt {V_iV_n}\,D_n = 0.028\ {\rm m}^2\ {\rm s}^{-1}$ and $Z_f/D_n = 20$. Corresponding bubble clouds are shown as well. (b,d) The radial variation of $\phi (r, z)$ at three different depths in the bubble cloud, for the $2.7$ mm and $8$ mm jets, respectively.

Note that the maximum void fraction $\phi (r = 0, z)$ is significantly smaller for the thinner jet (up to $18\,\%$ along the axis; see figure 4b) than for the larger jet (up to 27 % in figure 4d). The impact of the nozzle diameter on the maximum void fraction has already been demonstrated by Chanson et al. (2004). The larger void fraction, and hence buoyancy force, is consistent with the observation that the bubble cloud is smaller for the larger diameter. Furthermore, this increase in void fraction is very likely caused by the larger length scales associated with the corrugation and its amplitude occurring on the jet surface when the jet diameter $D_n$ is increased at a constant $Z_f/D_n = 20$ and constant impact momentum (Sene 1988; Liu, Gao & Hu 2022). Measurements of $\epsilon$ show that the roughness goes from 225 $\mathrm {\mu }$m for figure 4(b) to 445 $\mathrm {\mu }$m for figure 4(d). These points are indicated by labels III and IV in figure 2. It is clear that III belongs to the momentum-dominated regime, while IV belongs to the buoyancy-dominated regime.

Figure 5 illustrates the influence of a change in the dimensionless jet fall height $Z_f/D_n$, for a constant impact momentum and constant diameter $D_n = 8$ mm. Photographs of the jet before impact show that the jet issued at $Z_f/D_n = 100$ (figure 5c) is more corrugated than that issued at $Z_f/D_n = 20$ (figure 5a), for the same $\sqrt {V_iV_n}\,D_n=0.075\ {\rm m}^2\ {\rm s}^{-1}$. This is corroborated by the data in figure 6, which show the variation of roughness of the jet at impact ($\epsilon$) and the variation of $\bar {\phi }_{0}$, the average over depth of the maximum void fraction measured along the axis, as a function of $\sqrt {V_iV_n}\,D_n$ for different $Z_f$. The data show a strong increase in $\epsilon$ and $\bar {\phi }_{0}$ as $Z_f$ is increased for a constant momentum. The visualizations of figure 5(a,c) are illustrated more extensively in supplementary movies 7–10. The bubble cloud generated below the surface is smaller for the larger $Z_f/D_n$. Again, this is likely due to the fact that this jet captures more air within its cavities during free-fall, leading to a higher $\phi$. This is confirmed by the radial profiles given in figure 5(b,d), which show that $\phi _0$ for $Z_f/D_n = 100$ is approximately $3.2$ times that for $Z_f/D_n = 20$. Note that these cases are referred to by labels V and VI, respectively, in figure 2. They both correspond to the buoyancy-dominated regime. In addition, the inset displayed in figure 2 illustrates a zoomed-in view of the change in $H$ at higher $\sqrt {V_i V_n}\,D_n$, revealing that an increase in $Z_f /D_n$ results in a reduction of ${H}$, with a decrease of approximately 20 % observed when $Z_f /D_n$ is raised from 20 to 100. This ascertains that the observed dispersion for the cloud depth in figure 2 – and also in previous investigations reported in Guyot et al. (2020) for this regime, at higher impact momentum and larger diameter – is due to differences in jet fall height.

Figure 5. Images of the $D_n = 8$ mm jet just before impact for (a) $Z_f/D_n = 20$ and (c) $Z_f/D_n =100$, for a constant $\sqrt {V_iV_n}\,D_n = 0.075\ {\rm m}^2\ {\rm s}^{-1}$. Corresponding bubble clouds are shown as well. (b,d) The radial variation of $\phi$ at various depths in the bubble clouds for $Z_f/D_n = 20$ and 100, respectively.

Figure 6. Variation of $\epsilon$ and $\bar {\phi }_{0}$ with $\sqrt {V_iV_n}\,D_n$ for $D_n = 8$ mm, where $\bar {\phi }_{0}$ is the average over depth of the maximum void fraction measured along the axis. Both $\epsilon$ and $\bar {\phi }_{0}$ increase when $Z_f$ is increased for a constant jet momentum.

Finally, general trends of local void fraction profiles are depicted in figure 7 for the case of $D_n = 8$ mm. Each plot presents data at different vertical locations $z$ underneath the surface, normalized with respect to the cloud depth $H$. In all cases shown here, the data fit reasonably well with a Gaussian profile $\phi = \phi _0(z)\,\mbox {e}^{- r^2/\sigma ^2(z)}$ whose peak and width do not vary much as $z/H$ is changed. Plots in figure 7(a) correspond to data for the shortest jet fall height to diameter ratio $Z_f/D_n = 20$ at various $\sqrt {V_iV_n}\,D_n$, a measure of the jet impact momentum up to a factor of liquid density. As $\sqrt {V_iV_n}\,D_n$ increases, the centreline peak void fraction $\phi _0$ slightly decreases, while the width of the best-fitting Gaussian profiles increases. On the other hand, at a given $\sqrt {V_iV_n}\,D_n$, the peak void fraction $\phi _0$ is almost doubled when the jet's fall height is quintupled. In the next subsection, these observations on the air-to-water volume fraction in the biphasic region are developed further to capture properly the effect of buoyancy on the bubble cloud depth.

Figure 7. Measured profiles of air/water void fraction for the nozzle diameter $D_n = 8$ mm and different fall heights (a) $Z_f/D_n = 20$, (b) $Z_f/D_n = 50$ and (c) $Z_f/D_n = 100$, at various values of Q = $\sqrt {V_iV_n}\,D_n$. Gaussian fits are represented by continuous lines.

3.3. Prediction of bubble cloud depth

Guyot et al. (2020) proposed a prediction for the bubble cloud depth $H$ via (1.2) by assuming a uniform void fraction within the bubble cloud and a constant cone angle $\gamma$, based on single-phase turbulent jets. As mentioned earlier, the void fraction in the biphasic jet underneath the free surface can be well approximated by a Gaussian profile. These profiles are characterized by a maximum void fraction $\phi _0(z)$ on the axis and a variance $\sigma (z)$. It is then easy to show that the buoyancy force on a slice of bubble cloud of thickness ${\rm d}z$ at depth $z$ is ${\rm d}F_b = \phi _{0}(z)\,\rho g {\rm \pi}\,\sigma (z)^2 \,{\rm d}z$. We take $\phi _0(z)$ to be equal to its average value $\bar {\phi }_{0}$, since the peak void fraction $\phi _0(z)$ exhibits little variation with depth $z$, as already shown in figures 3–5. It now remains to provide $\sigma (z)$ in order to integrate this buoyancy force over $z$.

Indeed, it is expected that $\sigma (z)$ increases linearly with depth as the bubble cloud widens up as a cone, as evidenced by visualization and previous observations (Suciu & Smigelschi 1976; McKeogh & Ervine 1981; Clanet & Lasheras 1997; Guyot et al. 2020). We propose to measure the angle of this cone based on the experimental void fraction profiles. However, rising bubbles can be detected outside the conical jet region by the optical probe. This may lead to overestimation of the width of the void fraction profiles and hence the resulting buoyancy force on the bubble cloud. This is particularly true for the measurements at shallower depths, $z < 0.5H$, a problem also encountered by van de Donk (1981). In order to circumvent this difficulty, the conical jet hypothesis is maintained in accordance with previous authors so that $\sigma (z) = D_i/2+ z \tan \gamma _{0}$, where the cone angle $\gamma _0$ is computed as $\tan \gamma _{0} = (\sigma (\tilde {z}) - D_i/2 ) /\tilde {z}$ at a chosen reference depth $\tilde {z} = 0.8 H$. The latter depth was chosen to avoid both rising bubbles and the steep decrease in bubble void fraction in the neighbourhood of the cloud's tail at $z = H$.

Thereby, the net buoyancy force $F_b$ on the bubble cloud of depth $H_b$ can then be expressed as

(3.1)\begin{equation} F_b = \bar{\phi}_{0} \rho g {\rm \pi}\left( {\frac{1}{3}}\,H_b^3 \tan^2 \gamma_{0} + {\dfrac{D_i}{2}}\, H_b^2 \tan \gamma_{0} + {\dfrac{D_i^2}{4}}\,H_b \right)\!, \end{equation}

which is identical to the volume term in (1.2) if $\bar {\phi } = \bar {\phi }_{0}$ and $\gamma = \gamma _{0}$. The above expression indicates that the buoyancy force on a conical biphasic jet exhibiting a Gaussian void fraction profile can be interpreted as the buoyancy force exerted on an equivalent biphasic jet with a constant void fraction $\bar {\phi }$ equal to the maximum void fraction $\bar {\phi }_{0}$ of the Gaussian profile and with a width defined by the variance of the Gaussian profile at some reference depth, here chosen at $z = 0.8H$. This buoyancy force $F_b$ can now be injected into the momentum balance equation (1.2) in order to solve for $H_b$. The value of $\bar {\phi }$ in the outgoing momentum term can be taken equal to $\bar {\phi }_{0}$ for the sake of simplicity.

Experimental depths $H$ from the present study and predictions $H_b$ are compared in figure 8(a) for the different nozzle sizes and jet fall heights. Good agreement is found between the model and measurements for both the inertia-controlled bubble clouds ($D_n = 2.7$ mm) and the buoyancy-controlled ($D_n = 8$ mm and 10 mm) ones. A measurement for nozzle diameter $D_n=10$ mm (same length-to-diameter ratio 50), $V_n=5.5\ {\rm m}\ {\rm s}^{-1}$ and $Z_f/D_n=50$ has been included in the data for this figure. For comparison, two previous investigations, namely, those conducted by van de Donk (1981) and McKeogh & Ervine (1981), are presented in figure 8(a) as well. They provided both $H$ measurements and radial profiles of $\phi$ for jets with diameters of $D_n = 6$ and $9$ mm. The calculated $H_b$ values based on their measurements show very good consistency with the model.

Figure 8. (a) Comparison of the experimental depth $H$ with values predicted by the model $H_b$. (b) Normalized depth $H/H_i$ at various values of $Fr$ for injectors from current and past studies whose void fraction is known. Equation (3.2) is represented by the continuous line.

In what follows, the question of the role of the void fraction in the transition from the inertia-dominated to the buoyancy-dominated regime is discussed. This transition is expected to be dominated by the ratio of the outgoing momentum to the buoyancy force, respectively the second and third terms in (1.2). The square root of the ratio of these terms defines a dimensionless grouping similar to a Froude number, given by $Fr = \sqrt {3( 1-\bar {\phi }_{0} )/\bar {\phi }_{0}}\,U_t/\sqrt {g H_b}$. In fact, when the nozzle diameter is small relative to the cloud width $2 H_b \tan \gamma _0$, the cubic polynomial for $H_b$ in (1.2) can be rewritten in terms of $Fr$ as

(3.2)\begin{equation} \dfrac{H_b}{H_i} = \dfrac{Fr}{\sqrt{1 + Fr^2}} + \mathcal{O}\left( \dfrac{D_i}{2 H_b \tan \gamma_0}\, Fr^2\right)\!, \end{equation}

where $H_i$ is the cloud depth when the buoyancy force is absent, as given by (1.1). When $Fr$ is large, we expect the cloud to be in the inertia-dominated regime and hence $H_b$ to be close to $H_i$, the simple prediction by Clanet & Lasheras (1997). When, on the contrary, $Fr$ is small, $H$ is expected to be controlled by buoyancy and hence to be significantly smaller than $H_i$. Figure 8(b) illustrates the variation of $H/H_i$ with $Fr$ for data from figure 8(a). Indeed, all experimental data points fall on a single curve given by (3.2), for which $H/H_i$ falls steeply when $Fr$ is decreased. For a constant velocity, using larger nozzles will lead to bigger bubble clouds, thus larger $H$ and smaller $Fr$, thereby favouring the buoyancy-dominated regime. Note also that the influence of jet velocity is counterintuitive here, since a larger jet velocity for a given nozzle diameter will generate a larger $H$ and therefore also favour the buoyancy-dominated regime, instead of the inertial regime. This is because the relevant velocity scale in $Fr$ is the terminal bubble velocity $U_t$, which is a constant, and not the jet impact velocity. Note that (3.2) is obtained from (1.2) by assuming $\bar {\phi } = \bar {\phi }_{0}$ and $\gamma = \gamma _{0}$. This choice simplifies the expression of the Froude number, but obviously underestimates the jet angle and mean void fraction in the inertial contribution. This is partly the reason why $H/H_i$ does not reach the limit $H/H_i = 1$ in figure 8(b), even for data points that corresponded to the inertia-dominated regime in figure 2. Furthermore, previous authors have proposed adding a hemispherical dome to the truncated cone modelling the bubble cloud, as in figure 1 (Clanet & Lasheras 1997; Guyot et al. 2020). This small correction would introduce a factor of $1 + \tan \gamma _0$ (of the order of 10 %) to both the modelled heights $H_i$ and $H_b$, but it is not considered in the present data for the sake of simplicity.

For most studies for which bubble cloud depths data are available, the void fraction has not been measured concomitantly. We plot in figure 9(a) $H$ as a function of $\sqrt {V_iV_n}\,D_n$, the square root of the impact momentum over the liquid density $\rho$, for five past studies covering a wide range of scales, plus the present data. The scaling laws observed in figure 2 for the inertia-dominated and buoyancy-dominated regimes can be observed in the limits of low momentum and large momentum, respectively. The void fraction is not known for these past works, but we can still estimate the bubble cloud Froude number as $Fr^* = U_t/\sqrt {gH}$. Figure 9(b) shows that when $H/H_i$ is plotted as a function of $Fr^*$ for several past studies, the cloud sizes are clearly sorted into two regimes: (i) one for larger $Fr^*>0.2$, for which $H/H_i$ is close to 1 and for which there is little dispersion; and (ii) a second regime for $Fr^*<0.2$ where $H/H_i$ varies between $0.15$ and $0.5$, and for which significant dispersion is obtained. This dispersion is very likely due to the role of the void fraction, which has been omitted in $Fr^*$. These observations ascertain further that $Fr$ is a good parameter for monitoring the transition between the inertia-dominated and buoyancy-dominated regimes.

Figure 9. (a) Bubble cloud depth ($H$) data from previous investigations are presented here as a function of $\sqrt {V_iV_n}\,D_n$, a measure of the impact momentum, to illustrate the two distinct dynamical regimes identified in our study over a wide range of scales. (b) When the void fraction is not known, $Fr^*$ can still sort past experiments into inertia- or buoyancy-dominated regimes, even though more dispersion is observed.

3.4. Equivalent bubble cloud void fraction $\phi _{b}$

Whereas Guyot et al. (2020) took a uniform void fraction within the cloud to express the momentum balance, $\phi$ profiles were observed here to follow a Gaussian distribution in the radial direction. In this subsection, an equivalent constant void fraction $\phi _{b}$ is proposed in order to provide a bulk quantity for a bubble cloud and to discuss its variations with the jet fall height and impact momentum. At first, for a fixed depth $z$, the edge of this constant-void-fraction cone of bubble cloud is defined by assuming that its radius $R_{b}(z)$, beyond which the void fraction becomes zero, is the radius at which $80\,\%$ of the surface-integrated void fraction is attained based on a Gaussian profile $\phi = \bar {\phi }_0(z)\, {\rm e}^{-r^2/\sigma (z)^2}$. This simply gives the relation $R_{b}(z) = \sigma (z)\,\sqrt {\log {5}} \approx 1.27 \sigma$. Thereafter, in order to avoid overestimation of the bubble cloud volume due to rising bubbles, the truncated cone angle $\gamma _{b}$ for this constant-void-fraction cloud is taken as $\tan \gamma _{b} = (\tilde {R}_{b}- D_i/2)/\tilde {z}$, where $\tilde {R}_{b} \approx 1.27\, \sigma (\tilde {z})$ is this cloud's radius at the reference depth $\tilde {z} = 0.8H$. As already mentioned in § 3.3, the influence of rising bubbles is minimal at the reference depth $\tilde {z} = 0.8H$, measured close to the bottom of the cloud, while the peak void fraction ${\phi }_0$ is still comparable to those obtained in the bulk. Finally, the equivalent void fraction $\phi _b$ is then computed such that the net buoyancy force is the same on both the uniform-void-fraction cloud and the Gaussian-void-fraction cloud.

Figure 10(b,d) depict the evolution of this equivalent void fraction $\phi _{b}$ as a function of $Z_f/D_n$, for different values of impact momentum (represented by the colour) for the two injectors. Experimental results show that $\phi _{b}$ increases steeply for both injectors as $Z_f/D_n$ increases, as already illustrated in figure 5 for a particular case. As mentioned in § 3.2.2, this is likely due to the jet's capacity to entrain more air once perturbations have grown larger on its surface for higher fall heights $Z_f$ (Sene 1988). These plots also show that $\phi _{b}$ decreases for a constant $Z_f/D_n$ as the impact momentum is increased, in particular for low $Z_f/D_n$.

Figure 10. (a) Definition of $\phi _{b}$ and $R_{b}$. (b–e) Variation of $\phi _{b}$ and $\gamma _{b}$ with $Z_f/D_n$ for the two injector sizes at various impact momentum values. Two colour bars are provided for the two ranges of $\sqrt {V_iV_n}\,D_n$, corresponding to both $D_n$ values.

Figure 10(c,e) illustrate the variations of the cone angle $\gamma _{b}$ of the equivalent constant-$\phi$ jet with respect to $Z_f/D_n$, for different $\sqrt {V_iV_n}\,D_n$. Results for both nozzles show that $\gamma _{b}$ decreases with $Z_f/D_n$ for a constant impact momentum. This change in the bubble cloud shape may be related to the strong increase in the amplitude of surface perturbations at larger $Z_f$, and to the larger void fractions observed for these conditions ($\phi _0$ up to $50\,\%$ for $Z_f/D_n = 100$, compared with $20\,\%$ for $Z_f/D_n = 20$). The simple assumptions made regarding the shape of the bubble cloud, which are valid at moderate void fractions, probably no longer hold in such conditions.

4. Conclusion

Experiments on circular plunging jets have been carried out to characterize the two regimes for bubble cloud formation, namely the inertia-dominated and buoyancy-dominated regimes. Void fraction profiles were measured in the developed region of the cloud with novel optical probes for nozzle diameters $D_n = 2.7$, 8 and 10 mm. They can be well approximated by Gaussian distributions. When these measurements are incorporated into the momentum balance model, taking into account only the buoyancy force, the model predicts the experiments very well for both current and past studies. Even though the void fraction measurements have been carried out only for nozzle sizes smaller than 1 cm and bubble cloud sizes smaller than 50 cm, it is expected that this result still holds at larger scales, as already proposed by Guyot et al. (2020). This will have to be verified with void fraction measurements in such metric scale systems in future works.

The threshold between the inertia and buoyancy regimes is shown to be dependent on a characteristic Froude number $Fr$ composed of the terminal bubble velocity $U_t$, the cloud depth $H$ and the void fraction within the bubble cloud. For large values of this number, $H$ can be estimated with the simple model of Clanet & Lasheras (1997), while for low values of $Fr$ it is buoyancy that controls the bubble cloud size. The fact that a single equation can predict bubble cloud size independent of scale and of the wide zoology of jet dynamics, provided that the void fraction is known, is the major result of this work.

A constant equivalent void fraction $\phi _{b}$ is also introduced based on the integration of the Gaussian void fraction profiles. The variations of $\phi _{b}$ and of the corresponding jet cone angle $\gamma _{b}$ with the impact momentum and the jet fall height to diameter ratio $Z_f/D_n$ are discussed. This void fraction $\phi _{b}$ is expected to be a function of the jet diameter, height of fall and velocity (Wang et al. 2018). Data show that an increase in the height of fall $Z_f$ leads to a significant increase in the void fraction within the cloud, which in turn leads to a decrease in the bubble cloud size. The modelling of $\phi _{b}$ as a function of jet dynamics, which is necessary for practical purposes if one wishes to predict $H$ solely as a function of input parameters, is a question that future works will have to address.