# Significance of void fraction in bubble clouds generated by single and multi-plunging jets

Narendra Dev, John Soundar Jerome J, Hélène Scolan, Jean-Philippe Matas

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Narendra Dev, John Soundar Jerome J, Hélène Scolan, Jean-Philippe Matas. Significance of void fraction in bubble clouds generated by single and multi-plunging jets. Rhéologie, 2024, 45, pp.52-57. hal-04857687

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# Significance of void fraction in bubble clouds generated by single and multi-plunging jets

N. Dev, J. John Soundar Jerome, H. Scolan et J.-P. Matas

Univ Lyon, Université Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, LMFA, 6 UMR 5509, 69622 Villeurbanne

Reçu le 19 décembre 2023 - Version finale acceptée le 8 avril 2024

**Abstract:** Through laboratory-scale experiments using optical probes for precise measurement of the local bubble void fraction ($\phi$), we study the bubble cloud generated by circular plunging jets as they impact the free surface of a pool. We illustrate that a momentum balance, considering only liquid inertia and the buoyancy force resulting from the bubble cloud volume, offers a highly accurate estimate for the depth $H$, when $\phi$ is known. Furthermore, we utilize a straightforward configuration of closely packed multi-injectors as a model to explore air entrainment by large-scale plunging jets. Our results show that, even in such complex scenarios, the depth of the cloud can be determined primarily by the balance between the inertia imparted by the plunging liquid jet and buoyancy, which is itself a function of the volume of the bubble cloud and of the bubble void fraction.

**Keywords:** Plunging jets, Bubble cloud depth, Void fraction

## [Version abrégée en français à la dernière page]

### 1. Introduction

Entrainment of air in the form of a cloud of bubbles into the pool is often observed when a plunging jet impacts on a liquid surface [1, 2]. This phenomenon is prevalent in a wide array of industrial applications, including chemical stirring [3] and hydroelectric processes [4], as well as in natural occurrences like breaking waves and cascades [5, 6]. The maximum penetration depth of the bubble cloud, denoted as $H$ in Figure 1a is an important parameter in all such applications. Many empirical correlations delineating the cloud depth are available in current literature [6, 7] but a comprehensive understanding of the underlying physical principles governing these correlations has only been achieved in recent years. In their study, Clanet and Lasheras [8] proposed a momentum rate conservation to derive a straightforward prediction for $H$, denoted as $H_i$:

$$H_i = \frac{1}{2 \tan \gamma} \frac{V_i D_i}{U_t} \quad (1)$$

where $V_i$ is the impact velocity, $D_i$ is the impact diameter of the jet, $\gamma$ is the half-cone angle of the cloud and $U_t$ is the terminal velocity of the bubbles. They obtained very good agreement for their experiments with micro-jets as shown by open green

symbols in Figure 1c. The bubble clouds formed by micro-jets are known as inertial clouds. However, the principle of momentum conservation might not hold true for voluminous biphasic jets, where the influence of buoyancy cannot be ignored and may even become dominant. Such jets are buoyancy-dominated clouds. As shown by closed green symbols in Figure 1c, the model behind Eq. (1) fails to predict buoyancy-dominated clouds formed at higher $V_i D_i$ [9–11].

In a more recent development, Guyot et al. [11] expanded the aforementioned momentum balance model to incorporate buoyancy effects, assuming a uniform void fraction within the bubble cloud. These assumptions resulted in the following equation for the modelled cloud depth, $H_b$:

$$\rho V_i V_n \frac{\pi D_n^2}{4} = (1 - \bar{\phi}) \rho U_t^2 \pi \left( H_b \tan \gamma + \frac{D_i}{2} \right)^2 + \bar{\phi} \rho g \pi \left( \frac{1}{3} H_b^3 \tan^2 \gamma + \frac{D_i}{2} H_b^2 \tan \gamma + \frac{D_i^2}{4} H_b \right) \quad (2)$$

where $D_n$ is the nozzle diameter, $V_n$ is the velocity at the nozzle, and $\bar{\phi}$ is the void fraction, assumed to be constant in this model.

The model states that the input momentum flux of the liquid jet (first term) is consumed by the outgoing mo-
<figure><img src="image_2.png" /><figcaption>Figure 1. Schematics of (a) a typical bubble cloud and (b) the experimental setup and (c) correlation between bubble cloud depth *H* and *V*ᵢ*D*ᵢ, a measure of the jet impact momentum up to a factor of liquid density.</figcaption></figure>

mentum flux at the exit (second term) and by buoyancy (third term) induced by two-phase jet. In the small scale limit of micro-jets generating cloud sizes of 10 cm or less, the buoyancy force may become less significant compared to the outgoing momentum flux, ultimately simplifying the model of Guyot et al. [11] to Eq. (1). Conversely, in the case of very large jets or substantial bubble clouds (*H* > 20 cm), the contribution of the outgoing momentum flux becomes negligible, and their model leads to the case wherein the impact momentum is entirely consumed by buoyancy. In this limit, the cloud depth *H* being directly proportional to a volume term results in *H* being directly proportional to (VᵢVₙ)¹/³ Dₙ²/³. This closely aligns with the empirical correlations *H* ∝ (VᵢVₙ)⁰.⁶⁶ put forward by Van de Donk [12] and *H* ∝ (VᵢVₙ)⁰.⁷ proposed by McKeogh and Ervine [3]. Consequently, grounded in physical reasoning, the model of Guyot et al. [11] holds true across an extensive spectrum of jet diameters, encompassing all available data in the current literature [8-12].

All parameters in the cubic polynomial for $H_b$ in Eq. (2) are known *a priori*, with the sole exception of the average gas/liquid void fraction $\bar{\phi}$. Guyot et al. [11] found a good agreement of this model with experiments but a considerable dispersion in both their data and prior investigations were reported when the jet diameter was substantial. They ascribed this dispersion to variations in the void fraction across different experiments, as it is expected to be significantly influenced by both jet geometry and dynamics [1, 3, 13]. More recently, our work [14] confirmed the importance of the void fraction $\phi$ and its role in the observed dispersion in cloud depth. In

order to further elucidate the role of void fraction, we first compare in sections 3.1 and 3.2 experimental results where the void fraction is measured, to the predictions of the model of Guyot et al.[11]. We next present in section 3.3 results from a simple setup involving closely packed multi-injectors as a model to investigate air entrainment by large-scale plunging jets.

## 2. Methodology

The experiments are performed in a glass tank measuring 1.2 m × 0.6 m × 0.6 m. The tank is slightly tilted on one side to facilitate the drainage of overflowing water through a drainage pipe into a reservoir bucket, thereby maintaining a constant water level, as depicted in Figure 1b. Water from the drainage bucket is pumped to the straight injector, where it is discharged as a plunging jet. Three different nozzle diameters $D_n$ are employed: $D_n = 2.7$, 8, and 10 mm. Backlight imaging is used to film the jet just before the impact illuminated by a LED panel using a fast camera at 300 fps and 2496 × 2418 resolution. Images of the bubble cloud are captured using a digital camera at 50 fps and resolution 1920 × 1080 pixels. The mean penetration depth *H* is determined by averaging the results obtained from 3000 images.

The local void fraction $\phi$ is measured using an optical fibre probe (A2 Photonics Sensors) with a diameter of 125 µm, mounted upright on a slender plate as illustrated in the magnified view in Figure 1b. This horizontal plate is affixed to a vertical cylindrical rod. The probe assembly is vertically translated using two motorized linear stages. When a bubble pierces the tip of the probe, the electronic unit detects the change in refractive index, signalling the detection of the
bubble. Signals from the probe are sampled at 250 kHz for a duration of 60 seconds, ensuring adequate convergence of the void fraction for all conditions. The A2 Photonics software (SO6 v4.7) is utilized to analyse the signals. See reference [14] for more details on the experimental method.

# 3. Results and discussion

## 3.1 Local void fraction measurements

Figure 2c illustrates a typical bubble cloud formed by the jet from the 2.7 mm nozzle (Fig. 2a), while Figure 2d shows a typical bubble cloud for the jet formed by the 8 mm nozzle (Fig. 2b).

<figure><img src="image_3.png" /><figcaption>Figure 2. (a, b) Instantaneous images of the falling jet, (c, d) corresponding bubble clouds, (e, f) The radial variation of $\phi(r, z)$ at three different depths in the bubble cloud.</figcaption></figure>

Even though both jets have the same $V_i D_i$, which, up to a factor of liquid density $\rho$, is the square root of the impact momentum, a noticeable reduction in $H$ is observed for the 8 mm jet case. To understand this effect, the void fraction profiles for the 2.7 mm and 8 mm nozzles are pre-sented in Figures 2e and 2f, respectively, at three different depths. Radial profiles are captured at depths exceeding 4 times the nozzle diameter (4 $D_i$) to guarantee their location within the developed region, following the methodology outlined by Ervine and Falvey [15]. As seen in Figures 2e and 2f, $\phi$ for cloud formed by 8 mm jet is 1.5 times that formed by 2.7mm. This is mainly due to larger length scale of perturbation on 8 mm jet than that on 2.7 mm jet, which is responsible for entraining more air into the bulk. Also, increasing the jet diameter increases the length scale of perturbation, which results in the larger peak bubble void fraction [14] of Figure 2f. We interpret the smaller bubble cloud observed for figure 2d for a constant impact momentum as being caused by this larger void fraction.

## 3.2 Prediction of bubble cloud depth

As previously mentioned, Guyot et al. [11] proposed a method to predict the depth of the bubble cloud, $H$, assuming a uniform void fraction and a constant cone angle $\gamma$, based on single-phase turbulent jets. However, in biphasic jets beneath a free surface, the void fraction rather follows a Gaussian profile characterized by a maximum void fraction $\phi_0(z)$ along the axis and a variance $\sigma(z)$. The buoyancy force on a slice $dz$ of the bubble cloud at depth $z$ is given by $dF_b = \phi_0(z) \rho g \pi \sigma(z)^2 dz$. To integrate this buoyancy force over $z$, we consider $\phi_0(z)$ to be equal to its value averaged along $z$, $\bar{\phi}_0$, as the peak void fraction $\phi_0(z)$ shows little variation with depth (Fig. 2).

By maintaining the conical jet assumption, we set $\sigma(z) = D_i/2 + z \tan \gamma_0$, where the cone angle $\gamma_0$ is calculated as $\tan \gamma_0 = (\sigma(\bar{Z}) - D_i/2)/\bar{Z}$ at a chosen reference depth $\bar{Z} = 0.8 H$. This depth is selected to avoid rising bubbles and the sharp decrease in bubble void fraction near the bottom of the cloud if $\bar{Z} = H$ was considered. Consequently, the net buoyancy force $F_b$ on the bubble cloud with depth $H_b$ can be expressed as:

$$F_b = \bar{\phi}_0 \rho g \pi \left( \frac{1}{3} H_b^3 \tan^2 \gamma_0 + \frac{D_i}{2} H_b^2 \tan \gamma_0 + \frac{D_i^2}{4} H_b \right) \quad (3)$$

This expression is equivalent to the volume term in Eq. (2) if $\bar{\phi} = \bar{\phi}_0$ and $\gamma = \gamma_0$. The interpretation here is that the buoyancy force acting on a conical biphasic jet with a Gaussian void fraction profile is similar to the buoyancy force experienced by an equivalent biphasic jet with a constant void fraction $\bar{\phi}$ matching the maximum void fraction $\bar{\phi}_0$ of the Gaussian profile, and a width defined by the variance of the Gaussian profile at the selected reference depth [14].

The buoyancy force $F_b$ can be introduced into the momentum balance (Eq. (2)) to determine $H_b$. In Figure 3, the experimental depths $H$ from the current study are compared with the predicted depths $H_b$ for various nozzle sizes and jet fall heights. The model demonstrates good agreement with measurements for both inertia-controlled bubble clouds ($D_n = 2.7$ mm) and buoyancy-controlled ones ($D_n = 8$ mm and 10 mm). Additionally, data from two previous investigations by van de Donk [12] and McKeogh and Ervine [3], which provided both $H$ measurements and radial profiles of $\phi$ for jets with diameters of $D_n = 6$ mm and 9 mm, are included in Figure 3. The calculated $H_b$ va-
<figure><img src="image_4.png" /><figcaption>Figure 3. Comparing experimental depth *H* to predicted depth *H*<sub>b</sub> [14].</figcaption></figure>

lues based on their measurements exhibit strong
agreement with the model.

3.3 Multi-jet configuration

3.3.1 Bubble cloud depth

In the case of cascades, the jet is broken into several
segments. This scenario can be modelled by assuming
identical plunging jets in close vicinity of each other.
Figure 4 shows the liquid jet before impact and the
corresponding bubble cloud formed by single,
double, and triple jets. The distance between the jets,
*X* is such that *X/D*<sub>n</sub> = 5.

In the case of triple jets, the arrangement corresponds
to an equilateral triangle. X/D_n is small enough to
allow the interaction of the bubble clouds leading to
increase in H as the number of jets is increased, as
evidenced in Figure 4 for a nozzle velocity of 5.8 m/s,
jet diameter D_n = 2.7 mm and height of fall 20 D_n. The
observed elevation in H may be attributed to an
increased centreline velocity of the liquid, akin to the
similar phenomenon observed in the single-phase
configuration [16].

<figure><img src="image_5.png" /><figcaption>Figure 4. (a) Single, (b) double, and (c) triple bubble clouds arranged in equilateral triangular configuration. Nozzle velocity 5.8 m/s, jet diameter Dn = 2.7 mm.</figcaption></figure>

3.3.2 Radial void fraction measurement

The radial profiles of φ are measured along three
directions (0°, 30°, and 60°) intersecting through the
centroid, exhibiting a recurring pattern within the
cross-sections of three distinct bubble clouds (Fig.
5a). As shown in Figure 5b at low values of z/H, the
φ profiles are different in each direction, but with an
increase in z/H the profiles collapse on each other
(Fig. 5d). Their common profile is, as in the single jet
case of Figure 2, well predicted by a Gaussian fit. This
shows that the bubble cloud starts behaving like a
single bubble cloud at greater depth and a similar
analysis can therefore be performed to model its
depth.

<figure><img src="image_6.png" /><figcaption>Figure 5 (a) Cross-sectional area of three-jet bubble cloud illustrating the direction in which φ is measured, and (b-d) φ radial measurement at different values of z/H.</figcaption></figure>
### 3.3.3 Prediction of H formed by three plunging jets

The momentum balance equation (Eq. (2)) can be rewritten by considering the input momentum flux by three jets for the first term, the area out through which the outgoing momentum flux leaves the cloud ($S_{out}$) for the second term, and the union of the volume of the entire bubble cloud (V) for the buoyancy force, $F_b = \bar{\phi} \rho g V$. Thus Eq. (2) is converted for the three plunging jets case into:

$$3\rho V_i V_n \frac{\pi D_n^2}{4} = (1 - \bar{\phi}) \rho U_t^2 S_{out} + \bar{\phi} \rho g V \quad (4)$$

where $\bar{\phi}$ is the mean of maximum $\phi$ in three direction, measured at $z/H = 0.8$. Figure 6 shows that the experimental values and the model show a good match for different values of impact momentum, similar to the case of a single jet in Figure 3.

<figure><img src="image_7.png" /><figcaption>Figure 6. Comparison of experimental depth H for the bubble cloud created by three jets, against values predicted by the model $H_b$.</figcaption></figure>

## 4. Conclusion

Plunging jet experiments have been conducted which show two regimes of bubble cloud can be identified, inertia-dominated and buoyancy-dominated. Using an optical probe, void fraction profiles were measured in the developed region of the cloud for different nozzle diameters $D_n$ 2.7, 8, and 10 mm. The radial profiles can be accurately represented by Gaussian distributions. When these measurements are integrated into a momentum balance model to calculate the buoyancy force, the model predicts the experimental cloud depths remarkably well for both current investigations and prior studies. Similar experiments have been carried out for the bubble cloud generated by two and three plunging jets. The model similarly shows a very good prediction for

the multi plunging jets case. We plan to address in future works the role of the distance X between these jets, as well as configurations with a larger number of impacting jets.

## References

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[2] Lin T.J., Donnelly H.G. Gas bubble entrainment by plunging laminar liquid jets, AIChE J., 12, 563-71 (1966).

[3] McKeogh E.J., Ervine D.A. Air entrainment rate and diffusion pattern of plunging liquid jets, Chem. Eng. Sci., 36, 1161-72 (1981).

[4] Guyot G., Cartellier A., Matas J.P. Depth of penetration of bubbles entrained by an oscillated plunging water jet, Chem. Eng. Sci., X 2, 100017 (2019).

[5] Chanson H. The Hydraulics of Stepped Chutes and Spillways, CRC Press, Boca Raton (2002).

[6] Kiger K.T., Duncan J.H. Air-entrainment mechanisms in plunging jets and breaking waves Ann. Rev. Fluid Mech., 44, 563-96 (2012).

[7] Bin A.K. Gas entrainment by plunging liquid jets. Chem. Eng. Sci., 48, 3585-630 (1993).

[8] Clanet C., Lasheras J.C. Depth of penetration of bubbles entrained by a plunging water jet, Phys. Fluids, 9, 1864-6 (1997).

[9] Harby K., Chiva S., Muñoz-Cobo J.L. An experimental study on bubble entrainment and flow characteristics of vertical plunging water jets, Exp. Therm. Fluid Sci., 57, 207-20 (2014).

[10] Kramer M., Wieprecht S., Terheiden K. Penetration depth of plunging liquid jets - A data driven modelling approach, Exp. Therm. Fluid Sci., 76, 109-17 (2016).

[11] Guyot G., Cartellier A., Matas J.P. Penetration depth of a plunging jet: from microjets to cascades Phys. Rev. Lett., 124, 194503 (2020).

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[14] Dev N., John Soundar Jerome J., Scolan H., Matas J.P. Liquid inertia versus bubble cloud buoyancy in circular plunging jet experiments J. Fluid Mech., 978, A23 (2024).

[15] Ervine D., Falvey H. Behaviour of turbulent water jets in the atmosphere and in plunge pools, Proc. Instit. Civil Eng, 83, 295-314 (1987).

[16] Lin Y.F., Sheu M.J. Interaction of parallel turbulent plane jets, AIAA J., 29, 1372-3 (1991).
[Version abrégée en français]

Impact du taux de vide sur le nuage de bulles généré par un jet seul ou par un ensemble de jets

L'impact d'un jet plongeant sur la surface d'un liquide induit, au-delà d'une vitesse critique, le transport du gaz ambiant dans le liquide, donnant lieu à la formation d'un nuage de bulles [1, 2]. Ce phénomène est courant dans un large éventail d'applications industrielles, en présence notamment d'agitation chimique [3], dans les processus hydroélectriques [4], ainsi que dans des phénomènes naturels tels que les vagues déferlantes et les cascades [5, 6]. Une fois que les conditions d'entraînement des bulles sont réunies, la profondeur de pénétration maximale du nuage de bulles, désignée par *H* dans la Figure 1a, devient un paramètre essentiel à modéliser pour comprendre l'impact du nuage sur son environnement [8].

L'écoulement diphasique dans le nuage de bulles peut être modélisé en utilisant la conservation du flux de quantité de mouvement à l'impact et au bas du nuage [8] et la profondeur modélisée $H_i$ est alors donnée par l'équation (1). La profondeur $H_i$ est directement proportionnelle au produit $V_i D_i$, ce qui ressort clairement des données expérimentales pour les micro-jets (symboles verts ouverts sur la Figure 1c), mais ne correspond pas à ce qui est observé pour les jets millimétriques de taille ou de vitesse supérieure (symboles verts fermés sur la Figure 1c). Les nuages de bulles formés par les jets millimétriques sont à l'échelle $V_i D_i^{2/3}$. Guyot et al. [11] ont inclus la force de flottabilité dans le bilan de quantité de mouvement, obtenant l'équation (2) qui peut être résolue numériquement si l'angle du jet $(\gamma)$ et le taux de vide $(\phi)$ sont connus. Guyot et al. [11] ont montré que ce modèle de flottabilité fonctionne bien en considérant une seule valeur de $\phi = 15\%$, mais qu'il existe une forte dispersion à $V_i D_i$ plus élevé. En effet, $\phi$ dépend de la morphologie du jet et des conditions d'entrée [1, 3, 13]. Le premier objectif de cet article est de quantifier la robustesse de leur modèle à l'aide de mesures du taux de vide $\phi$. Des expériences à l'échelle du laboratoire sont réalisées à l'aide du dispositif illustré à la Figure 1b. Les images du jet et du nuage de bulles sont prises à partir d'une caméra à grande vitesse et d'un appareil photo numérique. Le taux de vide est mesuré à l'aide d'une sonde optique.

Les profils radiaux de $\phi$ montrent que, pour une quantité de mouvement d'impact constante et une hauteur de chute constante adimensionnée $H_f/D_n$ constante, plus d'air est entraîné pour un jet plus grand, conduisant à un nuage de bulles plus petit (Fig. 2). Le taux de vide supérieur sur le jet de plus grand diamètre est causé par l'échelle de longueur de rugosité plus grande que pour un jet plus petit, qui emprisonne l'air dans ces cavités. De plus, la Figure 2 montre que les profils radiaux de taux de vide $\phi$ sont bien ajustés par des gaussiennes et que la valeur maximale du taux de vide mesurée sur l'axe dépend peu de la profondeur. En utilisant les mesures de $\phi$, la force de flottabilité est calculée par intégration à l'aide de l'équation (3) et injectée dans le bilan de quantité de mouvement (Eq. (2)). Une bonne correspondance est observée entre la profondeur prédite par le modèle $H_b$ et la valeur expérimentale $H$ [14] (Fig. 3).

Dans les applications à grande échelle, les jets plongeants frappent l'interface liquide en étant brisés, comme un ensemble de jets à petite échelle. Afin de reproduire cette situation de façon synthétique, nous étudions un ensemble de jets plongeants. L'ajout d'un jet identique à proximité d'un jet plongeant existant conduit à une augmentation de $H$ (Fig. 4). Ceci s'explique par le fait que la vitesse de l'axe central est augmentée par l'ajout de jets, comme observé précédemment par Lin et Sheu [16] dans le cas monophasique. Le phénomène est amplifié lorsque trois jets plongeants proches génèrent un nuage de bulles. Le profil radial de $\phi$ pour les nuages de bulles triples montre que les profils de taux de vide se superposent à $z/H$ élevé (Fig. 5) et peuvent être ajustés à l'aide d'une gaussienne. Ainsi, la force de flottabilité peut à nouveau être calculée en utilisant l'équation (3) et $H_b$ peut être calculée, ce qui conduit à l'équation (4). La Figure 6 montre une bonne correspondance entre le modèle et les valeurs expérimentales.