# Liquid inertia versus bubble cloud buoyancy in circular plunging jet
experiments

Narendra Dev  Univ Lyon, Univ Claude Bernard Lyon 1, CNRS, Ecole Centrale de
Lyon, INSA Lyon, LMFA, UMR5509, 69622 Villeurbanne France  J. John Soundar
Jerome  Univ Lyon, Univ Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon,
INSA Lyon, LMFA, UMR5509, 69622 Villeurbanne France  Hélène Scolan  Univ Lyon,
Univ Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, LMFA,
UMR5509, 69622 Villeurbanne France  Jean-Philippe Matas  Univ Lyon, Univ
Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, LMFA, UMR5509,
69622 Villeurbanne France

###### 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 systematically measuring 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 the 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
different liquid above a critical velocity transports ambient gas into the
continuous liquid phase forming a cluster of bubbles, a bubble cloud (Lin and
Donnelly, 1966, Bonetto et al., 1994, Zhu et al., 2000, Lorenceau et al.,
2004). This phenomenon is widely encountered in industrial applications like
the stirring of chemicals (McKeogh and Ervine, 1981) or hydroelectric
applications (Guyot, 2019), and in nature, as in breaking waves and cascades
(Chanson, 2002, Kiger and 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 and (McKeogh and Ervine, 1981, Sene, 1988, El Hammoumi et al., 2002),
planar jets (Cummings and Chanson, 1999, Bertola et al., 2018) , and multi-
droplet streams (Speirs et al., 2018). Kiger and Duncan (2012) reviewed the
air-entrainment mechanism for laminar, turbulent and disintegrated jets of
different viscosity. Once the bubble entrainment conditions are reached, the
maximum penetration depth of the bubble cloud, denoted as $H$ in figure 1(a),
is an essential parameter to model in various applications (Clanet and
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 riverbed.
As can be inferred from Table 1, previous studies have addressed the bubble
cloud formation over a wide range of scales (Biń, 1993, Clanet and Lasheras,
1997, Chirichella et al., 2002, Roy et al., 2013, Qu et al., 2013, Harby et
al., 2014, Kramer et al., 2016, Miwa et al., 2018, Guyot et al., 2020).
Evidence for strong scale effects exist not only regarding bubble cloud size
but also for bubble count rate or void fraction (Chanson et al., 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 only been recently elaborated.

Figure 1: (a) Schematic of control volume of the bubble cloud, and (b) Schematic of the experimental setup illustrating flow lines to generate bubble cloud, back-lighting imaging setup and void fraction measurement setup with optical probes. Author(s) (year) | Jet and Nozzle | $D_{n}$ | $V_{n}$ | $Z_{f}$ | Re $={\rho V_{n}D_{n}}/{\mu}$
---|---|---|---|---|---
| geometry | (mm) | (m s-1) | (cm) | (max)
Lin and Donnelly (1966) |  $\downarrow$ $\medcirc$ | 2-8 | 0.8-2 | 1-17 | $2.5\times 10^{3}$
Van de Sande and Smith (1973, 1976) |  $\swarrow$ $\medcirc$ | 2.85-10 | 3-25 | 10-40 | $1.12\times 10^{5}$
van de Donk (1981) |  $\downarrow$ $\medcirc$ | 4-30 | 4-10 | 20-85 | $2.8\times 10^{5}$
McKeogh and Ervine (1981) |  $\swarrow$ $\downarrow$ $\medcirc$ | 6-30 | 1-7 | 50-500 | $1.1\times 10^{5}$
Clanet and Lasheras (1997) |  $\downarrow$ $\medcirc$ | 0.24-2.16 | 1.9-18 | 0.7-4 | $1.4\times 10^{4}$
Cummings and Chanson (1999) |  $\downarrow$ | $3-10$ | 1.1-2 | 0.5-18 | $1.16\times 10^{4}$
Chanson and Manasseh (2003) |  $\downarrow$ $\medcirc$ | 14-25 | 0.5-5 | 5-20 | $1.6\times 10^{5}$
Chanson et al. (2004) |  $\downarrow$ $\medcirc$ | 6.8-25 | 1.7-4.4 | 3-10 | $1.23\times 10^{5}$
Harby et al. (2014) |  $\downarrow$ $\medcirc$ | 4-14 | 2.5-13 | 2.5-30 | $9\times 10^{4}$
Kramer et al. (2016) |  $\downarrow$ $\medcirc$ | 13-81.9 | 5-7 | 20-100 | $6.4\times 10^{5}$
Bertola et al. (2018) |  $\downarrow$ | $10.4-11.5$ | 2.4-7.4 | 10.4-12.7 | $9.4\times 10^{4}$
Guyot et al. (2019) |  $\rightleftarrows$ $\medcirc$ | 0.3-2.4 | 1.89-20 | 0.6-4.8 | $2.4\times 10^{4}$
Guyot et al. (2020) |  $\downarrow$ $\medcirc$ | 0.3-213 | 3-28 | 20-950 | $3\times 10^{6}$
Present study |  $\downarrow$ $\medcirc$ | 2.7-10 | 2.5-10 | 5-80 | $1\times 10^{5}$
Table 1: Flow conditions in previous works on vertical ($\downarrow$),
inclined ($\swarrow$), and oscillating ($\rightleftarrows$) plunging jets from
circular ($\medcirc$) and planar ( ) nozzles. All authors used water. Data in
Lin and Donnelly (1966) refers also to oil and glycol jets.

In this context, Clanet and Lasheras (1997) developed a jet momentum
conservation argument, similar to Suciu and Smigelschi (1976), but by
postulating that (i) the biphasic jet expands with a constant cone half-angle
$\gamma=12.5^{\circ}$, as it is well known in turbulent jets (Horn and Thring,
1956, Ervine and Falvey, 1987, L’vov et al., 2008), and (ii) bubbles would
escape the biphasic zone as soon as the local advection speed, due to 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 $H_{i}$:

$H_{i}=\frac{1}{2\tan\gamma}\left(\dfrac{V_{i}D_{i}}{U_{t}}\right),$ (1)

where $V_{i}$ is the impact velocity and $D_{i}$ the impact diameter. Note
that there is no adjustable parameter nor any liquid or gas physical
properties in equation (1), except for $U_{t}$ which is taken equal to $22$
cm/s for all air bubbles in water larger than $1$ mm in diameter (Maxworthy et
al., 1996). Eq. (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 and Lasheras (1997) obtain a
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 which contains only a small void fraction of air,
say $\phi=10$%, the outgoing momentum
$\dot{Q}_{o}\sim(1-\phi)\rho{U_{t}}^{2}\pi H^{2}\tan^{2}\gamma$ in Fig. 1(a)
is already comparable with the net buoyancy force $F_{b}\sim\phi\rho
g{\pi}H^{3}\tan^{2}\gamma/3$ acting on it. Indeed, the largest cloud in
experiments by Clanet and Lasheras (1997) is about $16$ cm deep, and equation
(1) over-predicts the cloud depth data in the previous literature (Biń, 1993,
see Fig. 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 Fig. 1(a) to obtain the following
equation for the modelled cloud depth $H_{b}$:

$\displaystyle\underbrace{\rho
V_{i}V_{n}\dfrac{{\pi}{D_{n}}^{2}}{4}}_{\text{Incoming momentum flux}}$
$\displaystyle=$
$\displaystyle\underbrace{\left(1-\bar{\phi}\right)\rho{U_{t}}^{2}{\pi}\left(H_{b}\tan\gamma+\dfrac{D_{i}}{2}\right)^{2}}_{\text{Outgoing
momentum flux}}$ $\displaystyle+\underbrace{\bar{\phi}\rho
g\pi\left(\dfrac{1}{3}H_{b}^{3}\tan^{2}\gamma+\dfrac{D_{i}}{2}H_{b}^{2}\tan\gamma+\dfrac{D_{i}^{2}}{4}H_{b}\right)}_{\text{Buoyancy
force}},$

where $D_{n}$ is the nozzle diameter, $V_{n}$ the velocity at the nozzle, and
$\bar{\phi}$ 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, $V_{n}D_{n}^{2}=(1-\bar{\phi})V_{i}D_{i}^{2}$.
For the limiting case of micro-jets and bubble cloud depth of $10$ cm or
lesser, the buoyancy force might become less important compared to the
outgoing momentum flux in Eq. (1), which then reduces to Eq. (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 Eq. (1)
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 and Ervine (1981).
Hence, based on physical arguments, Eq. (1) elucidates the two most common
empirical relations, namely, $H\propto V_{n}D_{n}$ and
$H\propto\left(V_{n}D_{n}\right)^{0.68}$, 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) admitted $\bar{\phi}=15$ % based on results from past
studies (McKeogh and Ervine, 1981, van de Donk, 1981), and obtained a
relatively good agreement with large-scale experiments. However, they observed
a significant dispersion for the cloud depth both in their data and previous
investigations when the jet diameter is large. Guyot et al. (2020) attributed
this dispersion to the variations in the void fraction $\bar{\phi}$ between
various experiments since it is expected to depend strongly on both the jet
geometry and dynamics (McKeogh and 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 on
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
on 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-hold up techniques, and local gas/liquid void
fraction $\phi$ 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 and 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 minima at the jet axis (van de Donk, 1981, Bonetto and
Lahey, 1993, Cummings and Chanson, 1997, Brattberg and Chanson, 1998, Chanson
and Manasseh, 2003, Ma et al., 2010). McKeogh and 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 cloud depth $H$ along with measurements of
local void fraction $\phi$ in the developed zone of the plume. Such data over
various impact momentum and jet fall-height is necessary to help better
understand 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 and fall-heights
$Z_{f}$. Thereby, our aim is two-fold: we examine not only the robustness and
the relevance of the momentum balance models (Suciu and Smigelschi, 1976,
Clanet and Lasheras, 1997, Guyot et al., 2020) by incorporating in-situ void
fraction data into Eqn. (1), but also its role on the maximum penetration
depth of a two-phase plume, by independently varying 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 $1.2$ m $\times$ $0.6$ m
$\times$ $0.6$ m glass tank which 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 Fig. 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. Nozzle velocity $V_{n}$ varies between 1.5 to 12 m/s and is
controlled using a flowmeter and ball valve. Impact velocity $V_{i}$ is
deduced using equation $V_{i}=\sqrt{V_{n}^{2}+2gZ_{f}}$. The present work only
considers the cases where the jet is not broken into droplets before impact.

Backlight imaging is used to film the jet just before the impact
bisilluminated by LED panel using a CMOS fast camera from Ximea (CB262MG) at
300 FPS and 20 $\mu$s shutter speed at 2496 × 2418 resolution. Images of the
biphasic bubble cloud are filmed using a digital camera (Sony a7 III) and a
zoom lens (Sony FE 24-70mm F/4 ZA OSS Carl Zeiss, focal length kept at 35 mm)
at 50 FPS and bis200 $\mu$s exposure at a resolution of 1920 × 1080 pixels.
Using a MATLAB-integrated calibration application, lens distortion was found
negligible and a constant spatial resolution of 54 and 333 $\mu$m/pixel was
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 et al., 1985, Tsai,
1985), along with an 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 jet are extracted from images using machine-learning-based
object detection models, namely, Grounding DINO (Liu et al., 2023) and Segment
Anything Model (SAM) (Kirillov et al., 2023). The detected jet edges are then
analysed using an in-house MATLAB code to obtain the impact diameter ($D_{i}$)
and the roughness ($\epsilon$). The latter is defined as the average of the
RMS 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 fiber probe (A2
Photonics Sensors) of diameter 125 $\mu m$ mounted upright on a slender plate
(thickness = 4 mm), as shown in the magnified view in Fig. 1(b). This
horizontal plate is fixed to a vertical cylindrical rod (diameter = 1.5 cm).
The probe tip is located at 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 plane ($r$-$z$ plane) using two motorized linear stages (Igus)
having an accuracy of $0.01$ mm. The probe signals are sampled at $250$ kHz
for $60$ seconds, a duration which 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 centered at the jet axis and the
measurements are carried out in the radial direction of the cloud with a step
size of $2$ 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: Variation of $H$ with square root of impact momentum
$\sqrt{V_{i}V_{n}}D_{n}$ for $D_{n}$ = 2.7 and 8 mm at various $Z_{f}/D_{n}$.
The solid line shows the trend expected from Eq. (1). The dashed line shows
the 2/3 scaling law expected at large scales. The inset graph on the top-right
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 $H$. I - VI are the data
points corresponding to the flow conditions that are compared in upcoming
section.

Figure 2 illustrates the variation of $H$ for the $D_{n}=2.7$ mm and $D_{n}=8$
mm nozzles, as a function of $\sqrt{V_{i}V_{n}}D_{n}$, which is proportional
to the square root of the impact momentum flux, up to a factor $\rho$, as in
equation (1). 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 $\sqrt{V_{i}V_{n}}D_{n}$. This is consistent with the model proposed by
Clanet and Lasheras (1997) (solid line), equation (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), Eq. (1) accurately predicts the behaviour of $H$ up to
$\sqrt{V_{i}V_{n}}D_{n}\approx 0.02$ m2/s. Beyond this threshold, a transition
occurs, and $H$ scales as $(V_{i}V_{n})^{1/3}D_{n}^{2/3}$, 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 to
what equation (1) would predict. Also, Fig. 2 shows the coexistence of two
regimes in a narrow range $\sqrt{V_{i}V_{n}}D_{n}=0.02$ – $0.03$ m${}^{2}/$s,
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.

###  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 first discuss $\phi$ 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 air to water volume fraction at the jet axis,
$\phi(r=0,z)$, denoted here as $\phi_{0}$. 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 sharply decrease at the end of the cloud, as
previously reported by van de Donk (1981), McKeogh and Ervine (1981), Chanson
and Manasseh (2003), and Hoque and Aoki (2008). When $Z_{f}/D_{n}$ is
increased, a similar axial profile is observed, but $\phi_{0}$ is increased by
almost $26$ %.

The jet morphology just before impact is illustrated in Fig. 3(b) and (c) for
$Z_{f}/D_{n}=20$ and $Z_{f}/D_{n}=50$, and it shows a slightly larger
amplitude of the jet undulations for the larger $Z_{f}/D_{n}$ case. Using the
algorithm introduced in Section 2, we measure that the undulations on the jet
surface exhibit a roughness of $\epsilon=170$ $\mu$m for Figure 3b and
$\epsilon=225$ $\mu$m for Figure 3c. As discussed by McKeogh and 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 provide additional visualizations of
the jets of figure 3(b) and (c) respectively (frame rate of 300 images/s).
These two conditions are indicated by labels I and II in Fig. 2. They are both
situated in the inertia-dominated regime, where buoyancy effects are still
negligible. As a result, the effect of void fraction on $H$ is negligible,
even though $\phi$ increases.

Figure 3: (a) Axial evolution of void fraction $\phi_{0}(z)$ for two different
$Z_{f}/D_{n}$ at a fixed jet diameter $D_{n}=2.7$ mm and
$\sqrt{V_{i}V_{n}}D_{n}=0.016\pm 0.0002$ m2/s. Instantaneous images of the
freely-falling jet just before impact are given in (b) and (c).

####  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 they are in the
developed region (Ervine and Falvey, 1987). Fig. 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 larger $D_{n}=8$ mm, as shown in Fig. 4(c),
is significantly smaller (see also, labels III and IV in Fig. 2). The
visualizations of figure 4(a) and (c) are illustrated more extensively in
supplementary movies 3 to 6. The void fraction profiles for the 2.7 and 8 mm
injectors are shown in Fig. 4(b) and (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 when an air jet is injected into the pool
(Kobus, 1968, Freire et al., 2002). bisWhen our data is compared with fits
based on air/bubble diffusivity and air-to-water volume flux ratio, as in
Cummings and Chanson (1997), the best fit was found for the air-to-water
entrainment ratio of O(10)! This unrealistic value results perhaps from their
constant-diffusivity-assumption for advection-diffusion process in the two-
phase mixing layer. In our case, and as described by Clanet and Lasheras
(1997), the bubbles are convected by the large-scale eddies dominating the
evolution of the submerged jet.

Note that the maximum void fraction $\phi(r=0,z)$ is significantly smaller for
the thinner jet (up to $18$ % along the axis, Fig.4b) when compared with that
of the larger jet (up to 27 % in Fig. 4d). The impact of the nozzle diameter
on the maximum void fraction has already been evidenced 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 jet diameter $D_{n}$ is increased at a
constant $Z_{f}/D_{n}=20$ and constant impact momentum (Liu et al., 2022,
Sene, 1988). Measurements of $\epsilon$ show that the roughness goes from 225
$\mu$m for Fig. 4b to 445 $\mu$m for figure 4d. These points are indicated by
labels III and IV in Fig. 2. It is clear that III belongs to the momentum-
dominated regime, while IV belongs to the buoyancy-dominated regime.

Figure 4: (a) and (c) present instantaneous pictures of the falling jet just
before impact for $D_{n}=2.7$ and $8$ mm, respectively, at constant
$\sqrt{V_{n}V_{i}}D_{n}=0.028$ m2/s and $Z_{f}/D_{n}=20$. Corresponding bubble
clouds are shown as well. (b) and (d) illustrate 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. Figure 5: Images of $D_{n}$ = 8 mm jet just
before impact for (a) $Z_{f}/D_{n}$ = 20 and (c) 100, for a constant
$\sqrt{V_{n}V_{i}}D_{n}=0.075$ m2/s. Corresponding bubble clouds are shown as
well. (b) and (d) illustrate the radial variation of $\phi$ at various depths
in the bubble clouds for $Z_{f}/D_{n}$ = 20 and 100 respectively.

Figure 5 illustrates the influence of a change of 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
$Z_{f}/D_{n}=100$ (Fig. 5c) is more corrugated than that issued at
$Z_{f}/D_{n}=20$ (Fig. 5a), for the same $\sqrt{V_{i}V_{n}}D_{n}=0.075$ m2/s.
This is corroborated by the data of figure 6, which shows 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 $\sqrt{V_{i}V_{n}}D_{n}$ 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.

Figure 6: Variation of $\epsilon$ and $\bar{\phi}_{0}$ with
$\sqrt{V_{i}V_{n}}D_{n}$ 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. Figure 7: Measured profiles of air/water void fraction
for the nozzle diameter $D_{n}=8$ mm and different fall-height (a)
$Z_{f}/D_{n}=20$, (b) $Z_{f}/D_{n}=50$, and (c) $Z_{f}/D_{n}=100$ at various Q
= $\sqrt{V_{i}V_{n}}D_{n}$. Gaussian fits are represented by continuous lines.

The visualizations of figure 5(a) and (c) are illustrated more extensively in
supplementary movies 7 to 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 figures
5(b) and (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 as 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 zoom view of the change in $H$ at higher
$\sqrt{V_{n}V_{i}}D_{n}$, 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 provided in Guyot et al. (2020) for this regime at higher
impact momentum and larger diameter, is due to differences in jet fall height.

Finally, general trends of local void fraction profiles are depicted in figure
7 for the case of $D_{n}=8$ mm. Each figure presents data at different
vertical locations $z$ underneath the surface, normalized with respect to the
cloud depth $H$. In all cases shown here, data fits reasonably well with a
Gaussian profile
$\phi=\phi_{0}(z)\mbox{e}^{-\left(r^{2}/\sigma^{2}(z)\right)}$ whose peak and
width do not vary much as $z/H$ is changed. The first row in figure 7
corresponds to data for the shortest jet fall height to diameter ratio
$Z_{f}/D_{n}=20$ at various $\sqrt{V_{i}V_{n}}D_{n}$, a measure of the jet
impact momentum up to a factor of liquid density. As $\sqrt{V_{i}V_{n}}D_{n}$
increases the centreline peak void fraction $\phi_{0}$ slightly decreases
while the width of the best-fitted Gaussian profiles increases. On the other
hand, at a given $\sqrt{V_{i}V_{n}}D_{n}$, the peak void fraction $\phi_{0}$
is almost doubled when the jet’s fall height is quintupled. In the next
section, these observations on the air/water volume fraction in the biphasic
region are further developed to properly capture the effect of buoyancy on the
bubble cloud depth.

###  3.3 Prediction of bubble cloud depth

Guyot et al. (2020) proposed a prediction for the bubble cloud depth $H$ via
Eq. (1) by assuming a uniform void fraction within the bubble cloud and a
constant cone angle $\gamma$, based on single-phase turbulent jets. As
mentioned just before, the void fraction in the biphasic jet underneath the
free surface can be well-approximated by a Gaussian profile. These profiles
are characterised 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 $dz$ at a depth $z$ is
$dF_{b}=\phi_{0}(z)\rho g\pi\sigma(z)^{2}dz$. 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 to 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 and Smigelschi, 1976, McKeogh and Ervine, 1981, Clanet and
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 overestimating 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 $\tan\gamma_{0}=\left(\sigma(\tilde{z})-D_{i}/2\right)/\tilde{z}$
at a chosen reference depth $\tilde{z}=0.8H$. 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

$F_{b}=\bar{\phi}_{0}\rho
g{\pi}\left({\frac{1}{3}}H_{b}^{3}\tan^{2}\gamma_{0}+{\dfrac{D_{i}}{2}}H_{b}^{2}\tan\gamma_{0}+{\dfrac{D_{i}^{2}}{4}}H_{b}\right),$
(3)

which is identical to the volume term in Eq. (1), 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) 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.

Figure 8: (a) Comparison of experimental depth $H$ with those predicted by the
model $H_{b}$ and (b) Normalized depth $H/H_{i}$ at various $Fr$ for injectors
from current and past studies whose void fraction is known. Eq. (4) is
represented by the continuous line. Figure 9: (a) Bubble cloud depth ($H$)
data from previous investigations are presented here as a function of
$\sqrt{V_{i}V_{n}}D_{n}$, 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 void fraction is not known, $Fr^{*}$ can still sort past
experiments into inertia or buoyancy-dominated regimes, even though more
dispersion is observed.

Experimental depths $H$ from the present study and predictions $H_{b}$ are
compared in Fig. 8(a) for the different nozzle sizes and jet fall heights. A
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 of 50), $V_{n}=5.5$ m/s and $Z_{f}/D_{n}=50$
has been included in the data of this figure. For comparison, two previous
investigations, namely, those conducted by van de Donk (1981) and McKeogh and
Ervine (1981) are presented in Fig. 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 a very good consistency with the model.

In what follows, the question of the role of 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 buoyancy force, respectively second and third terms in Eq. (1). The square
root of the ratio of these terms defines a dimensionless grouping similar to a
Froude number, referred to as
$Fr=\sqrt{3\left(1-\bar{\phi}_{0}\right)/\bar{\phi}_{0}}U_{t}/\sqrt{gH_{b}}$.
In fact, when the nozzle diameter is small compared to the cloud width
$2H_{b}\tan\gamma_{0}$, the cubic polynomial for $H_{b}$ in Eq. (1) can be
rewritten in terms of $Fr$ as

$\dfrac{H_{b}}{H_{i}}=\dfrac{Fr}{\sqrt{1+Fr^{2}}}+\mathcal{O}\left(\dfrac{D_{i}}{2H_{b}\tan\gamma_{0}}Fr^{2}\right),$
(4)

where $H_{i}$ is the cloud depth when the buoyancy force is absent, as given
by Eq. (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 and 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 Fig. 8(a). Indeed, all experimental data points fall on a
single curve given by Eq. (4), for which $H/H_{i}$ falls steeply when $Fr$ is
decreased. For a constant velocity, using larger nozzles will lead to bigger
bubble clouds, and so larger $H$ and smaller $Fr$, thereby favouring the
buoyancy-dominated regime. Note also that the influence of jet velocity is
counter-intuitive 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 expression (4) is obtained from Eq.
(1) 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 which corresponded to the
inertia-dominated regime in Fig. 2. Furthermore, previous authors have
proposed to add a hemispherical dome to the truncated cone modelling the
bubble cloud, as in figure 1 (Clanet and Lasheras, 1997, Guyot et al., 2020).
This small correction would introduce a factor $1+\tan\gamma_{0}$ (of the
order of 10 %) to both the modelled heights $H_{i}$ and $H_{b}$, but 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 on Fig. 9(a) $H$ as a
function $\sqrt{V_{i}V_{n}}D_{n}$, square-root of the impact momentum over
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 limit 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 in two regimes: (i) One for larger $Fr^{*}>0.2$ for which $H/H_{i}$ is
close to one, 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 a 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 further ascertain that $Fr$ is a good parameter to monitor
the transition between the inertia-dominated and the buoyancy-dominated
regime.

###  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 section, 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 admitting 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 $\phi=\bar{\phi}_{0}(z)e^{-r^{2}/\sigma(z)^{2}}$. This simply
gives the relation $R_{b}(z)=\sigma(z)\sqrt{\log{5}}\simeq 1.27\sigma$.
Thereafter, in order to avoid over-estimation 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 $\tilde{R}_{b}\simeq
1.27\sigma(\tilde{z})$ is this cloud’s radius at the reference depth
$\tilde{z}=0.8H$. As already mentioned in section 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: (a) Definition of $\Phi_{b}$ and $R_{b}$. (b) Variation of
$\Phi_{b}$ and $\gamma_{b}$ with $Z_{f}/D_{n}$ for the two injector sizes at
various impact momentum. Two color bars are provided for the two ranges of
$\sqrt{V_{i}V_{n}}D_{n}$, corresponding to both $D_{n}$ values.

Figures 10(b) and (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 subsection 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 height $Z_{f}$ (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(c) and (e) illustrate the variations of the cone angle $\gamma_{b}$
of the equivalent constant-$\phi$ jet with regards to $Z_{f}/D_{n}$, for
different $\sqrt{V_{i}V_{n}}D_{n}$. 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 to $20$% 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 do not hold anymore 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
the buoyancy-dominated regimes. Void fraction profiles were measured in the
developed region of the cloud with novel optical probes for nozzle diameter,
$D_{n}$ = 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, it predicts the
experiments very well for both current and past studies. Even though the void
fraction measurements have only been carried out 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 $Fr$ built with 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 and 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 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 jet
diameter, the 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 the question future works will have to
address.

Acknowledgements. We acknowledge the technical support of G. Geniquet, S.
Martinez and A. Buridon on the experimental set-up. We also thank A2 Photonic
Sensors for the support with the optical probes. This research was funded by
the French Agence Nationale de la Recherche ANR under grant no. JETPLUME
ANR-21-CE05-0029.

Funding. \- This research was funded by the French Agence Nationale de la
Recherche ANR under grant no. JETPLUME ANR-21-CE05-0029.

Declaration of interest. The authors report no conflict of interest.

Author ORCIDs. Narendra Dev, 0009-0007-4626-3585; J. John Soundar Jerome,
0000-0003-2148-9434; H. Scolan, 0000-0003-2739-2297; J.-P. Matas,
0000-0003-0708-1619;

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