2021

[1]Simon Beelen

[1]Dominik Krug

[1]Physics of Fluids group, University of Twente, Drienerlolaan 5, Enschede,
7522NB, Netherlands

2]Department, Organization, Street, City, 10587, State, Country

# In-situ measurements of void fractions and bubble size distributions in
bubble curtains.

<EMAIL_ADDRESS>Martijn van Rijsbergen  Miloš Birvalski  Fedde Bloemhof
<EMAIL_ADDRESS>*  [

###### Abstract

We report the development of a novel measurement system designed to measure
bubble properties in bubble curtains (i.e. planar bubble plumes)
simultaneously with acoustical measurements. Our approach is based on
electrical, contact-based needle sensors in combination with an optical
system. The latter is used for calibration and validation purposes. Correcting
for the insensitive distance of the needle tips yields very good agreement
between the two approaches in terms of the local void fraction and bubble size
distributions. Finally, the system is employed to study bubble plumes evolving
from three different nozzle types. All cases display consistent self-similar
behaviour with spreading rates increasing with increasing gas flow. The
spreading is further found to be significantly higher when the plumes
originated from a porous hose compared to the two other configurations
featuring either discrete holes or nozzle elements.

###### keywords:

Bubble curtain, Bubble size distribution, void fraction, Electrical probe,
Image analysis

##  1 Introduction

Bubble curtains are widely employed across diverse engineering applications
such as controlling the movement of fish (Noatch and Suski 2012), limiting the
spreading of floating debris (Spaargaren 2018) or liquid spills (Lo 1997), or
to mitigate salt water intrusions (Abraham and van der Burgh 1962). Besides
these, the use of bubble curtains to reduce underwater noise pollution is of
particular interest, especially with the recent push to increase offshore
windfarm capacity. The pile driving during the installation of such windfarms
generates significant noise emissions. If unmitigated, these can harm marine
mammals within a radius of 100 m and still cause significant disturbance at
distances up to 50 km around a pile driving site (Bailey et al. 2010; Duarte
et al. 2021). The use of bubble curtains to reduce the noise impact during
construction is a proven concept already that is also frequently employed in
practice (Merck et al. 2014; Würsig et al. 2000; Tsouvalas 2020). However, the
operation of bubble curtains is expensive and costs can easily surpass
€100.000 per pile depending on parameters such as the depth of the water and
local currents among others (Strietman et al. 2018). There is hence
significant interest to achieve performance improvements to reduce these
expenses during construction. Another related aspect is the need for validated
models to ensure that the pile-driving operation reliably complies with the
relevant regulations, e.g. regarding the maximum permitted sound pressure
level in order to protect marine life

The interaction of the sound critically depends on parameters of the bubbles
such as their size, their distribution and the resulting variations in void
fraction (Commander and Prosperetti 1989). These parameters, in turn, can vary
drastically depending on local conditions, such as water depth and current,
but importantly may also change in response to variations in the water quality
(e.g. Zhang et al. 2014), e.g. the concentrations of dissolved gas,
surfactants and salt (Winkel et al. 2004; Craig et al. 1993; Firouzi et al.
2015). Within the bubble curtain these parameters can vary due to for example
coalescence and break up (Camarasa et al. 1999) and due to the pressure drop
the bubbles experience when rising (Zheng et al. 2010). Experimental
investigations aimed at a better understanding of the relevant physics of the
sound-bubble interaction therefore ideally combine both acoustic and
hydrodynamic measurements simultaneously in the same setup and location. The
practical challenge in doing so lies in the fact that the emitted noise of
pile driving typically peaks between $100$–$500\,\text{Hz}$ (e.g. Bailey et
al. 2010; Matuschek and Betke 2009). As a consequence, the relevant wave
lengths are in the order of 10–3m, which necessitates the use of large basins
to avoid spurious confinement effects on the acoustics. Given these
difficulties, the acoustic insertion loss of bubble curtains is commonly
measured outdoors in lakes or the sea, where a detailed bubble
characterisation of the bubble field is either lacking entirely (Dähne et al.
2017; Würsig et al. 2000; Stein et al. 2015; Lucke et al. 2011) or
complemented from a separate laboratory test (e.g. Rustemeier et al. 2012). In
an effort to push beyond these limitations, this present paper presents the
development, calibration and validation of a novel measurement setup to
characterize bubble curtains with good accuracy and at large scale up to 1.8m
in width. Crucially, the new device can be operated in situ to characterize
the bubble curtain alongside acoustic tests.

There exists a multitude of different approaches to measure bubble properties
(see the overviews by Boyer et al. (2002) and Mudde (2010)). At the most
general level, these can be classified into intrusive and non-intrusive
measurement techniques. Non-intrusive measurement techniques based on image
analysis are typically used in configurations where the camera can be placed
outside the flow domain (e.g. Ferreira et al. 2012; Besagni and Inzoli 2016;
Wang et al. 2018) and require sophisticated algorithms to deal with
overlapping bubbles (e.g. Lau et al. 2013; Zou et al. 2021; De Langlard et al.
2018). Severe limitations in terms of void fraction and/or depth of field
remain however, such that using an underwater cameras outside of the bubble
curtain is not a feasible option for large bubble curtains. Image analysis can
be used semi-intrusively and locally if a small camera unit is placed within
the bubble curtain. Intrusive methods such as optical fibre probes (e.g.
Luther et al. 2004; Pjontek et al. 2014; Magaud et al. 2001; Enrique Julia et
al. 2005), hot film anometry (e.g. Rensen et al. 2005; Wang and Ching 2001)
and electrical probes (e.g. Steinemann and Buchholz 1984; Muñoz-Cobo et al.
2017; Huang et al. 2018; Revankar and Ishii 1993; Tompkins et al. 2018; Tyagi
and Buwa 2017) are generally capable of handling higher void fractions. In
particular, electrical probes have been employed in bubble curtains before,
e.g. by Chmelnizkij et al. (2016) who performed measurements in a tank with a
diameter of 5 $m$ and a depth of 4.8 $m$. A total of 16 probes were spaced
across the width of to the bubble curtain at distances of 20–240 $mm$
depending on the height above the nozzles. At 14 out of 16 sensor locations, a
second sensor was placed at a short vertical distance (6.4 $mm$) to provide
estimates on bubble rise velocities and sizes via the contact times at
individual sensors and the delay of the bubble hits between them. Such a two
sensor arrangement is attractive and simple in principle. However, in practice
a calibration is required to account for interactions between bubble and
sensor. And even then interpretation at the level of a single bubble remains
very difficult since the measured chord length will vary even for the same
bubble depending on if it is pierced closer to its edge or its center. Besagni
et al. (2016) proposed a clever calibration method for dual optical fibre
probes that focuses on determining the statistics of the bubble size
distribution from the measured chord length distribution. They use image
analysis to establish a relation between the aspect ratio of bubbles as a
function of their equivalent diameter, which serves as an input to the
calibration. The procedure then consists of updating an estimated (log-normal)
bubble size distribution iteratively until the predicted chord length
distribution matches the experimentally obtained one. In a similar fashion,
other methods reported in literature such as the max entropy method have been
developed to transform the chord length distribution to a bubble size
distribution (e.g. Tyagi and Buwa 2017; Santana et al. 2006).

In the following, we will describe the hardware aspects of our measurement
device and the test rig in section 2 and provide details on the methods for
calibration and data analysis in section 3. This will be followed by
validation and initial measurement results in section 4 and finally our
conclusions (section 5).

##  2 Experimental setup

###  2.1 General considerations

As mentioned previously, typical void fractions (in the percent range) and
lateral dimensions (on the order of meters) do not lend themselves to the
application of optical techniques if the camera is to be placed outside the
bubble plume. Immersing the optical system inside the bubbly flow region can
mitigate some of the issues. However, doing so introduces disturbances to the
flow and can only capture a small region of the flow per camera unit. In order
to capture the bubble distribution across the full height and width of the
bubble curtain, conductivity based electrical sensors are therefore the most
appealing choice. Even though each individual sensor only represents a point
measurement, their low cost and simple operation renders combining many of
them into a larger array to achieve a high spatial resolution. For our
purpose, we opted to combine the electrical sensors with an underwater camera
that simultaneously captures a part of the bubble field in the center of the
curtain. Doing so has three main benefits: (i) The images provide a way to
independently verify the results from the electrical sensors – seeing is
believing. (ii) The optical measurements allow us to obtain the input required
for the sensor calibrations under exactly the same conditions. This greatly
improves the robustness and accuracy of the method. (iii) The camera
recordings provide a means to gauge to what extent very small bubbles, which
may not be picked up by the electrical sensors, are of relevance in a
particular configuration. This is an important aspect in interpreting the
sensor results, that would otherwise be unaccounted for.

###  2.2 Electrical probes

####  2.2.1 Probe design and operation

The electrical probes are made of coated stainless steel acupuncture needles.
The acupuncture needles have a diameter of $D_{n}=0.12\,\text{mm}$ and length
$L_{n}=40\,\text{mm}$. In order to insulate the needle shaft, a black coating
is applied by dipping the needle in paint which also acts as a metal primer.
The thickness of the coating ($\sim 20\,\mu\text{m}$ ) is controlled by the
speed at which the needle is lifted out of the bath. Finally, approximately 1
mm of the coating covering the tip of the needle is scraped off to expose this
part (see blowup in figure 1a). The needle probes are then soldered onto a 300
mm wide printed circuit board (PCB). The PCB serves to keep the probes in
place but also compactly hosts the required circuitry. Afterwards the PCB is
fully coated in transparent epoxy to render it waterproof. Figure 1a shows a
single PCB, which holds a total of 40 needles spaced by 8.5mm. Most locations
contain only a single needle (labelled ’A’ in figure 1a), but every 10th
position is equipped with a vertically staggered double needle (’B’) to also
measure the bubble velocity.

Figure 1: a) One of the PCB’s used for the measurements consisting of single
needles (A) and double needles (B). b) Overview of multiple available PCB’s

A potential (typically 3V) is applied between the needles and a common
electrode in the water and each electrical probe is connected to a comparator
circuit. Gas encapsulating the tip of the probe increases the resistance
between the tip of the probe and the common electrode resulting in a switch of
the output signal of the comparator. It is most straightforward to drive the
circuits in direct current (DC) mode and this is used for the results
presented here. We noticed, however, that this leads to the build up of
tarnish on the tip of the probe, which significantly hampers operation and
eventually results in an isolated tip. It is possible to restore functionality
by cleaning between experiments and this has been done for the results
reported here. Yet, a more elegant and sustainable solution is, to operate the
circuits in alternating current (AC) mode. Doing so proved to effectively
minimize the impact of tarnish on the measurement results. A diagram of a
circuit suitable for both DC as well as for AC operation (with an alternating
square wave) is shown in appendix A.

The complete system allows for simultaneous operation of up to 240 needles
across 6 PCB’s covering a range of 1.8 m in total (see Figure 1b). Each sensor
is read out with 10kHz in order to resolve a relevant timescale of about 1ms
(based on a bubble velocity of 1$ms^{-1}$ and a size of 1 mm). To achieve the
data transfer, the binary signal of each comparator circuit is serialized on
the PCB, such that the outputs of 40 needles are sent consecutively to one
common circuit above water, that gathers the data of 6 PCBs and combines the
bits of each board into 40 bytes per sample.

####  2.2.2 Characterisation of the probe tips

The stripped part of the tip has a finite extent $L\approx 1\,\text{mm}$ and
there is therefore some uncertainty to the precise position of the gas-water
interface relative to the needle, at which the signal will switch going in and
out of the bubble. In order to characterise this accurately, tests have been
performed simulating the piercing process of a bubble by moving the needle
through a water surface (see figure 2). The needles were moved using a micro
stage at a velocity of $V_{m}=50\,\mu\text{ms}^{-1}$ and the time at which the
tip hit the surface was determined optically.

Figure 2: A schematic of the DC and AC signal of the needle probes for a) Air
to water and b) water to air

Results from these tests for both the DC actuation and the AC actuation are
shown in figure 2 along with the filtering of the AC signal. The probe signal
is plotted vs the tip location $s$, with $s=0$ corresponding to the water
surface. From these data, it becomes clear that the probe detects water,
corresponding to ”0” signal, as soon as only a small part of the tip is
exposed to water. Concretely, this is reflected in the ‘switching distances’
$A$ from the probe tip being short for the air to water transition in figure
2a, whereas the counterpart $B$ for the reverse direction is comparable to
$L$. In the ideal case, the switching position would be the same for both
directions, i.e. $A=B$. We can define a ‘lag distance’ based on the difference
$\Delta=|A-B|$. A non-zero $\Delta$ implies that bubbles smaller than this
value cannot be detected. Practically, this turns out to be of little
relevance as from our experience such small bubbles tend to not get pierced in
the first place. However, the lag also results in an underestimation of the
bubble contact time and hence the void fraction, which needs to be accounted
for. We found that the values of $\Delta$ from the present quasi-static tests
were significantly affected by the menisci forming at the needle and results
will likely not transfer to the dynamic bubble interaction with curved
surfaces. We therefore decided to determine the precise value of $\Delta$ as
part of a calibration procedure. Obviously, reducing the stripped length $L$
would also help mitigate this issue, but we found that decreasing the exposed
surface area significantly increased the problems with tarnishing during
operation.

The double probes (see Figure 1a group B) are used to determine the bubble
rise velocity and the chord length of the pierced bubble. The rise velocity
can be found by $V_{b}=\frac{h}{t_{2}-t_{1}}$ where $h$ is the difference in
height between the two probes (see Figure 3a) and $t_{1}$ and $t_{2}$ are the
different arrival times of the bubble on the two needles. The chord length is
determined by $C_{b}=V_{b}T_{con}$ where $T_{con}$ is the contact time of the
bubble. For an accurate measurement $h$ is a vital parameter for which we can
obtain an experimental estimate by slowly moving the double probes out of the
water (see figure 2b). From the time difference in the signals and the known
velocity $V_{m}$ we found $h=0.85\,mm$, which is in good agreement with values
obtained from imaging the probe tips.

Figure 3: a) Electrical probes, I a bubble approaches a set of double needles
both needles are in full contact with water (signal 0 0), II Bubble
encapsulates the tip of the longer needle (signal 1 0), III bubble left longer
needle and encapsulates the shorter needle (signal 0 1). b) Optical system. c)
Experimental setup with the bubble curtain hose, air supply, optical system
and electrical probes

###  2.3 Optical system

The optical system is schemetically shown in Figure 3b and consists of a Magma
G-235 camera with a KOWA 16 mm lens in a watertight casing. The framerate of
the camera is set to $\sim 30\,Hz$ and the resolution is $1920\times
1200\,\text{pix}^{2}$. The bubbles pass through the measurement volume which
is enclosed by 2 transparent plexiglass plates leading to a measurement volume
of $V_{meas}=95.0\times 59.7\times 49.5\,\text{mm}^{3}$. The measurements have
been calibrated using a transparent ruler which has been held against the
front and back plate of the volume. Backlight illumination is provided through
a costum made LED plate that is placed behind a diffusor plate. In order to
keep the optical path free of bubbles, the distance between the camera and the
measurement volume is shielded by a screening pipe, which has a narrow opening
at the top to let entrapped air escape. The optical system is combined with a
single PCB located above the measurement volume spanning half the measurement
volume (see Figure 3b) to enable direct comparison of the results. The entire
arrangement is then placed in the center of the plume such that the optical
measurement domain aligns with the hose generating the bubble screen as shown
in Figure 3c.

###  2.4 Bubble curtain test facility

Experiments with a bubble curtain have been carried out in the so-called
concept basin (CB) at MARIN (Maritime Research Institute Netherlands). The CB
measures LxWxD: 220x4x3.6 $\text{m}^{3}$. A 4 m long aerator hose connected to
the air supply was placed over the width of the tank on the bottom. In most
experiments a ‘Japanese’-type aerator hose with porous walls was used.
Additional test were performed with a PVC pipe fitted with nozzles (Festo
UC-1/8 silencers) every 100 mm and a conventional PVC pipe with holes of 1 mm
in diameter drilled every 50 mm (see Appendix B for images). Pressurised air
was supplied by a compressor coupled to a Festo MS6-LFM-1 filter at flow rates
in the range of $0.55-1.67Lm^{-1}s^{-1}$, which were measured using a thermal
mass flow sensor (Bronkhorst F-203AC). The entire measurement system was
attached to a platform that could be moved vertically enabling measurements
between 0.2 and 3.6 m above the bottom of the tank. An overview of the entire
setup is given in Figure 3c. The figure also introduces the coordinate system
used, where the height $z$ is measured from the top of the hose, $y$ runs
along the hose and $x$ indicates the transversal direction of the bubble
curtain.

##  3 Post processing and analysis

###  3.1 Electrical probes

####  3.1.1 Void fraction

The output signal of an individual probe, $S_{i}$, is either 0 (water) or 1
(air). This means that the most straight forward estimate of the void fraction
of probe $i$ is the average of the signal in time

$\overline{\alpha}^{*}_{n,i}=\frac{1}{T}\int_{T}S_{i}(t)dt,$ (1)

where $T$ is the averaging period. Subscript $n$ is used to indicate results
from the needle measurements and the overline denotes time averaging
throughout the paper.

Due to the finite lag distance $\Delta$, Eq. 1 will result in a slight
underestimation of the actual void fraction. A correction can be made based on
the number of bubbles $N_{T}$ that hit the probe during $T$, since for every
pierced bubble a small part is not measured. Provided that $\Delta$ and the
average bubble rise velocity $V_{r}$ are known, the total void fraction,
$\bar{\alpha}_{i}$ is given by

$\overline{\alpha}_{n,i}=\frac{1}{T}\left(\int_{T}S_{i}(t)dt+N_{T}\frac{\Delta}{V_{r}}\right).$
(2)

which represents our measurement of the local void fraction at the location of
probe $i$. As a caveat, it should be noted again that Eq. 2 does not include
contributions of bubbles with chord lengths smaller than $\Delta$ as these
will not be picked up in the signal $S_{i}$.

####  3.1.2 Bubble size distribution

Figure 4: Schematic of calibration algorithm

The probes measure a chord length distribution $C_{m}$ by using the rise
velocity measured by both needles and the contact time of the bubble on the
longer needle (generally the one that will be hit first). The chord length
which depends on the size of the bubbles, their shape but also on the location
at which a particular bubble is pierced. The bubble size distribution is
therefore not measured directly by the electrical probes but needs to be
reconstructed based on $C_{m}$ and other parameters. In order to achieve this,
we adopt a method previously outlined in Besagni et al. (2016).

The main steps of the procedure are outlined in Figure 4. Besides $C_{m}$,
other required input parameters are an empirical relation for the typical
bubble aspect ratio $\phi$ as a function of the bubble size, and the lag
distance $\Delta$.

The algorithm assumes that the distribution of the volume equivalent bubble
diameter $d_{eq}$ can be approximated by a log-normal distribution. This
choice is motivated by findings of Mandal et al. (2005), who showed that a
log-normal distribution is the most appropriate in flows where the bubble size
is determined by break up and coalescence of bubbles, e.g. bubbles produced by
nozzles in bubbly flows. The log-normal distribution is given by

$f=\frac{1}{d_{eq}\sigma\sqrt{2\pi}}\exp{\left(-\frac{(\ln{d_{eq}}-\ln{d_{\mu}})^{2}}{2\sigma^{2}}\right)}$
(3)

with the free parameters $\sigma$ and $d_{\mu}$. The goal is then to determine
the combination of these two parameters that is most consistent with the
measured chord length distribution $C_{m}$. To this end, $51\times 51$
variations of both variables are tested in the relevant range (as estimated by
the results from the camera images) of $0.2<\sigma<0.3$ and
$2\,mm<d_{\mu}<4\,mm$. For each combination, $N=100$ bubbles are considered
with sizes $d_{eq}$ equally spaced between the lower measurement limit (0.96
mm) and 4 times the bubble radius where the log normal distribution has a
maximum such that the $i$-th bubble appears with a probability

$f_{i}=\frac{1}{d_{eq,N}-d_{eq,1}}\int_{d_{eq,i}}^{d_{eq,i+1}}f\textrm{d}d_{eq}.$
(4)

In the next step, for each of these bubbles a corresponding ellipsoidal shape
is determined based on the aspect ratio $\phi\equiv b/a$ (with $a$,$b$
denoting the semi major and semi minor axes, respectively), which is given
empirically in the form $\phi(a)$. We determine the ellipse properties based
on equivalence of the cross sectional area resulting in $d_{eq}=2\sqrt{ab}$.
Note that this differs from the choice in Besagni et al. (2016) who employed
$d_{eq}=2\sqrt[3]{a^{2}b}$. The difference between these two approaches can be
interpreted as a different choice for the out of plane half axes, which is
equal to $a$ in the case of Besagni et al. (2016), while our definition
implies that this dimension equals $\sqrt{ab}$. The main reason for our choice
was that the large spread in the $\phi$ data appears inconsistent with
assuming rotational symmetry in the horizontal plane for the bubbles (as
implied by setting the out-of-plane dimension equal to $a$).

Next, the vertical chord length is determined at 100 random locations for each
bubble assuming a horizontal orientation of the major axis. Finally, the
expected chord length distribution for that specific bubble $C_{i}$ is
obtained after subtracting $\Delta$. The probability of a bubble to get hit
scales with its projected area $A_{i}=\pi\sqrt{a^{3}b}$. The simulated chord
length distribution $C_{sim}$ across all $N$ bubbles can therefore be
constructed by summing the chord length contributions of the $N$ individual
bubbles (${C_{i}}$) weighted by their projected area $A_{i}$ and by their
probability $f_{i}$ according to

${C_{sim}}=\frac{\sum_{i=1}^{N}{C_{i,j}}A_{i,j}f_{i,j}}{W\sum_{j=1}^{N_{b}}\sum_{i=1}^{N}{C_{i,j}}A_{i,j}f_{i,j}},$
(5)

where $N_{b}$ and $W$ respectively denote the number and width of the bins
used for the chord length distributions. Finally the difference between the
simulated and measured chord length distribution is quantified as the error

$E=\sum_{j=1}^{N_{b}}|{C_{sim,j}}-{C_{n,j}}|.$ (6)

and the combination of $\sigma$ and $d_{\mu}$ that minimizes $E$ is selected
as the bubble size distribution $B_{n}$ corresponding to $C_{n}$. The entire
procedure runs within minutes on a typical desktop computer, such that there
was no need to go beyond the present brute-force approach.

The two required input parameters $\Delta$ and $\phi(a)$ are determined with
the use of the optical system. For the case of $\Delta$ this was done by
performing the above steps for varying values of the lag distance and
comparing the output to a bubble distribution obtained from the camera images.
We observed the best agreement for $\Delta=0.96mm$, which is also consistent
with the considerations in section 2.2.2. This value was therefore adopted
throughout.

The aspect ratio relation is determined by analyzing the camera images of the
non-overlapping bubbles for every measurement. A moving average over the
aspect ratio data of 1000 bubbles is then fitted with a function of a similar
form to that proposed by Besagni et al. (2016). Details on this fit and
corresponding results will be presented in section 4.2.

###  3.2 Optical system

####  3.2.1 Image processing

The camera images are analysed in order to extract the void fraction, bubble
size distribution and aspect ratio. The procedure used in doing so is outlined
graphically in Figure 5.

Figure 5: Graphical overview of the used image analysis algorithm

Starting from the original image $I_{org}$, the lighting is equalized using a
background image $I_{bg}$ according to

$I_{light}=I_{org}\frac{\text{max}(I_{bg})}{I_{bg}}.$ (7)

The equalised image $I_{light}$ is then binarized by comparing the local pixel
value to $\text{max}(I_{bg})$. At this point, small spots ($d_{eq}\leq 1\,mm$)
are removed from the image since these almost exclusively correspond to small
dirt particles in the water. This is confirmed by visual inspection and by
comparison to recordings made with no bubbles present in the flow.
Consequently, bubbles touching the outer edge of the image are eliminated to
yield the cleaned up image $I_{clean}$. In order to ensure that this step does
not affect the void fraction estimate, we compensate for the removed bubbles
by adjusting the effective measurement volume per image by

$V_{adj}=V_{meas}\frac{\langle{I}_{clean}\rangle}{\langle{I}_{bin}\rangle},$
(8)

where $V_{adj}$ is the volume used for determining the void fraction of the
processed image, $V_{meas}$ is the total measurement volume,
$\langle{I}_{clean}\rangle$ is the average value of the binarized clean image
and $\langle{I}_{bin}\rangle$ is the average value of the binarized image
after removing the described areas. The clean image will be split up into
‘dust’, ‘non-overlapping bubbles’ and ‘overlapping bubbles’. The dust can be
identified readily since the lack of reflection leads to lower greyvalues in
this case. The non-overlapping bubbles are recognized by the solidity
$\mathcal{S}$ of the patches, i.e. the ratio of the patch surface area to that
of its convex hull. The criterion for non-overlapping bubbles is
$\mathcal{S}>0.97$ and an ellipse with equal surface area is fitted to these
cases. The remaining patches are then considered as clusters of two or more
overlapping bubbles and their treatment is detailed in the following.

####  3.2.2 Cluster deconstruction

The goal of the cluster deconstruction is to describe the overlapping bubbles
as a group of overlapping ellipses. In practice, there is a limit to what
extent this is possible. In particular, the method presented here only
identifies bubbles that form part of the outer contour of the cluster such
that larger clusters of many bubbles can not be dealt with appropriately.

To illustrate our method, two examples are given in Figure 6. We make use of
the fact, that the bubble images are not uniform but feature a brighter
reflection spot. We identify these spots via simple thresholding (Figure 6b)
and then use them as starting points (minima) for the watershedding technique
(Meyer 1994). The watershedding technique fills the contour of the cluster
pixel by pixel starting from the outline of the reflections. Once adjacent
watershedded areas touch, the areas will no longer grow in the direction in
which they touched leading to a segmented cluster.

Figure 6: Illustration of the bubble cluster deconstruction algorithm. 2
examples have been given: example 1 shows an almost perfect fit, example 2
shows a less success full application of the algorithm. a) The original
images. b) Recognition of the bubble reflections. c) Watershedded areas
constructed using the reflections. The coordinates of the contours are shown.
d) The resulting ellipses

In Figure 6c an example cluster and part of the deconstructed cluster is
shown. We denote the coordinates of the segment of the contour pertaining to a
single watershedded region as $(\mathbb{\hat{y}_{w}},\mathbb{\hat{z}_{w}})$
and use $(\mathbb{\hat{y}_{c}},\mathbb{\hat{z}_{c}})$ to refer to the
remainder of the contour of the cluster, the hat denotes the use of the local
image coordinate system. The task at hand is then to fit an appropriate
ellipse to $(\mathbb{\hat{y}_{w}},\mathbb{\hat{z}_{w}})$. A general ellipse
with arbitrary orientation and position is defined by
$c_{1}y^{2}+c_{2}z^{2}+c_{3}yz+c_{4}y+c_{5}z-1=0$, where $c_{1},\dots,c_{5}$
denote the 5 independent parameters. The most straightforward way to fit an
ellipse is therefore to minimize the residual $||\mathbb{P}_{w}||_{2}$ of the
system

$\mathbb{c}\mathbb{M}_{w}-\mathbb{1}=\mathbb{P}_{w},$ (9)

where
$\mathbb{M}_{w}=[\mathbb{{\hat{y}_{w}}}\circ\mathbb{{\hat{y}_{w}}},\,\mathbb{{\hat{z}_{w}}}\circ\mathbb{{\hat{z}_{w}}},\,\mathbb{{\hat{y}_{w}}}\circ\mathbb{{\hat{z}_{w}}},\,\mathbb{{\hat{y}_{w}}},\,\mathbb{{\hat{z}_{w}}}]$
and $\mathbb{c}=[c_{1},\,c_{2},\,c_{3},\,c_{4},\,c_{5}]$. However, this simple
method does not penalize ellipses exceeding the contour of the cluster and
therefore often leads to unphysical results. This issue can be resolved noting
that the values of $\boldsymbol{P}_{w}$ are negative if the fitted ellipse
lies outside the watershedded contour. Ellipses exceeding the contour can
therefore be penalized by putting a higher weight on negative values of the
residual ($\boldsymbol{P}_{w}^{-}$) compared to their positive counterparts
($\boldsymbol{P}_{w}^{+}$). To also restrict ellipses from exceeding the
contour at other regions of the of the cluster, we additional consider
$\boldsymbol{P}_{c}$ defined analogous to equation 9. The full residual used
in fitting the ellipses is therefore given by

$||\boldsymbol{P}^{*}||_{2}=a_{1}||\mathbb{P_{w}^{-}}||_{2}+a_{2}||\mathbb{P_{w}^{+}}||_{2}+a_{3}||\mathbb{P_{c}^{-}}||_{2},$
(10)

where we used the weights $a_{1}=40$, $a_{2}=1$ and $a_{3}=30$. Note that
$a_{2}>0$ is required to optimize to the contour shape, but since this does
not apply for $\boldsymbol{P}_{c}^{+}$ it can be omitted. The ellipses fitted
to the sample images with this method are shown in Figure 6d.

####  3.2.3 Camera-based void fraction

The void fraction averaged across the $i$-th image is determined by summing
the volume of the $N(i)$ individual bubbles contained in it and dividing it by
the volume associated to that image, such that

$\alpha_{c}(i)=\frac{\sum_{j=1}^{N(i)}V_{bub}^{i}(j)}{V^{i}_{adj}},$ (11)

with subscript ‘$c$’ denoting quantities obtained from the camera images. Note
that in line with the discussion in section 3.1.2, the bubble volume
$V_{bub}=\frac{4}{3}\pi\left(\frac{d_{eq}}{2}\right)^{3}$ is again based on
$d_{eq}=2\sqrt{ab}$ with the semi-axes $a,b$ determined by the ellipse fits.

##  4 Results and validation

Here, we present sample measurements to cross-validate camera and needle
results and to illustrate the capabilities of the new system. To enable a
meaningful comparison between camera and needle results, we consider a case
with a moderate void fraction. This ensures that uncertainties in the image
analysis due to overlaps remain limited while the number of bubbles is high
enough to consider the mixture locally homogeneous. This trade-off was met
best for a case with an air flow rate of $0.55\,\text{Lm}^{-1}\text{s}^{-1}$
at a height of $z=3.4\,\text{m}$, which is therefore used as the test case in
the following unless specified otherwise.

###  4.1 Void fraction

The void fraction is the most basic quantity extracted from the measurements.
Results measured via the camera are shown in Figure 7a as a function of time.
We also show the contributions of non-overlapping bubbles ($\alpha_{c,no}$)
and of overlapping bubbles ($\alpha_{c,o}$) individually. Even at the modest
overall void fraction $\overline{\alpha}_{c}=0.49\,\%$, clusters are seen to
contribute more than half of the total void fraction underlining the
challenges in the optical approach. For reference, we note that assuming
rotational symmetry out of plane following Besagni et al. (2016) results in a
higher estimate of $\overline{\alpha}_{c}=0.63\,\%$.

Figure 7: a) Contribution of individual bubbles and bubbles in clusters to the
void measured by the camera. b) Comparison of void fraction measured by the
camera and by the electrical probes

An appropriate reference to the camera results for the void fraction is the
spatial average $\langle\alpha_{n}\rangle_{cl}$ over the 7 needle sensors
positioned along the centerline of the bubble curtain directly above the
camera measurement volume. A direct comparison between $\alpha_{c}$ (shown in
red) and $\langle\alpha_{n}\rangle_{cl}$ (blue) is presented in Figure 7b.
Even though the measurement regions do not fully overlap for the two methods,
there is a very good agreement between the two methods. Some minor
discrepancies are seen to occur whenever $\langle\alpha_{n}\rangle_{cl}$ is
high. These are likely a consequence of the difficulties in disentangling
bubble clusters properly at these instances and we therefore expect the needle
measurement to be more reliable in these conditions. The temporal mean
$\langle\overline{\alpha}_{n}\rangle_{cl}=0.50\%$ matches the camera result
closely which validates the needle measurements. In particular, we note that
with $\langle\overline{\alpha}_{n}^{*}\rangle_{cl}=0.37\%$ (see also grey line
in Figure 7b) the needle based estimate for the void fraction would be
significantly lower without accounting for the lag distance $\Delta$. Here we
used $\Delta=0.96\,\text{mm}$ based on fitting the bubble size distribution
(see Section 3.1.2) and $V_{r}=1\,\text{ms}^{-1}$.

###  4.2 Bubble aspect ratio

As described in section 3.1.2, knowing the bubble aspect ratio as a function
of the major axis ($\phi(a)$) is required to determine the bubble-size
distribution. We can determine $\phi$ for individual bubbles from the camera
images and results for the test case (accumulated over a set of 1920 images)
are shown in Figure 8a. Note that only results for non-overlapping bubbles are
considered here as these data are more reliable. There is a considerable
spread in $\phi(a)$ for individual bubbles and we therefore consider the
moving mean $\phi_{m}$ over 1000 bubbles in order to reduce the scatter. This
quantity can be fit using a composite expression similar to that used in
Besagni et al. (2016):

$\phi_{f}=\begin{cases}\kappa_{1}a^{2}+\kappa_{2}a+\kappa_{3}&\text{if
}a\leq\kappa_{4}\\\ \kappa_{5}a^{\kappa_{6}}&\text{if
}a>\kappa_{4}\end{cases}$ (12)

Small bubbles with $a\to 0$ are not deformable and we therefore set
$\kappa_{3}=1$ while the remaining parameters are fit to the data. For the
results shown in Figure 8a, we obtain $\kappa_{1}=0.0057$,
$\kappa_{2}=-0.1378$, $\kappa_{4}=2$, $\kappa_{5}=0.9810$ and
$\kappa_{6}=-0.3753$, which captures the variation in $\phi_{m}$ very well.

Figure 8: a) Aspect ratio of individual bubbles along with moving mean
$\phi_{m}$ and fit $\phi_{f}$ according to Eq. 12. b) Variation of $\phi_{m}$
for hose configurations, measurement heights $z$, and flow rates $V$. Grey
lines represent additional case for the ‘Japanese’ configuration with
$2.0m\leq z\leq 3.4$ and flow rates ranging within $0.55Lm^{-1}s^{-1}\leq
V\leq 1.67Lm^{-1}s^{-1}$

We note, however, that the results for $\phi_{m}$ vary for different
experimental conditions as shown in Figure 8b. For the case of the Japanese
hose, there is a considerable dependence on the flow rate $V$, while the
variation of $\phi_{m}$ at different heights is less pronounced. Additionally,
also the details of the bubble generation are seen to affect the results.
Especially aspect ratio results for the ‘conventional’ hose with drilled holes
differ from the two other configurations in particular for larger bubbles with
$a\gtrsim 5mm$. Similar effects have been noted in the literature before and
in particular Besagni and Deen (2020) proposed an expression to relate bubble
aspect ratios to flow properties. Their prediction $\phi_{BD}$ requires the
bubble Reynolds number and the Eötvös number (relating surface tension and
gravitational forces) as inputs. A result for $\phi_{BD}$ based on typical
values of these quantities in our experiments is included in Figure 8b for
reference. It agrees well with the ‘conventional’ hose case, but does not
represent the results for the other two configurations well. It is therefore
highly beneficial to have the simultaneous optical measurements for every test
case. This allows us to fit Equation 12 for every configuration individually
for the best accuracy.

###  4.3 Bubble size distribution

With $\phi(a)$ known, the procedure outlined in Section 3.1.2 can be employed
to obtain a bubble size distribution based on the measured chord length
distribution $C_{n}$. We can cross-check these results vs. the bubble size
distribution $B_{c}$, which is extracted from analysing the camera images.
This comparison is presented in Figure 9 for the test case. A striking
observation from this figure is by how much the distributions $C_{n}$ and
$B_{c}$ differ, which underlines the need for appropriate postprocessing of
the chord length data. How important it is to account for the lag distance
$\Delta$ of the needle sensors becomes clear when considering
$B_{n}(\Delta=0\,\text{mm})$, which represents the calibration result when
ignoring this effect. The predicted size distribution in this case differs
significantly from the reference $B_{c}$. To improve on this, we
systematically vary $\Delta$ and monitor the residual (squared difference) $R$
between $B_{n}(\Delta)$ and $B_{c}$. This residual is plotted as black line in
the inset of Figure 9 and displays a minimum for a value of $\Delta=0.96\,mm$,
for which then also $B_{n}(\Delta=0.96\,mm)$ is found to be in very good
agreement with $B_{c}$. Similar trends are also observed for other cases
(shown as grey lines in the inset, see also Figure 10), for which $R$ could be
computed. Moreover, this value for $\Delta$ also matches the outcome of the
probe tests in Section 2.2.2. Additionally, we also found that
$\overline{\alpha}_{c}$ and $\langle\overline{\alpha}_{n}\rangle_{cl}$ are in
good agreement for this value and $\Delta=0.96\,mm$ is therefore adopted for
all measurements.

Figure 9: Measured chord length distribution $C_{n}$ and the resulting bubble
size distribution as measured using the needles $B_{n}$ compared to the bubble
size distribution obtained via the camera $B_{c}$. Measured at z=3.4 m and
V=0.55 Lm-1s-1

In Figure 10, we present results on the bubble size distribution for different
gas flow rates $V$ and heights $z$ for both the Japanese hose but also for the
conventional hose. For the other cases, no comparison to camera results was
possible, either due to the high void fraction (below $z=2\,m$) or due to
insufficient data (nozzle hose). The three samples for the conventional hose
at $V=1.12L/ms$ in Figure 10a show similar agreement between $B_{n}$ and
$B_{c}$ as observed in Figure 9. A more systematic assessment is possible from
Figures 10b,c,e,f where the parameters of the log-normal distribution obtained
from the calibration ($d_{\mu,n}$, $\sigma_{n}$) are compared to those
resulting from fitting $B_{c}$ ($d_{\mu,c}$, $\sigma_{c}$). The agreement for
$d_{\mu}$ (Figure 10b) is very good and also the slight trend of increasing
$d_{\mu}$ with increasing $V$ is captured faithfully for the most part. The
correspondence is somewhat worse for $\sigma$ (Figure 10c), where $\sigma_{n}$
is seen to underpredict $\sigma_{c}$ consistently by about 15-20%. We believe
that part of this discrepancy is due to the fact that very small bubbles
($d_{eq}\leq 1mm$) are not picked up by the needle measurements. The bubble
distributions originating from the conventional hose (see samples in Figure
10d) differ significantly from those of the Japanese hose, which was used for
calibration. Nevertheless, the agreement between camera and needle
measurements overall remains equally as good for this case illustrating the
robustness of our method. This can be judged by the results for $d_{\mu}$
(Figure 10e) and $\sigma$ (Figure 10f), where the former match somewhat less
well compared to the Japanese hose, whereas the agreement for the latter is
found to be better for the most part.

c

Figure 10: Bubble size distribution obtained via the double probes compared to
the camera images for two the Japanese hose (a,b,c) and the conventional hose
(d,e,f). For every case $\phi_{f}(a)$ has been determined separately and
$\Delta=0.96\,\text{mm}$. a) Resulting bubble size distributions for a
flowrate of $V=1.12\,\text{L}\text{m}^{-1}\text{s}^{-1}$, solid lines
calibration result and dashed lines camera result b) $d_{\mu}$ for all
considered cases compared to the camera. c) $\sigma$ for all considered cases
compared to the camera d) Resulting bubble size distributions for a flowrate
of $V=1.12\,\text{L}\text{m}^{-1}\text{s}^{-1}$. e) $d_{\mu}$ comparison. f)
$\sigma$ comparison

###  4.4 Characterization of sample bubble plumes

In the following, we present some representative results of plume
characteristics based on the needle probes mounted perpendicularly to the air-
injection hose (see Figure 3a).

In Figure 11, we chose three different configurations to illustrate
differences in plume behaviour. The most basic quantity obtained from the
needle sensors is the distribution of contact times $T_{con}$ in space and
time. These are shown in Figure 11(a-c) for the three cases. Already from
these data, it is evident how the plume close to the nozzle in Figure 11a is
narrow and without significant lateral movement. At larger $z$ and $V$, the
plume in Figure 11b is somewhat wider but additionally also exhibits a slow
waving motion in time. These fluctuations are most pronounced for the case
shown in Figure 11c, which is recorded even further from the nozzle and at a
lower gas flow rate. Note that the ‘white stripes’ in these figures are due to
a fault of the needle probes at the corresponding locations. These data were
recorded in DC mode and such probe failures were significantly reduced after
switching to AC operation of the sensors.

Figure 11: Measurement results for three different configurations: at (a,d,g)
$z=0.8\,m$ and $V=1.12\,Lm^{-1}s^{-1}$, (b,e,h) $z=2\,m$ and
$V=1.67\,Lm^{-1}s^{-1}$, and (c,f,i) $z=2.8\,m$ and $V=0.55\,Lm^{-1}s^{-1}$.
(a,b,c) Spatiotemporal distribution of contact times $T_{con}$ at a needle
(corrected for $\Delta$). (d,e,f) Local void fraction $\alpha_{n}$ as a
function of time. (g,h,i) Time averaged void fraction based on conventional
($\alpha_{n}$ plotted vs. $x$) and conditioned ($\alpha_{rel}$ plotted vs.
$x_{rel}$) averaging procedure. Note that origin for $x_{rel}$ in (g,h,i) is
chosen to align with the peak location of $\alpha_{n}$

In order to allow for a more quantitative evaluation, we determine the local
void fraction $\alpha_{n}$ by averaging in time according to Eq. 2 with
$T=5\,\mathrm{s}$ and taking the mean over 5 adjacent needle probes
(corresponding to a distance of 30 mm). In doing so, we employ linear
interpolation using the 2 adjacent needles to each side to fill in for the
broken probes to avoid large gaps in the distribution of $\alpha_{n}$. The
contours of $\alpha_{n}(x,t)$ corresponding to the data shown in Figure 11a-c
are displayed in Figure 11d-f as a function of the measurement time. On this
basis, it is possible to determine the location of the center of the bubble
curtain ($x_{cen}$) from

$x_{cen}(t)=\frac{\int_{\Lambda}\alpha_{n}(x,t)xdx}{\int_{\Lambda}xdx},$ (13)

where the integration is over the width $\Lambda$ of the measurement rake. The
location of $x_{cen}(t)$ is indicated by dashed lines in Figure 11d-f.
Additionally, we define the ‘top-hat’ scales for void fraction and plume width
by

$\hat{\alpha}(t)=\frac{M(t)}{Q(t)}$ (14)

and

$\hat{w}(t)=\frac{Q(t)}{\hat{\alpha}(t)}$ (15)

based on the integrals $Q(t)=\int_{\Lambda}\alpha_{n}dx$ and
$M(t)=\int_{\Lambda}\alpha_{n}^{2}dx$. Results for $\hat{w}$ are included in
Figure 11d-f as dashed lines located at $x_{cen}(t)\pm\hat{w}(t)/2$. We check
for instances where the bubble curtain leaves the measurement region by
verifying that $|x_{cen}(t)\pm\overline{\hat{w}}|\leq\Lambda/2$ and discard
times where this does not hold from the processing (indicated by dotted
outlines in Figure 11d-f).

Finally, we present two types of temporal averages over the entire measurement
time of 120s in Figure 11d-f). One is the conventional average in the
laboratory frame of reference resulting in $\overline{\alpha}_{n}(x)$ as
defined in Eq. 2. In addition, we consider the conditioned average relative to
the instantaneous centerline location $x_{cen}$, which we define as
$\alpha_{rel}(x_{rel})$ where $x_{rel}=x-x_{cen}$ denotes the lateral
coordinate relative to the centerline location. Naturally, the spatial
distributions of $\overline{\alpha}_{n}$ and $\overline{\alpha}_{rel}$ agree
closely in the first case (Figure 11g), where there is very little movement of
the centerline. Noticeable differences are visible for the case in Figure 11h
and in the most extreme case (Figure 11i) the peak value of
$\overline{\alpha}_{rel}$ is about twice that of $\overline{\alpha}_{n}$,
which is significant even in view of the relatively poor statistical
convergence in this case. These differences are of high relevance when trying
to deduce acoustical properties of the bubble curtain as the sound interacts
with the instantaneous distribution for which $\overline{\alpha}_{rel}$ is
more representative. As pointed out by Milgram (1983) already, the slow
movement of the centerline (or ‘wandering’ as they called it) renders
convergence of conventional averages poor and our measurements here are also
certainly too short to fully characterise this wandering motion.

In Figure 12a, we present results for $\alpha_{rel}$ at different heights
(with the red curve corresponding to the case shown in Figure 11(b,e,h). Close
to the nozzle, the peak void fraction exceeds 2% and this value drops to about
0.5% as the flow spreads upwards. The width of the plume at the highest
measurement station ($z=3.4m$) exceeds the measurement domain and we also
noticed that the presence of the free surface (located at $z=3.6m$) influenced
the flow at this stage. The same data is re-plotted rescaled by
$\hat{\alpha}_{rel}$ and $\hat{w}_{rel}$ (defined analogously to Eq 14 and Eq
15) in Figure 12b. Within the limits of the statistical convergence, there is
a reasonable self-similar collapse for all data points except for the one at
$z=3.4m$. The distribution is well described by a Gaussian (dashed black line)
and the same curve also fits the data at other values of $V$ and for other
hose types very well (not shown). In Figure 12c, time averaged values of the
top-hat width as a function of the height for three different flow rates are
shown. For all three cases, the data in the range $0.4\,\mathrm{m}\leq z\leq
2.8\,\mathrm{m}$ can be well approximated by a linear fit of the form
$\overline{\hat{w}}=\beta z+\zeta$, which is consistent with the self-similar
scaling for planar plumes (i.e. neglecting the effect of slip velocity of the
bubbles, see e.g. Paillat and Kaminski 2014). Presumably due to interaction
with the free surface, the last data point at $z=3.4m$ deviates slightly from
the linear trend, in particular for the two higher gas flow rates.

The spreading parameter $d\overline{\hat{w}}/dz=\beta$ (see inset of Figure
12c) is seen to increase with $V$. This trend as well as the magnitude of
$\beta$ is consistent with results by Cederwall and Ditmars (1970) for planar
plumes based on velocity measurements. Similar findings (in terms of a Froude
number dependence) are also reported for round plumes by Kobus (1968) and
Milgram (1983). Finally, results for $\overline{\hat{\alpha}}_{rel}$ are shown
in Figure 12d. The decay with $z$ is reasonably approximated by a $1/z$
dependence for these data, again consistent with the self-similar scaling for
this quantity. The inset of Figure 12d shows the ratio between
$\overline{\hat{\alpha}}_{rel}$ and $\overline{\hat{\alpha}}_{n}$. We expect
this ratio to be large in cases with pronounced centerline movement. Results
should be interpreted with care especially for the largest value of $V$ due to
limited statistical convergence. It appears clear, however, that the relevance
of centerline movement increases with increasing height, and presumably also
with increasing $V$.

Figure 12: a) Measured instantaneous void fraction at different heights for
$V=1.67\,Lm^{-1}s^{-1}$. b) Same data as in (a) rescaled by
$\overline{\hat{w}}_{rel}$ and $\overline{\hat{\alpha}}_{rel}$. The dashed
black line is a fitted Gaussian which is characterized by the standard
definition $\sigma_{g}=0.2845$ and the mean $\mu_{g}=0.01$. c)
$\overline{\hat{w}}_{rel}$ as a function of $z$ for different air flow rates,
markers show measured values and the solid lines show the best fit in the form
of $\overline{\hat{w}}_{rel}=\beta z+\zeta$. d)
$\overline{\hat{\alpha}}_{rel}$ as a function of $z$ for different air flow
rates. The fit of the solid lines is in the form of
$\overline{\hat{\alpha}}_{rel}=\upsilon z^{-1}+\xi$

Finally, the spreading of the bubble plumes created with the three different
hoses is compared in Figure 13. Consistent with result at other flow rates
(not shown), it can be seen from Figure 13a that the plume originating from
the Japanese hose is the widest at all heights, while differences remain small
between the nozzle and the conventional designs. These trends are also
reflected in Figure 13b, where the growth rate of the bubble column width is
shown for the three types. The growth rate of the conventional hose and the
hose with nozzles is very similar and significantly higher values of $\beta$
are observed for the Japanese hose. This difference does not appear to be
related to differences in the bubble size distribution because there the
largest deviations occur for the conventional hose (see Figure 10), while
distributions for the other two cases are very similar. A key difference
between the configurations is that the Japanese hose is a continuous line
source emitting air around its circumference, whereas the bubbles originate
from discrete sources separated by respectively 5cm and 10cm for the
conventional and nozzle hoses. Even though these individual plumes merge
within the first 1m above the nozzle, especially the different plume evolution
between the Japanese and nozzle hose suggest that these differences might
affect the flow even at much larger heights.

Figure 13:  a) Top hat width of the three different configurations at
$V=1.67\,Lm^{-1}s^{-1}$. b) spreading parameter for all hose types at varying
gas flow rates

##  5 Conclusions and Discussion

In summary, we report the development of a system capable of measuring the
void fraction and the bubble size distribution in bubble curtains _in-situ_. A
key feature of our approach is the combination of a contact based sensor array
with an optical system. This enables the validation of the needle based
measurements and allows us to calibrate the aspect ratio dependence $\phi(a)$
during the actual measurement. Our measurements revealed that accounting for
the insensitive length $\Delta$ of the needle tip was critical to obtaining
accurate and reliable data from the electrical probes. When correcting for
this effect, we found very good agreement between camera and needle based
measurements of the local void fraction as well as of the bubble size
distributions. We employ a statistical model that assumes a log-normal
distribution for the latter, which describes the present data well. It appears
possible, however, that modifications to this assumption might become
necessary for other nozzle configurations and it is useful to monitor this via
the optical measurements.

The system is then employed to measure the characteristics of bubble plumes
originating from three different nozzle configurations. Our results show that
especially at higher flowrates and further from the nozzle, it is critical to
account for the meandering of the plume in order to obtain representative
distributions as an input to an acoustic model. This can be achieved by
conditioning on the instantaneous center. We find that for all three nozzle
types, such conditioned void fraction distributions evolve in a self-similar
manner up to the vicinity of the free surface. The spreading rates increase
with increasing gas flow rate, which is consistent with previous reports on
round (Milgram 1983; Kobus 1968; Fraga and Stoesser 2016) and planar bubble
plumes (Kobus 1968). However, we find that the spreading rate also depends on
the nozzle geometry. In particular, the spreading parameter $\beta$ is
substantially higher when the Japanese hose is employed. When interpreting
these results, it should be kept in mind that here the analysis is based on
the void fraction distribution while most other studies base the spreading on
the velocity field. The velocity and the void-fraction based width are not
necessarily equal and their ratio can depend on factors such as the nozzle
configuration as Wu et al. (2021) have shown for single round plumes.

With the proposed measurement technique a data set with the hydrodynamical
properties of different bubble curtains can be constructed. Future research
should extend the use of this system to longer measurement times and larger
arrays and focus on combining the measurements with acoustical measurements.

## Declarations

### Ethical Approval

Not applicable

### Competing interests

The authors report no conflict of interest.

### Authors’ contributions

Simon Beelen wrote the main manuscript, carried out the data processing, made
the figures, helped develop the system and carried out the experiments.
Martijn van Rijsbergen helped develop the system and carried out the
experiments.
Miloš Birvalski helped develop the system and carried out the experiments.
Fedde Bloemhof helped develop the system and carried out the experiments.
Dominik Krug wrote the main manuscript, and made the figures.
All authors reviewed the manuscript.

### Funding

This publication is part of the project AQUA (with project number P17-07) of
the research programme perspectief which is (partly) financed by the Dutch
Research Council (NWO).
This project is supported by the Netherlands Enterprise Agency (RVO) and TKI
Wind op Zee.
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 950111, BU-PACT).

###  5.1 Availability of data and materials

The data will be uploaded to arxiv.

### Code availability

The code is available upon request.

##  Appendix A Circuitry electrical probes

Figure 14: The circuitry used by the needle probes, can be actuated either by
a constant input(DC) or by an alternating input (AC)

##  Appendix B Pictures hoses

Figure 15: The three different hoses used in the experiments

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