# Micro-droplet nucleation through solvent exchange in a turbulent buoyant jet

You-An Lee1  Chao Sun1,2  Sander. G. Huisman1  Detlef Lohse1,3  1Physics of
Fluids Group and Max-Planck, Center for Complex Fluid Dynamics, Faculty of
Science and Technology, J.M. Burgers Center for Fluid Dynamics, University of
Twente, 7500 AE Enschede, Netherlands 2Center for Combustion Energy, Key
Laboratory for Thermal Science and Power Engineering of Ministry of Education,
Department of Energy and Power Engineering, Tsinghua University, 100084
Beijing, China 3Max Planck Institute for Dynamics and Self-Organization, Am
Faßberg 17, 37077 Göttingen,Germany

###### Abstract

Solvent exchange is a process involving mixing between a good solvent with
dissolved solute and a poor solvent. The process creates local oversaturation
which causes the nucleation of minute solute droplets. Such ternary systems on
a macro-scale have remained unexplored in the turbulent regime. We
experimentally study the solvent exchange process by injecting mixtures of
ethanol and trans-anethole into water, forming a turbulent buoyant jet in
upward direction. Locally, turbulent mixing causes oversaturation of the
trans-anethole following turbulent entrainment. We optically measure the
concentration of the nucleated droplets using a light attenuation technique
and find that the radial concentration profile has a sub-Gaussian kurtosis. In
contrast to the entrainment-based models, the spatial evolution of the
oversaturation reveals continuous droplet nucleation downstream and radially
across the jet, which we attribute to the limited mixing capacity of the jet.
Though we are far from a full quantitative understanding, this work extends
the knowledge on solvent exchange into the turbulent regime, and brings in a
novel type of flow, broadening the scope of multicomponent, multiphase
turbulent jets with phase transition.

##  1 Introduction

Solvent exchange can be employed to extract specific components from mixtures.
The process finds various industrial applications, such as liquid-liquid
extraction (Lohse, 2016), drug delivery (Lepeltier et al., 2014), resource
recycling (Zimmermann et al., 2014), and oil recovery (Sun et al., 2017). Such
solvent exchange processes consist of a ternary liquid system, where mixing
between a good solvent and a poor one takes place, leading to oversaturation
of the solute originally dissolved in the good solvent. This in turn triggers
nucleation of the solute droplets. Such droplet nucleation is called
spontaneous emulsification, or is known by its popular name ‘ouzo effect’. In
this process metastable droplets are formed in the liquid, creating a milky
appearance due to Mie scattering for the case of watered-down ouzo beverage.
In recent decades, numerous efforts have been made to understand such a
complex multicomponent and multiphase flow, see the review by Lohse & Zhang
(2020). Vitale & Katz (2003) and Sitnikova et al. (2005) were among the
earliest studies of the ouzo effect, using divinylbenzene and trans-anethole
as the solute, respectively. They both located the position of the metastable
region in the ternary phase diagram, and experimentally revealed the diffusive
growth process of the droplets and their size distribution.

To quantitatively study the diffusion dynamics of the multicomponent droplets,
Su & Needham (2013) extended the classic work of Epstein & Plesset (1950) on
the dissolution and growth of gas bubble in surrounding liquid to the case of
a droplet. Chu & Prosperetti (2016) further analytically investigated the
droplet growth and dissolution of the multicomponent droplets, broadening the
theoretical framework of droplet diffusion dynamics, making it more applicable
to the real-world applications, such as liquid-liquid extraction.
Corresponding molecular dynamics simulation was done by Maheshwari et al.
(2017). Tan et al. (2019) studied the micro-droplet nucleation of the water-
ethanol drop in host anise oil. They combined the multicomponent diffusion
model, thermodynamic equilibrium, and a scaling analysis of Marangoni and
buoyancy forces. With this they could successfully predict the spontaneous
emulsification by the convection-enhanced diffusion process.

While the aforementioned cases are homogeneous nucleation in the bulk, the
ouzo effect has also been widely investigated on the surface of microfluidic
channels, so-called surface droplet formation via heterogeneous nucleation.
Hajian & Hardt (2015) studied the droplet nucleation and its radial migration
in a micro-channel, measuring the concentration field formed by the solvent
exchange and the following diffusive mass transfer. For controlled solvent
exchange in a microfluidic setups, Zhang et al. (2015) found that the volume
of the nucleated droplets scales as $Pe^{3/4}$, with Peclet number
$Pe=\frac{Q}{wD}$, where $Q$ is the volume flow rate, $w$ the channel width,
$D$ the diffusion coefficient. Following this finding, the universality of the
phenomena has been demonstrated, including the effect of the solution
composition (Lu et al., 2015, 2016), confining the flow geometry (Lu et al.,
2017; Zeng et al., 2019), and formation (Li et al., 2018) and growth (Dyett et
al., 2018) of the surface nanodroplets. For related work, Li et al. (2021)
implemented a Hele-Shaw like thin cell with a porous section to better collect
the nucleated droplets. For this geometry, they formulated a theoretical model
based on the ternary diagram and the diffusion equation, and found and
explained the scaling of the volume of the nucleated droplets in the
oversaturation front to be proportional to $Pe^{1/2}$.

While solvent exchange is thus reasonably and in part even quantitatively
understood, the scope of quantitative studies has up to now been limited to
small scales and the laminar flow regime. For industrial-scale applications
such as liquid-liquid extraction, however, processes on larger scales are
generally less controlled than those in microfluidic devices due to
turbulence. It is therefore highly desirable to unveil the diffusion dynamics
of the ternary liquid system in turbulent environments, where intense mixing
and nucleation appear. The turbulent jet is one of the most fundamental and
well-studied form of turbulent shear flow (see e.g. Pope (2000)), which makes
it an ideal candidate to bridge the field of solvent exchange and turbulence.

Built upon the classic single phase studies for turbulent jets and plumes
(List, 1982; Morton et al., 1956; Fischer et al., 1979; Turner, 1986; Pope,
2000), the case of multiphase flow has drawn extensive interest due to its
common occurrence in environmental fluid mechanics (Woods, 2010) and
industrial applications such as combustion (Raman & Fox, 2016) and aerosol
formation (Lesniewski & Friedlander, 1998; Neuber et al., 2017). The Deepwater
Horizon Oil Spill further triggered a series of investigations into such types
of multiphase flow (K.McNutt et al., 2012; Reddy et al., 2012; Ryerson et al.,
2012). With turbulent entrainment, bubbles or droplets can grow and dissolve
in a jet, inducing significant buoyancy variation in the flow. Chu &
Prosperetti (2019) analytically identified key parameters for the multiphase
plume dynamics, including the droplet dissolution rate, the density change
rate due to dissolution, the droplet velocity, and the plume velocity.

In addition to the dissolution, considerations for chemical reaction have
advanced the line of research further to tackle real-world complex problems.
There are two major perspectives of research on reactive jets and plumes. On
the one hand, the ambient fluid reacts with the dispersed phase in the jet or
plume. This category serves as an extension of the works on bubbles or
droplets dissolution, focusing on the buoyancy variation accompanying the
reactions, see Cardoso & McHugh (2010) and Domingos & Cardoso (2013). These
authors also discussed the competition between stratification and reaction as
the dominant source of buoyancy variation.

On the other hand, the ambient fluid reacts with the carrier phase of the jet,
or simply with the single phase injected fluid. Studies on such type of flow
shed new light on the interplay among entrainment, mixing, and reactions as
the flow develops, enabling the disentanglement of these effects by comparing
the corresponding length and time scales. Ülpre et al. (2013) injected a
single phase acidic jet into an alkaline environment, reporting a model
combining chemistry and the fluid dynamics of turbulent plumes to predict the
neutralization distance. Domingos & Cardoso (2015) studied single phase
reacting thermals, distinguishing fast and slow reactions. Their experimental
findings show that the reactions only occurred preferentially in part of the
thermals for the instantaneous case, while for the slow reactions the thermals
were homogeneous in composition. Mingotti & Cardoso (2019) experimentally
compared the slow and instantaneous reaction in turbulent plumes, discovering
preferential reaction for the instantaneous case as well, which appeared on
the edge of the eddies. They formulated a theoretical prediction for the
concentration evolution in reacting plumes using scale analysis of
entrainment, mixing, and reactions. Solvent exchange in our case is
fundamentally different from a chemical reaction in terms of driving
mechanism. The product of solvent exchange, namely the nucleated droplets, is
driven by the degree of oversaturation, while the products of chemical
reactions result from chemical kinetics. Also, speaking of chemical reaction
and mixing, Guilbert & Villermaux (2021) experimentally studied the effect of
reaction on the concentration distribution of the product in a randomly
stirred mixture, distinguishing between diffusion-controlled and reaction-
controlled regimes (Guilbert et al., 2021), and showing a compressed
distribution profile for reactive mixtures (Guilbert & Villermaux, 2021). In
spite of such difference, the cases of instantaneous reactions in the
aforementioned studies are highly insightful for our attempt to understand the
solvent exchange process in turbulent buoyant jet, not only because the
timescale to generate the product are both extremely small, but also because
the turbulent entrainment and mixing might play key roles in both processes.

The present experimental study aims to investigate the solvent exchange
process in a multiphase turbulent buoyant jet in the upward direction,
bridging the gap between the ouzo effect and turbulent shear flow. We will
employ trans-anethole and refer to it as $oil$ throughout the paper as it is
the oil phase in the ternary liquids system. We will use a vertical ethanol-
oil jet being injected in a quiescent water bath, creating an ’ouzo’ mixture
close to the needle. We utilize a novel method to construct the averaged
oversaturation (concentration) field for the ouzo solution using titration,
light attenuation technique, and minimization using an axisymmetric
discretization. We compare the ouzo case with the nucleation-free case of
injecting only dyed ethanol, revealing the distinct features of the ouzo case
in the radial concentration profile, the centerline evolution, and the
oversaturation flow rate. Adapting the mixing-limited arguments for the
prolonged depletion of the reacting plume of Mingotti & Cardoso (2019), the
extended lifetime of the droplet-laden respiratory puff of Chong et al. (2021)
and of the evaporating spray of de Rivas & Villermaux (2016) and Villermaux et
al. (2017), we provide a qualitative explanation for the continuous nucleation
downstream and the wide-spread location of the micro-droplet nucleation across
the jet.

The paper is structured as follow: In §2 we detail the experimental setup and
the methods to measure the velocity field and the concentration field, and to
resolve the nucleated droplets. The concentration fields can be reconstructed
by combining the information from two cameras, as presented in §3. The results
extracted from the concentration fields are presented in §4, with supporting
discussion and qualitative analysis in §5. Conclusions and future prospects
are presented in §6.

##  2 Experimental method

###  2.1 Setup

We have conducted several experiments to study the solvent exchange process in
a turbulent buoyant jet (ouzo jet). The jet experiments were operated in a
glass tank with dimensions $25\text{\,}\mathrm{cm}$ $\times$
$25\text{\,}\mathrm{cm}$ $\times$ $50\text{\,}\mathrm{cm}$ (W $\times$ L
$\times$ H) as in Fig. 1. The tank was filled with decalcified water before
each run of experiments. The injected fluid consisted of mixtures of ethanol
(Boom, 100%) and trans-anethole (Sigma Aldrich, $\geq$99%) with varying weight
ratio, and the fluid was injected by a Harvard 2000 syringe pump at selected
flow rates, forming an upwards turbulent buoyant jet through a round needle
with inner diameter $0.51\text{\,}\mathrm{mm}$, outer diameter
$0.82\text{\,}\mathrm{mm}$, and length $12.7\text{\,}\mathrm{mm}$. Reference
cases were also conducted by injection of dyed ethanol, offering comparison
between the common passive scalar jet and the ouzo jet. The water and the
injected liquid were kept at room temperature
($$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$\pm$1\text{\,}\mathrm{K}$$) for
all the experiments to minimize the effect of the temperature-dependence of
the solubility in the ternary liquids system. The experimental conditions are
listed in Table 1.

To measure the local concentration we use a light attenuation technique. For
this we use a LED back lighting with diffuser and recorded using two Photron
FASTCAM Mini AX200 high speed cameras with 1024 $\times$ 1024 pixels
resolution at 50 fps, one from afar with a Sigma $50\text{\,}\mathrm{mm}$
objective to capture the entire jet evolution, and the other with a Zeiss
$100\text{\,}\mathrm{mm}$ objective placed closer to the tank and zoomed in to
the region right above the injection needle. The injection and the recording
lasted for 40–50 s, providing 2000–2500 frames for each experiment. To obtain
the velocity field of the turbulent buoyant jets, we relied on particle image
velocimetry (PIV) with laser activation and using different cameras, which
will be detailed in §2.5.

Figure 1: Experimental setup. The inset shows the photo of the needle for injection with inner diameter (ID), outer diameter (OD), and length (L). Note that the laser was only activated for PIV. No. | $w_{e}/w_{o}$ |  $C_{\text{dye}}$ (ppm) |  $Q$ $(\text{m}^{\text{3}}/\text{s})$ $\times 10^{-7}$ |  $L_{m}$ (m) $\times 10^{-2}$ |  $u_{0}$ (m/s) | $Re_{0}$
---|---|---|---|---|---|---
1 | 100 | - | 3.33 | 2.48 | 1.63 | 555
2 | 100 | - | 8.33 | 6.19 | 4.08 | 1387
3 | 33 | - | 3.33 | 2.48 | 1.63 | 555
4 | 33 | - | 8.33 | 6.19 | 4.08 | 1387
5 | - | 4000 | 3.33 | 2.48 | 1.63 | 555
6 | - | 4000 | 8.33 | 6.19 | 4.08 | 1387
Table 1: Experimental conditions. Each experiment was repeated three times to
confirm the repeatability of the data. Experiments 1–4 correspond to the ouzo
jets with empty $C_{\text{dye}}$ column, and experiments 5–6 correspond to the
dyed ethanol jets with empty $w_{e}/w_{o}$ column. $w_{e}/w_{o}$ denotes the
weight ratio between ethanol and the oil in the injected fluid. $Q$ is the
volume flow rate regulated by the syringe pump, leading to different
characteristic length $L_{m}$, initial velocity $u_{0}$, and initial Reynolds
number $Re_{0}=Qd/(\frac{1}{4}\pi d^{2}\nu)$, where $d=0.51$ mm is the inner
diameter of the needle. Here, $L_{m}=M^{3/4}/B^{1/2}$, where
$M=Q^{2}/(\frac{1}{4}\pi d^{2})$ is the initial momentum flux, and
$B=Qg(\rho_{\text{jet}}-\rho_{\text{amb}})/\rho_{\text{amb}}$ is the initial
buoyancy flux. $B=1.635\times 10^{-7}$
$\text{m}^{\text{4}}/\text{s}^{\text{3}}$ is constant throughout our
experiments as the density of the oil is negligible, i.e., we consider the jet
density $\rho_{\text{jet}}$ to be solely determined by the ethanol. The
viscosity of the oil was also neglected, leading to a constant viscosity
$\nu=\nu_{\text{e}}=1.5\times 10^{-6}$ $\text{m}^{\text{2}}/\text{s}$, which
is used to calculate $Re_{0}$.

###  2.2 Titration and oversaturation

To study the nucleation and the solvent exchange process quantitatively,
knowledge of oversaturation of the nucleated component is required (Li et al.,
2018, 2021). This means that the first step is to obtain the binodal curve,
namely the saturation curve by titration. Normally, the titration results of
such ternary system are determined visually by checking the change of
appearance. However, this is extremely difficult since we operated in the
regime with extremely small amounts of oil. We therefore opted for cameras for
accurate determination of the point of saturation. The camera-assisted
titration was conducted by step-wise adding water into the oil-ethanol
solution, followed by one minute thorough mixing, and then idle time for
around 15 minutes to ensure the temperature stabilizes to
$$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$\pm$1\text{\,}\mathrm{K}$$ and the
light intensity variation is less than $1\%$ . Note that the ethanol-water
mixing is an exothermic process, which due to the local heating affects the
solubility and thus the oversaturation and the light intensity. We took 100
frames to obtain the averaged light intensity of the mixture. The water-adding
process lasted until the solution turned opaque enough, after which we
obtained the intensity-to-concentration graph, see the black points in Fig.
2(a). Based on the data points we performed a $2^{nd}$ order polynomial curve
fitting to obtain the titration curve, from which we define the 1$\%$ drop of
the light intensity as the point of saturation, as marked by the red star in
Fig. 2(a). We repeated such titration procedure for 25 different initial
compositions of the ethanol-oil mixture, with weight ratios $w_{e}:w_{o}$
ranging from $10:1$ to $300:1$, forming the binodal curve as shown in Fig.
2(b).

(a)

(b)

(c)

(d)

Figure 2: (a) Camera assisted titration to determine the saturation point for
a specific $w_{e}/w_{o}$. The black data points are obtained by averaging the
light intensity of the entire domain, and the error bar denotes the standard
deviation of the light intensity within the domain. The star marks water
concentration ($C_{\text{water}}$) causing 1% drop of the light intensity. (b)
A simplified ternary diagram to illustrate the titration process and the
constructed binodal curve.

(c) show the binodal curve and the theoretical diffusion path for the case
$w_{e}/w_{o}=33$. (d) The relation between the water concentration and oil
oversaturation can then be built from (c) for the following calibration.

Together with the diffusion path, the acquired binodal curve as in Fig. 2(b)
was then used to quantify the amount of oversaturation, see Fig. 2(c). Ruschak
& Miller (1972) first formulated the diffusion path theory, which was later
applied to the study of nanodroplets formation on surfaces (Lu et al., 2017;
Li et al., 2021). The diffusion path can be understood as the trajectory of
the local fraction of all three phases during the mixing process. As
rigorously evaluated in Li et al. (2021), we assume the diffusion coefficient
of water and oil to be the same, meaning that the diffusion path becomes a
straight line in the ternary diagram, namely the dashed arrows in Fig. 2(b).
For the ease of interpretation, Fig. 2(c) can be obtained by coordinate
transformation from the water-rich side of Fig. 2(b) to a Cartesian coordinate
system, mapping the binodal curve and the diffusion path. With the straight-
line assumption, the diffusion path depends solely on the initial ethanol-oil
ratio. While there is only one binodal curve for all the different
compositions, the diffusion path varies with the composition $w_{e}/w_{o}$.
For a specific water fraction, its saturation concentration is first
identified on the binodal curve in Fig. 2(c). Then we connect the saturation
point on the binodal curve to the end point of the pure oil phase (Li et al.,
2021), at coordinate (0,1), shown as the light green dashed line in Fig. 2(c).
The connection between the saturation point and the pure oil phase stems from
the consideration that the ethanol-to-water ratio in the mixture remains
unchanged in a fluid parcel during oil nucleation (Lu et al., 2016). From the
light green dashed line we can extract the length of the segment between the
diffusion path and the binodal curve as the oversaturation, displayed as the
green solid segment in Fig. 2(c).

Then the oil oversaturation is related to the water concentration as in Fig.
2(d). Fig. 2(d) not only displays the significant dependence of the
oversaturation on the initial composition, $w_{e}/w_{o}$, but also reveals the
existence of an upper bound for the oversaturation. The existence of the upper
bound might seem counter-intuitive in the first place, as one might expect a
monotonic increase with more water introduction. It is, however, not the case
in such a ternary liquid system. Further mixing with water beyond the critical
fraction makes the nucleated oil re-dissolve into the solution, lowering the
oversaturation. While Fig. 2(c) already reveals that the segment length seems
to decrease at larger water fraction, Fig. 2(d) clearly displays this fact,
which is also later confirmed by the recorded intensity in the concentration
calibration as shown in Fig. 3. The water concentration $C_{\text{water}}$ is
the selected value used in the concentration calibration, which will be
described below.

###  2.3 Light attenuation technique and calibration

To characterize the concentration of the nucleated oil in the flow, we
implemented a light attenuation technique. Unlike the passive scalar studies,
the droplet nucleation makes the flow almost opaque, causing the commonly used
laser induced fluorescence (LIF) not feasible. We therefore use a light
attenuation technique to measure the local concentration. This technique has
been widely adapted in turbulent jet and plume studies based on the pioneer
work of Cenedese & Dalziel (1998), from single phase cases with dye as in
Kikkert et al. (2007), Allgayer & Hunt (2012), Van Sommeren & Woods (2012),
and Mingotti & Cardoso (2019), to multiphase ones as in Leppinen (2001), and
Mingotti & Woods (2015). The method is based on the Lambert-Beer law, relating
the recorded depth-integrated intensity to the concentration field, namely

$\log\left(\frac{I}{I_{\text{ref}}}\right)=-2\gamma Cd,$ (1)

where $I$ is the recorded intensity of the averaged images, $I_{\text{ref}}$
the reference intensity of the background, $\gamma$ the light absorption
property of the dye determined from calibration, $d$ the depth of the
substance in the line-of-sight, and $C$ the desired line-of-sight average
concentration. The line-of-sight averaged concentration used in the
aforementioned papers, however, is not a solution for the nonlinear response
we see here. In a pioneering work, Sutherland et al. (2012) analyzed the
averaged image based on the axisymmetric discretization, converting the depth-
integrated recorded image to the concentration field as a function of radius
and height, which serves as a very good strategy for our interests to obtain
the oversaturation field as a function of radius and height in the domain.

We performed our calibration in a $10\text{\,}\mathrm{mm}$ thick cell as shown
in Fig. 3(a). The calibration cell is immersed into the tank filled with
water, and the illumination and the camera magnification are exactly the same
as in the jet experiments, ensuring an in-situ calibration. In spite of the
effort to regulate the fluid temperature during the calibration and before the
real experiments, the exothermic nature of mixing between water and ethanol in
the jet will rise the local temperature. Although the $local$ temperature
variation might alter the oversaturation, leading to less accurate
calibration, such deviation is simply beyond control or even monitoring during
the experiments. With more than $35\text{\,}\mathrm{L}$ of water in the
ambient and less than $60\text{\,}\mathrm{ml}$ injected fluid, it is safe to
assume that the large volume of water at
$$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$\pm$1\text{\,}\mathrm{K}$$ is
sufficient to mitigate the temperature increase quickly, and wait for future
numerical efforts to validate our assumption. As the reference case, the
calibration curve for the dyed ethanol is shown in Fig. 3(b), which was used
to calculate the concentration field for the dyed ethanol jet. Red food dye
(JO-LA) was added step-by-step followed by taking 200 frames to obtain the
averaged light intensity for every $5\times 5$ pixels working window. Although
the linear relation indeed holds within a certain range of dye concentration,
we fit a $2^{nd}$ order polynomial to increase its applicability. The
nonlinear calibration curve does not appear for the first time. Van Sommeren &
Woods (2012) also observed a nonlinear response in their experiment, which
they fitted with polynomials, confirming that in-situ calibration also
functions well with nonlinear optical response.

(a)

(b)

(c)

(d)

(e)

Figure 3: (a) Calibration cell. The red frame ($x=\ $$0\text{\,}\mathrm{cm}$,
$z=\ $$10\text{\,}\mathrm{cm}$) denotes the selected $5\times 5$ pixels
working unit whose calibration curves are shown in (b)–(e). (b) Calibration
curve for the reference dye case. $\Phi$ denotes the degree of light
attenuation per unit depth, $\log(\frac{I_{\text{ref}}}{I})/d$, where $d$ is
the cell thickness. (c,d) The calibration curve for the two ouzo cases. The
abscissa $C_{\text{oil,sat}}$ is the oversaturation of the oil. For (b)–(d)
the black error bar represents the standard error of the mean $\Phi$, which is
small throughout the calibration. (e) The rescaled calibration curves as in
(c), (d).

On the other hand, for the ouzo cases, the cell was first filled with a
mixture of ethanol and oil at the specific $w_{e}/w_{o}$ ratio, and water was
then added gradually. Averaging the intensity over 200 frames for every 5
$\times$ 5 pixels working window, the calibration curves as in Fig. 3(c) and
Fig. 3(d) were built for every working window locally in the entire domain,
which, however, reveals a completely difference calibration curve from the dye
case. The $x$-axis in these two figures, namely the oil oversaturation, were
derived from Fig. 2(d). The upper bounds for the ouzo calibration curve in
Fig. 3(c) and Fig. 3(d) are imposed by the theoretical maximum oversaturation
shown in Fig. 2(d), namely 0.0023 for $w_{e}/w_{o}=100$, and 0.0087 for
$w_{e}/w_{o}=33$. Such theoretical upper bounds are only valid if the
diffusion path does not deviate too much from the straight line assumption as
in Fig. 2(c). In the calibration, we did observe that the light attenuation
level and the oversaturation decreased as more water was added, see the
cluster of data points close to the peak in Figs. 3(c) and 3(d).

For the case of ouzo, the combination of a logistic function and a linear
function fits our calibration data well, written as

$\Phi(C)=\frac{\log\left(\frac{I_{\text{ref}}}{I}\right)}{d}=\frac{a_{0}}{1+a_{1}e^{a_{2}(C-a_{3})}}+\frac{a_{4}(C-a_{5})}{1+a_{1}e^{a_{2}(C-a_{3})}},$
(2)

where $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, and $a_{5}$ are the fitting
parameters, and $C$ is the oil oversaturation, equivalent to the $C_{oil,sat}$
in Fig. 3(c). Note that other fits are possible, but for our purpose we simply
need a good approximation of the experimental curve $\Phi(C)$. To compare the
calibration curve for the two $w_{e}/w_{o}$ ratios, the oversaturation and the
attenuation were both rescaled, showing a reasonable collapse, see Fig. 3(e).
The abscissa $\widetilde{C}_{\text{oil,sat}}$ in Fig. 3(e) is the rescaled
oversaturation, which is obtained from
$\widetilde{C}_{\text{oil,sat}}=(C_{\text{oil,sat}}-C_{\text{thresh}})/(C_{\text{max}}-C_{\text{thresh}})$,
where $C_{\text{thresh}}$ is the oversaturation where the attenuation reaches
0.1% of the highest attenuation as indicated by the black star in (c), (d),
and $C_{\text{max}}$ is the theoretical upper bound discussed earlier. The
rescaled oversaturation also eliminates the flat section of the logistic
function for low oversaturation, creating a one-to-one injective map for the
calibration. Note that the reason behind such nonlinear optical response is
beyond the scope of this research and remains unclear. Fig. 4(a) shows a
snapshot of the turbulent jet with oil droplet nucleation, while Fig. 4(b) is
the averaged image over 2000–2500 frames as that in Fig. 4(a). The intensity
field of a certain height is only included in the averaging operation after
the jet has reached that height. The averaged image contains the upstream
section only with $z\leq$ $30\text{\,}\mathrm{cm}$ since the low density dark
layer accumulated on top of the tank grew and interfered with the ascending
jet, causing the intensity field above $z=$ $30\text{\,}\mathrm{cm}$ to be
averaged relatively poorly (less than $20\text{\,}\mathrm{s}$). Fig. 4(c)
shows a horizontal cross-section of the jet, illustrating the discretization
scheme similar to Sutherland et al. (2012), where $dy_{mn}$ can be directly
obtained as a function of $dr$ solely based on geometric relations, where $m$
denotes $x$ position on the recorded images, and $n$ the axisymmetric elements
in $r$ direction. Note that $m\leq n$ always holds, and we calculate the left
half ($x<0$) and right half ($x>0$) of the jet separately, that is $C_{n}\neq
C^{*}_{n}$. While the concentrations $C_{n}$ remain as unknowns, the recorded
intensity $I_{m}$ at a specific location from Fig. 4(b) can then be expressed
as the integration of the converted intensity calculated from the calibration
curve over every discretization element on the light path. For the reference
dye case with the $2^{nd}$ order polynomial calibration curve, the
concentration can be obtained by minimization of $m$ equations with $n$
variables, written as

$\log{\left(\frac{I_{ref}}{I_{m}}\right)}=\sum_{n=0}^{N-1}dy_{mn}\left(a_{0}+a_{1}C_{n}+a_{2}C_{n}^{2}\right),$
(3)

while for the ouzo cases with calibration curve as Eq. (2), the equations for
minimization are written as

$\log{\left(\frac{I_{ref}}{I_{m}}\right)}=\sum_{n=0}^{N-1}dy_{mn}\left(\frac{a_{0}}{1+a_{1}e^{a_{2}(C_{n}-a_{3})}}+\frac{a_{4}(C_{n}-a_{5})}{1+a_{1}e^{a_{2}(C_{n}-a_{3})}}\right)$
(4)

The calculation procedure is repeated for every height, leading to the
oversaturation fields as in Fig. 7. We detail the procedures of the
optimization process and the evaluation of the results in Appendix A.

We add a note of care: The superposition of attenuation level across the
layers of axisymmetric unit illustrated in Fig. 4(c) might be questionable for
a scattering medium. The summation in eq. (4) is fully applicable if the level
of attenuation is proportional to the depth of the medium; however, such
algorithm does not hold when the light propagates through a scattering medium.
Despite that our scattering is relatively low, the proposed method and the
obtained results in this paper should be further evaluated in the future. We
will discuss this issue in more detail in Appendix C.

(a)

(b)

(c)

Figure 4: (a) The instantaneous image obtained by dividing the recorded frame
by the background frame. (b) The mean image derived by averaging the frames as
in (a) over $30\text{\,}\mathrm{s}$. $Re_{0}=1387$ in the image.

(c) An horizontal cross-section of the axisymmetric discretization, similar to
that in Sutherland et al. (2012). Here we do not impose symmetry for the left
half ($x<0$) and the right half ($x>0$) of the jet. The averaged image in (b)
can be converted to a concentration field by the calibration curve in Fig. 3
for every 5 $\times$ 5 pixels, together with the optimization algorithm
constructed based on the scheme in (c).

###  2.4 Droplet size detection

The nucleated droplets through solvent exchange are generally referred to as
micro- or even nano-sized droplets. However, experimentally measured sizes are
based on cases with no flow (Tan et al. (2019)) or laminar flow (Hajian &
Hardt, 2015; Lu et al., 2015). To measure the sizes of the nucleated droplets
in the case of a turbulent jet, we utilize a Navitar 12 $\times$ objective on
a Photron FASTCAM Mini AX200 camera, achieving a resolution of $3$
$\mu$m/pixel, see Fig. 5. Fig. 5(a) clearly shows the nucleated droplets in
the plume after the injection stopped, which is similar to the laminar flow
scenario in the aforementioned literature. Comparing Figs. 5(b) and 5(c), we
see that the droplets were indeed micro-sized ($\mathcal{O}(10$ $\mu$m)) for
both turbulent cases. The droplets for $w_{e}/w_{o}=33$ are larger and thus
easier to identify, which is in line with the findings of Vitale & Katz
(2003).

(a)

(b)

(c)

Figure 5: Identification of nucleated droplets. For comparison, we show a
droplet of 25 $\mu$m diameter in all three cases. (a) The nucleated droplets
for $w_{e}/w_{o}=100$ when the injection just stopped (i.e.,$u_{z}$ approached
$0$). (a) serves as a reference laminar case where the droplet nucleation can
be easily identified. (b) The nucleated drops of a jet with $w_{e}/w_{o}=100$
in the turbulent regime for $Re_{0}=555$. The black thin sheets are the fluid
interfaces due to refraction, while the black dots in the inset are the
nucleated droplets with 17 $\times$ magnification, which are obviously smaller
than $25$ $\mu$m. (c) The nucleated droplets for $Re_{0}=555$ and
$w_{e}/w_{o}=33$, which are larger and easier to identify

###  2.5 PIV

While the light attenuation technique resolved the mean oversaturation field,
we relied on particle image velocimetry (PIV) to measure the velocity field.
Ethanol seeded with 20–$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ fluorescent
particles was injected into the water-filled tank, which was also seeded with
the same particles in order to prevent measuring a velocity that is solely
based on entrained or injected liquid. Note that we used the obtained velocity
field of the ethanol jet to represent that of the ouzo jet, assuming that the
micro-sized nucleated oil droplets can be considered as tracer particles which
barely affect the velocity field. The Photron cameras were replaced with
LaVision Imager sCMOS cameras with resolution 2560 $\times$ 2160 pixels,
capturing the image pairs on the $1\text{\,}\mathrm{mm}$ thick laser sheet
created by a dual cavity laser (Quantel Evergreen 145 laser,
$532\text{\,}\mathrm{nm}$). The velocity fields are calculated using the
multi-pass method with a starting window size 64 $\times$ 64 pixels to a final
size 16 $\times$ 16 pixels with 25$\%$ overlap.

Considering that the velocity decreases drastically as the jet travels
downstream, the experiments were performed with two different time intervals
$dt$ between image-pairs to resolve both the far field and the near field,
followed by averaging over 1000 frames to obtain the mean velocity fields.
Note that the region below $z=$ $3\text{\,}\mathrm{cm}$ cannot be resolved
because the jet is thinner than the $1\text{\,}\mathrm{mm}$ laser thickness.

Obtained from the velocity field, Figs. 6(a), 6(c) show the mean centerline
evolution of velocity in $z$-direction, $u_{m}$. Theoretical models prediction
for a pure jet and a pure plume are also presented:

$\displaystyle u_{\text{m,jet}}=4.2M^{1/2}z^{-1},$ $\displaystyle
u_{\text{m,plume}}=3.2B^{1/3}z^{-1/3},$ (5)

where $M=Q^{2}/(\frac{1}{4}\pi d^{2})$ is the initial momentum flux, and
$B=Qg(\rho_{\text{jet}}-\rho_{\text{amb}})/\rho_{\text{amb}}$ is the initial
buoyancy flux, see the caption in Table 1.

The coefficients obtained from our measurements are lower than the values in
the fully turbulent jet and plumes, which are 5–7.5 for jet and 3.4–3.9 for
plume (Fischer et al., 1979; List, 1982). We attribute such difference to the
relatively low Reynolds number in this study, where the jet regime lies right
after the laminar-to-turbulent transition.

(a)

(b)

(c)

(d)

Figure 6: The mean velocity field obtained by PIV measurements at (a,b)
$Re_{0}=555$, and (c,d) $Re_{0}=1387$ . (a,c) The centerline velocity in
$z$-direction, $u_{m}$, from the theoretical model and the valid measurements
of the both fields. (b,d) The measured and fitted radial profile of the
streamwise velocity $u_{z}$. Kur in the legend is abbreviation for kurtosis.

Such comparison with the models reveals that the initial ethanol jet quickly
transitioned toward a plume under strong buoyancy effect. Figs. 6(b), 6(d)
show the radial flow profile of the streamwise velocity $u_{z}$, which are
later used to calculate the oversaturation flow rate as in Fig. 13.

##  3 Oversaturation field

Following the procedure detailed in §2.3, the optimization problem was solved
to construct the oversaturation field, Fig. 7. As shown in Figs. 3a–3d, the
calculated $\widetilde{C}_{\text{oil,oversat}}$ in most of the domain fall
below 1. Such results suggest that the aforementioned procedures in §2 work
decently, including titration, straight diffusion path assumption,
calibration, and the optimization. As for the region very close to the needle
in the near field, the local oversaturation exceeds the theoretical upper
bound, leading to $\widetilde{C}_{\text{oil,oversat}}>1$. While the actual
cause of such results remains uncertain, we believe that the most probable
cause is the violation of the straight diffusion path assumption in that
region. Such violation might have resulted from the intense mixing and cross-
component diffusion upon the onset of turbulence in the initial shear layer.

As the far view results at the initial shear layer might be biased by the
resolution constraint, the results obtained from zoomed-in recordings are
presented in Figs. 3e–3h, corresponding to the regions marked by the black
frame in Figs. 3a–3d. Although the oversaturation might not be quantitatively
correct in this region, the zoomed-in view results further confirm the intense
nucleation in the initial shear layer. In the study of aerosol formation in
turbulent gas jets, Lesniewski & Friedlander (1998) also pointed out that the
nucleation of the target substance is largely confined to the initial shear
layer, especially when the vapor mole fraction of the target substance is low.
While the system of the current work is different from that of the aerosol
study, the two processes both couple the variation of thermodynamic states and
ambient turbulent shear flow, leading to the nucleation of droplets or
particles.

Considering the opaque nature of the ouzo jet flow, the calculated results
cannot be verified by other concentration detection methods such as LIF.
Bearing the limitation in mind, what we can do is to check the consistency of
the results by repeatability tests and comparing between the far view and the
zoomed-in view results obtained from two different cameras. While the
repeatability will be demonstrated in §4, Fig. 16 in Appendix B shows a
quantitative comparison between the two recordings using the centerline
evolution and the radial profile. As detailed in Appendix B, for region with
$\widetilde{C}_{\text{oil,oversat}}\leq 1$, the proposed method delivers
consistent results between the far view and zoomed-in view recordings.

Note that the turbulence started to develop after a certain laminar length
(Hassanzadeh et al. (2021)), which is subtracted to obtain the virtual origin
in $z$-direction for the analysis in §4. The concentration fields for the
dyed-ethanol injection were also obtained by the same method, which are not
shown here but will be exploited as reference cases in the analysis in §4.

Figure 7: Calculated rescaled oversaturation fields for different $Re_{0}$ and
$w_{e}/w_{o}$. Panels (a–d) show the results derived from the far view camera,
while (e–h) exhibit the results obtained from the zoomed-in camera, which
correspond to the black square in (a–d), respectively, with the same color
scheme below $\widetilde{C}_{\text{oil,oversat}}$=1. The white spots in the
contour map are caused by the local minima found during optimization, which
poses little effect on the analysis in §4 as we exclude those defects for
curve-fitting. Note that the abscissa is converted from $x$ to $r$ using the
axisymmetric transformation in Fig. 4(c)

.

##  4 Results

Figure 8: The radial oversaturation profiles for different $Re_{0}$ and
$w_{e}/w_{o}$. The dots are the data points and the dashed lines are curves
fitted with Eq. (6). The table next to the figure gives the obtained fitting
parameters $\sigma/z$ & $\kappa$ for a variety of heights.

###  4.1 Generalized normal distribution

To better capture the radial profile of oversaturation field in Fig. 7, we
fitted the results by a generalization of a normal distribution,

$C(r)=C_{m}e^{-\frac{|r|^{\kappa}}{\kappa\sigma^{\kappa}}}+C_{0},$ (6)

where $C_{m}$ is the centerline oversaturation, where $\sigma$ is the width of
the jet, $\kappa$ the parameter to characterize the potential non-Gaussian
profile, and $C_{0}$ the background offset of the oversaturation, which is
also very close to zero. When $\kappa=2$ the profile becomes the standard
Gaussian case. Fig. 9 showcases the quality of the fitting, where the center
received less weight than the outer region based on the amount of information
contained, that is, the area of the corresponding discretized annulus region
in Fig. 4(c). Also, we exclude the data at any specific height with less than
5 data points to optimize the quality of the fitting. The fitted variables and
the fitted profile are then used in the following sections to analyze the
evolution of the centerline oversaturation and the jet profile.

As discussed, $\widetilde{C}_{\text{oil,oversat}}>1$ in Fig. 7 suggests that
the data there is less reliable. We hope to obtain the profile and centerline
value only based on the reliable data points, Therefore, data points with
$\widetilde{C}_{\text{oil,oversat}}>1$ are also excluded from curve-fitting,
but larger fitted $\widetilde{C}_{\text{oil,oversat}}$ is allowed. Such
fitting strategy somehow affects the centerline value in the near field, where
$\widetilde{C}_{\text{oil,oversat}}>1$. We present and discuss the difference
in centerline value between the minimization and the following curve-fitting
in Appendix B. The current method cannot work properly in the initial shear
layer; we would like to remind the readers that the results in that region
cannot be interpreted by our curve-fitting procedures nor our minimization
procedure, we therefore should keep this in mind when interpreting the
analysis in §4.

###  4.2 Radial profile

In the classic jet and plume studies, the jet width of the concentration field
$b_{T}(z)$ is usually determined by the position where the value drops to
$1/e$ of the centerline value. For a self-similar flow, $b_{T}(z)/z$ is a
constant at around $0.127$ for pure jet, and $0.12$ for pure plume (Fischer et
al., 1979). When the radial flow profile is Gaussian, the jet width determined
from the definition above, $b_{T}(z)$, and $\sigma$ obtained from Eq. (6) are
the same. However, from Fig.9 we can tell the radial profile is non-Gaussian
for the ouzo cases, therefore we implemented the traditional definition of
$b_{T}(z)$ to characterize the jet width and plot the evolution of
$b_{T}(z)/z$ in Figs. 9(a) and 9(b). While the reference dye cases match well
with the constant value reported in the literature (Fischer et al., 1979), the
ouzo cases exhibit a continuously decreasing trend after the initial shear
layer, starting with twice the value of the reference case and approaching the
reference value as the jet evolved downstream. The larger jet width suggests a
wider spread of the nucleation front, following the turbulent entrainment and
mixing at the meandering turbulent/non-turbulent interface (TNTI) (Westerweel
et al., 2009; Watanabe et al., 2015a). Note that we study the averaged density
profile, nucleation visible in instantaneous images is very likely to not be
in the center and continuously moving around. Neuber et al. (2017) numerically
investigated the nucleation and condensation of aerosol in gas turbulent jets,
showing that the largest supersaturation level and nucleation rate are both
not located in the centerline. While we do believe that the nucleation is most
intense at the TNTI, the radial dispersion of the droplets nucleated there and
the streamwise dispersion of the dense droplets nucleated in the initial shear
layer contribute to the relatively higher centerline value in the mean
profile. As the oil, originally dissolved in the injected fluid, nucleates and
gradually gets consumed with increasing $z$ the ratio $b_{T}(z)/z$ also
approaches the reference dye case. Note that the spatial evolution of
$b_{T}(z)/z$ is nearly independent of $Re_{0}$ and $w_{e}/w_{o}$ upstream,
while the dependence appears downstream. Higher initial oil fraction and
higher $Re_{0}$ can both extend the wider profile further downstream.

Figs. 9(c) and 9(d) show the jet width ratio between the oversaturation
profile and the velocity profile. The ratio for the reference cases reveals
that the concentration spreads slightly wider than the momentum, namely
$b_{T}/b_{u}$ slightly larger than 1, which is consistent with the literature
value. For the ouzo jets, the jet width ratio is roughly two times larger
upstream, and gradually decreases as the flow evolves downstream.

(a)

(b)

(c)

(d)

Figure 9: The radial oversaturation profile of the buoyant jets. The errorbars
represent the uncertainty of the curve fitting discussed in §4.1. Panels (a)
and (b) show the spatial evolution of the jet width ($b_{T}(z)$) divided by
the vertical distance $z$ by probing the radial position where the
concentration reach 1/e of the fitted peak. For each parameter set we show the
results of two repeated experiments, denoted by subscripts a, and b. (c,d)
display the jet width ratio between the oversaturation (concentration) profile
$b_{T}$ and the velocity profile $b_{u}$.

(a)

(b)

Figure 10: Kurtosis calculated from the fitted parameter $\kappa$ for (a)
$Re_{0}=555$ and (b) $Re_{0}=1387$

As the ratio $b_{T}(z)/z$ for the ouzo jet is larger than the dye case, we
wonder how the nucleation changes the distribution of the radial profile. To
characterize the radial profile, we calculate the spatial evolution of the
kurtosis from the fitted parameter $\kappa$ of Eq. (6), see Figs. 10(a) and
10(b) for the results. We can interpret the concentration profile in Eq. (6)
as a probability density function of the concentration $\text{PDF}(C)$ for
which we find that $C_{m}$, $\sigma$, and $\kappa$ are constraint in such a
way that $\int\nolimits_{-\infty}^{\infty}\text{PDF}(C)dr=1$, as per the
definition of the PDF. From that we can express $C_{m}$ in terms of $\kappa$
and $\sigma$. The kurtosis can then be calculated from its definition in terms
of the 2${}^{\text{nd}}$ and 4${}^{\text{th}}$ central moments:

$\displaystyle\text{PDF}(C)$
$\displaystyle=\frac{\kappa^{-1/\kappa}}{2\sigma\Gamma\left(1+\frac{1}{\kappa}\right)}e^{-\frac{|r|^{\kappa}}{\kappa\sigma^{\kappa}}}$
(7) kurtosis
$\displaystyle=\frac{\int\limits_{-\infty}^{\infty}r^{4}\text{PDF}(C)dr}{\left(\int\limits_{-\infty}^{\infty}r^{2}\text{PDF}(C)dr\right)^{2}}=\frac{\Gamma\left(\frac{1}{\kappa}\right)\Gamma\left(\frac{5}{\kappa}\right)}{\Gamma\left(\frac{3}{\kappa}\right)^{2}},$
(8)

where $\Gamma(x)\equiv\int\nolimits_{0}^{\infty}\eta^{x-1}e^{-\eta}d\eta$ is
the (complete) gamma function. While the kurtosis for the dye cases is around
3 as expected for a Gaussian distribution, for the ouzo cases it starts with a
very high value below $z=$ $3\text{\,}\mathrm{cm}$, which corresponds to the
initial shear layer in Fig. 7 with $\widetilde{C}_{\text{oil,oversat}}>1$, and
thus we remain doubtful about claiming a regime with super-Gaussian kurtosis.
Then the kurtosis sharply decreases to a value close to 2, entering a regime
with sub-Gaussian kurtosis, and only slightly increases and approaches 3 as
the flow develops.

The spatial evolutions of the jet width and of the kurtosis in the far field
both reveal the radially spread nucleation front following the TNTI in the
ouzo jets, whose effect becomes weaker downstream with the consumption of the
dissolved oil. Compared with the reference case, the profile with sub-Gaussian
kurtosis suggests a wider radial distribution, leading to a more uniform
concentration distribution in the domain. Such distribution is in line with
the finding of reactive mixing study reported in Guilbert & Villermaux (2021),
where the reaction makes the gamma probability distribution of the product
narrower, expediting the mixing process toward uniformity.

###  4.3 Centerline scaling

The centerline evolution of the concentration has always been the focus of
turbulent jet and plume studies, as the scaling between the dilution and
streamwise length is crucial to the self-similarity of the flow. In Fig. 11 we
present the centerline evolution for the dyed ethanol jet and the ouzo jet for
different $Re_{0}$. As the dye concentration dropped monotonically due to
dilution, the oversaturation of the ouzo jets experienced several stages, from
increasing sharply right above the needle, tightly followed by a sudden
decrease, and finally entering a mild dilution regime. In Figs. 11(a) and
11(b) we normalized the height $z$ with the characteristic length scale for
the jet-plume transition, $L_{m}=M^{3/4}/B^{1/2}$, where
$M=Q^{2}/(\frac{1}{4}\pi d^{2})$ is the initial momentum flux, and
$B=Qg(\rho_{\text{jet}}-\rho_{\text{amb}})/\rho_{\text{amb}}$ is the initial
buoyancy flux, see the caption of Table 1. For $z/L_{m}<1$, the flow is
characterized as momentum dominated jet, followed by a gradual transition to
the buoyancy-dominated plume as the flow reaches $z/L_{m}=5$. Figs. 11(a) and
11(b) display the centerline evolution for $Re_{0}=555$ and $Re_{0}=1387$. For
the ouzo cases, we can clearly identify three different regimes: (I) a fast
nucleation stage where the concentration climbs up quickly. (II) a fast
dilution stage, which is only absent for the $Re_{0}=555,$ $w_{e}/w_{o}=100$
case, and (III) the mild dilution stage in the downstream location, where the
nucleation is competing with the dilution. A higher initial oil composition
(smaller $w_{e}/w_{o}$) induces more intense nucleation in the near field,
contributing to the sharper variation in stage (I) and (II) for both $Re_{0}$.
In stage (III), however, the centerline evolution is nearly independent of
$w_{e}/w_{o}$.

$Re_{0}$ determines the momentum-buoyancy competition, and in turn the regime
where different stages occurs, that is for $Re_{0}=555$ the jet develops into
a pure plume regime quickly ($z>5L_{m}$), while the initial momentum plays a
major role for $Re_{0}=1387$. Despite the difference in the regime where the
three stages are located, the evolution of the centerline oversaturation only
depends weakly on $Re_{0}$ if the abscissa is not normalized, see Fig. 11(c).
The figure reveals that the effects of $Re_{0}$ and $w_{e}/w_{o}$ are only
visible in stage (I) and (II) , while the data from various conditions nearly
collapse in stage (III), as $Re_{0}$ determines the momentum-buoyancy
competition which affects the strength of turbulent entrainment, it seems that
the centerline oversaturation cannot fully characterize the turbulent
entrainment effect on the nucleation. This is not surprising as we believe
that the nucleation front follows the TNTI, located at the rim of the jet. In
§4.4 we will show that the effect of the turbulent entrainment on the
nucleation is more prominent in the evolution of the oversaturation flow rate
which covers the entire domain.

(a)

(b)

(c)

Figure 11: Normalized centerline evolution of the rescaled oversaturation for
dyed ethanol, ouzo $w_{e}/w_{o}=100$, and ouzo $w_{e}/w_{o}=33$. (a) and (b)
shows results for $Re_{0}=555$ and $Re_{0}=1387$, respectively, with
normalized height as abscissa. (c) contains all the results in one plot with
original height as abscissa. $C_{\text{max}}$ is the initial dye concentration
for dye case and the theoretical peak for ouzo cases. The errorbars represent
the uncertainty of the curve fitting discussed in §4.1.

To retrieve quantitative information from the centerline evolution, it is
crucial to know how the oversaturation scales with the height $z$. Therefore
the curves with different conditions in Fig. 11 are separated, while repeated
experiments with the same control parameters are ensemble-averaged, leading to
Figs. 12(a) through 12(f), which are then fitted locally using a moving window
to derive the local scaling exponent $\beta$ as
$C_{m}(z)/C_{\text{max}}\propto(z/L_{m})^{\beta}$. Figs. 12(g) and 12(h) show
that the dye cases match the value in the literature, starting with $\beta=-1$
for the pure jet and then gradually approached the pure plume value
$\beta=-1.66$. For the ouzo jets, the multiple stages mentioned above are
clearly visible, with $\beta>0$ in stage (I) initial production stage, then
stage (II) a sharp drop to $\beta<-2$ in a short range, in the end stage (III)
reaching a mild dilution regime with $\beta$ slightly below $0$. These
transitions reveal the competition between the nucleation and the dilution.
Focusing on stage (III), the smaller magnitude of $\beta$ for ouzo jets
clearly demonstrates the ongoing and prolonged nucleation even at downstream
location. Such finding is similar to the studies in aerosol formation in a
turbulent jet by Lesniewski & Friedlander (1998). They attributed the less
steep centerline decay to the nucleation beyond the initial shear layer. We
note that the drastic change and the extreme values in the stage (I) and (II)
might not be quantitatively accurate, as they are located in the intense
nucleation region shown in Fig. 7, indicating the violation from the straight
diffusion path assumption.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 12: Centerline evolution of the oversaturation. Panels (a)–(f) present
the centerline evolution in all cases in Fig. 11. The curve in each plot is
then locally fitted with moving window to deliver the best fitting results.
The length of the fitting window is defined as $h_{\text{fit}}$, which starts
with $h_{\text{fit}}=0.1$$z/L_{m}$ in the fast nucleation regime, changing to
$h_{\text{fit}}=$ $z/L_{m}$ between the peak and $z=5L_{m}$, then ending with
$h_{\text{fit}}=3$$z/L_{m}$ for $z>5L_{m}$. The results of the fitting are
shown in (g) and (h). $\beta$ denotes the local scaling exponent of
$C_{m}/C_{\text{max}}\propto z^{\beta}$. The ouzo cases can be segmented into
three stages as detailed in the caption of Fig. 11. In the dilution stage, the
exponent $\beta$ ($\beta<0$ there, representing decaying) is identical for
both $w_{e}/w_{o}$, and is significantly smaller (i.e., less decaying) than
the reference dye case. The fluctuation in the end (large $z$) were probably
caused by insufficient averaged frames downstream as described in §2.3.

###  4.4 Oversaturation flow rate

(a)

(b)

Figure 13: Mass flow rate evolution along the height. The errorbars show the
propagated uncertainty originating from the concentration field. (a) shows the
mass flow rate $Q_{dye}(z)$ of reference dye case calculated using Eq. (10),
which is reasonably conserved within our domain. The gray dashed lines are the
expected values with the initial dye concentration and the volume flow rates.
(b) displays the rescaled oversaturation flow rate for the ouzo cases with two
different $w_{e}/w_{o}$ and two different $Re_{0}$, calculated using Eq. (9).
The height $z$ is normalized by $L_{m}$ to show the variation with jet-plume
transition. Each legend represents the case with 2 repeating experiments. The
results are reproducible while the dependence on $Re_{0}$ is significant. For
$z/L_{m}\leq 5$ we see the curves between the two compositions are very close,
while the composition makes a difference downstream, especially for
$Re_{0}=555$.

To make use of the oversaturation data in the entire domain (see Fig. 7), we
calculated the rescaled oversaturation flow rate by multiplying the
oversaturation field, the velocity field from Fig. 6, and the corresponding
area within the discretized annulus region, namely the rescaled oversaturation
flow rate

$\displaystyle\tilde{Q}_{oil}(z)=\sum_{r=0}^{(n-1)dr}\tilde{C}_{\text{oil,sat}}(r,z)u_{z}(r,z)A_{\text{annu}}(r)$
(9)

Similarly, we can obtain the mass flow rate for the dye reference cases,

$\displaystyle
Q_{dye}(z)=\sum_{r=0}^{(n-1)dr}C_{\text{dye}}(r,z)u_{z}(r,z)A_{\text{annu}}(r)$
(10)

Here $\tilde{C}_{\text{oil,sat}}$ is the rescaled oversaturation from Fig. 7
(dimensionless) for the ouzo cases and $C_{\text{dye}}$ is the concentration
field for dye cases, $u_{z}(z)$ the streamwise velocity measured by PIV as in
Fig. 6, and $A_{\text{annu}}(r)$ the area of the axisymmetric elements as in
Fig. 4(c).

For the standard scenario without any nucleation or depletion, the mass flow
rate should be conserved in the streamwise ($z$) direction, $Q(z)=const$,
which is confirmed in our reference dyed ethanol cases in Fig. 13(a). The mass
flow rates also match well with the expected values calculated by initial dye
concentration (4000 ppm) and the volume flow rates, which are shown by the
gray dashed lines. Fig. 13(b), on the other hand, shows the continuous
increase of the oversaturation flow rate before reaching the peak. This very
nicely demonstrates and quantifies the nucleation. The repeatability of the
results across different trials is good, firmly indicating that the nucleation
from the solvent exchange process outperformed the dilution within the domain,
as both effects resulted from turbulent entrainment. Such finding contrasts
the behavior of the centerline scaling exponent $\beta(z)$ in Fig. 12, as
$\beta(z)$ did not become larger than $0$ downstream. The increasing
oversaturation flow rate evolution is consistent with the findings in the
previous sections, namely the wider radial oversaturation profile and the
reduced dilution downstream. Also, the significant dependence of
$\tilde{Q}_{oil}(z)$ on $Re_{0}$ demonstrates the crucial role of the
entrainment in the solvent exchange process in the jet, which is not evident
in the centerline evolution. We can therefore infer that the oversaturation
flow rate characterizes the mean concentration field better than the
centerline evolution for this study.

For $z/L_{m}\leq 2$, that is, before the buoyancy dominates over the momentum,
the experimental data for the two employed compositions $w_{e}/w_{o}$ for both
$Re_{0}$ almost collapse, suggesting a potentially universal evolution for the
solvent exchange process in this regime. Despite the collapse in the jet
regime, the $w_{e}/w_{o}=100$ cases deviate from the $w_{e}/w_{o}=33$ ones
with increasing buoyancy dominance, reaching a peak, and even start to decline
somewhere downstream for $Re_{0}=555$. The peak signals the end of nucleation,
and the later decline of the curve reflects that the dissolution takes over
from nucleation. As discussed in §2.2 and in Fig. 2(d), when sufficient amount
of water is entrained and mixed with the jet fluid, the local water fraction
exceeds the critical value corresponding to the upper bound of oversaturation,
leading to the dissolution of the nucleated droplets back into the ternary
liquid system.

##  5 Discussion

With the experimental results above, we aim to build up a theoretical
framework to capture the observed features, or at least to enable a
qualitative explanation of the findings. In the buoyant jet, the turbulent
entrainment of water into ethanol-oil mixture is a crucial mechanism for the
solvent exchange process, as the nucleation will be activated once the
entrained water starts to mix with the injected fluid, forming the ternary
system consisting of water, ethanol, and trans-anethole. Incorporating the
established turbulent entrainment model, the $z$-direction evolution of the
compositions of the elements in the ternary system can easily be obtained. The
volume flow rate for a jet ($i=j$) or a plume ($i=p$) is obtained by

$\displaystyle q_{i}(z)=\pi u_{m,i}(z)(b_{w,i}(z))^{2},$ (11)

where $b_{w,i}=0.115z$ is the jet width for the velocity field from the PIV
measurements, and $u_{m,j}(z)=4.2M^{1/2}z^{-1}$ for the jet, and
$u_{m,p}(z)=3.2B^{1/3}z^{-1/3}$ for the plume from Eq. (5). This results in
the volume flow rates

$\displaystyle q_{j}(z)=0.17M^{1/2}z,$ $\displaystyle
q_{p}(z)=0.13B^{1/3}z^{5/3},$ (12)

for the jet and plume, respectively. Here $M$ and $B$ are the initial momentum
flux and buoyancy flux, respectively. Note that we directly combined the two
models in the middle of the transitional range, $z/L_{m}=3$. This procedure is
far from perfect but sufficient for a qualitative investigation. With the
volume flow rate of the entrained water $q_{i}(z)$, $i=j,p$, and the volume
flow rate of the injected ethanol-oil mixture $q_{0}$, the oversaturation
$C_{oil,sat}(z)$ can be calculated as

$\displaystyle C_{w}(z)=(q_{i}(z)-q_{0})/(q_{i}(z)-q_{0}+\rho_{e}q_{0}),$
$\displaystyle C_{o}(z)=\frac{1-C_{w}(z)}{1+w_{e}/w_{o}},$ $\displaystyle
C_{\text{oil,sat}}(z)=C_{o}(z)-C_{\text{o,binodal}}(z),$ (13)

where $C_{w}(z)$ is the weight fraction of water in the three phase mixture
during the process, $C_{o}(z)$ is the weight fraction of oil in the mixture,
and $C_{\text{o,binodal}}(z)$ denotes the saturation concentration of the oil
from the binodal curve shown in Fig. 2(c). Fig. 14(a) shows the spatial
evolution of the oil oversaturation. Although the sharp variation upstream
also appeared in the experimental results in Fig. 11, the variation here is
more pronounced. Also, the oversaturation shows a continuous sharp decrease
throughout the entire domain, reflecting that the so-called mild dilution
stage in the experimental data is absent in our simplified model. Multiplying
the results in Fig. 14(a) with the corresponding volume flow rate, as the
procedure in Eq. (9), the overall oversaturation flow rate as in Fig. 13(b)
can be examined from a theoretical perspective, which is shown in Fig. 14(b).
Unlike the (nearly) monotonic increase in Fig. 13(b), the modelled cases here
exhibit extremely intense nucleation right above the needle, and the
oversaturation flow rate is nearly conserved for almost the entire domain,
which indicates the absence of either nucleation or dissolution. The modelled
results are similar to findings of Lesniewski & Friedlander (1998) regarding
aerosol formation. They identified that the mass flow rate is conserved after
the initial shear layer if the nucleation is confined in that area, which is
obviously not the case in our study, where we find that nucleation continues
to persist throughout the flow.

(a)

(b)

Figure 14: Analytical results obtained using the entrainment-only model. (a)
is determined by the scale analysis, while (b) is obtained by multiplying
results of (a) with the volume flow rate, and thus sharing the same plot
legend. (a) can be a reference case to the centerline evolution in Fig. 11. If
the entrained water mixed fully with the jet fluid, the nucleation would
complete not far from the needle, which is not observed in the experimental
findings. Note that the sharp drop in the figure results from the direct
bridging of the volume flow rates for a jet and a plume as mentioned earlier,
see Eq. (12). On the other hand, (b) can be compared with Fig. 13(b), showing
the plateau instead of the monotonic increase using the entrainment-only
model.

As the entrainment is the only mechanism we include in the model, the
significant deviation from the experimental results unveils that additional
factors must contribute to the observed phenomena. Mingotti & Cardoso (2019)
considered mixing as the bottleneck for the fast reactive plume, which led to
the prolonged depletion of the injected liquid before reaching stoichiometric
level. Mingotti & Cardoso (2019) pointed out that mixing was constrained
within the Batchelor scale located at the rim of the eddies in the reactive
plume, causing the incomplete reaction for those entrained fluid. Despite the
simplicity of our model lacking the mixing effect for the solvent exchange
process, we argue that the weak dilution stage in Fig. 11 and the monotonic
increase in Fig. 13(b) are direct consequences of the mixing limitation,
considering that the time scale for the fast reaction in Mingotti & Cardoso
(2019) and the micro-droplet nucleation in our study are both extremely short
(Tan et al. (2019)). We note that the mixing-limited phenomena not only
appeared in a fast reactive plume, but also in other types of multiphase flow,
for example the evaporation of droplets in dense sprays (de Rivas & Villermaux
(2016), Villermaux et al. (2017)) and in a respiratory puff (Chong et al.
(2021)), where the lifetime of the droplets were controlled by the mixing in
the ambient humidity field. In our experiments, the limited mixing prevented
the oil in the injected fluid from completely nucleating in a short range
above the virtual origin, allowing the dissolved oil to be widely distributed
by the eddies before nucleation, leading to the wide-spread nucleation across
the jet, which has been identified and emphasized throughout our experimental
results.

##  6 Conclusions and Outlook

We have experimentally studied the solvent exchange process in a turbulent
buoyant jet. We calculate the oversaturation field from the recorded images
using a light attenuation technique, which consists of titration, calibration,
axisymmetric discretization, and an optimization algorithm. These experimental
procedures involve the background knowledge in the solvent exchange process
and the turbulent buoyant jet. The fundamental assumption we made is the
straight diffusion path, which leads to the evaluation of the oversaturation,
and in turn the nonlinear calibration curve. Except for the region right above
the needle, the constructed oversaturation fields gave values below the
theoretical upper bound for the oversaturation, consistent with the diffusion
path assumption. In addition, the very similar results between the two cameras
further support the validity of the assumption, the implemented methods, and
hence the calculated oversaturation fields.

Analyzing the rescaled oversaturation fields, the ouzo jets exhibited wider
jet spreading $b_{T}(z)/z$ and a radial profile with sub-Gaussian kurtosis,
reflecting enhanced nucleation in the entrainment region. In the streamwise
direction, the centerline oversaturation demonstrated a three-stage evolution,
consisting of fast nucleation, sharp dilution, and finally mild dilution, with
the scaling exponent $\beta$ of the centerline oversaturation being almost
independent of $Re_{0}$ and the initial compositions of the ethanol-oil
mixture, $w_{e}/w_{o}$, in the final stage. In contrast, integrating the
velocity field, the oversaturation flow rates in the entire domain show
monotonic increase before reaching the peak, again reflecting the wide-spread
nucleation of the oil droplets across the jet in the entrainment region.

In addition to the experimental findings, we have formulated a simplified
model, only considering the turbulent entrainment. It serves as the first step
to build the theoretical framework for the ouzo jet. The differences between
the experimental results and the model suggest the existence of other
mechanisms, which we attributed mainly to the rate-limiting mixing constraint
within the Batchelor scale. The entrainment rate exceeded the mixing rate,
leading to the prolonged nucleation downstream and across the jet.

To the authors’ knowledge, this work is the first attempt to experimentally
and theoretically tackle the ouzo jet, a complex fluid mechanics problem
combining solvent exchange and turbulent shear flow. The approach to quantify
the concentration of the nucleated droplets, namely the oversaturation, seems
a promising way for future efforts in this line of research. However, we are
aware of the limitation of the proposed method, which is a lack of direct
measurements for droplets size distributions and for the local concentration.
To the authors’ knowledge, estimation of nucleated droplet size and number
distribution in the early studies is limited to small scale system, allowing
the use of microscopic instruments, and thus free from the difficulties such
as struggling between depth of field and resolution. With the limitations in
large scale and turbulent flow, we have no choice but to focus on temporally
averaged and depth integrated images first before diving into local and
fluctuating micro-scale characteristics.

Attempting to see through the opaque flow induced by solvent exchange, we plan
to conduct future research in a laterally confined quasi two-dimensional jet,
hoping to reveal the time-dependent process and the local characteristics
missing in the current study. Also, as TNTI is mentioned several times, we
hope to visualize the phenomena by reducing the opaqueness of the flow in the
quasi two-dimensional jet (Giger et al., 1991; Watanabe et al., 2015b; Rocco &
Woods, 2015). Last but not least, an analytical framework including the mixing
rate would be highly valuable towards a thorough understanding of the topic.

## Acknowledgement

The authors acknowledge the funding by ERC Advanced Grant Diffusive Droplet
Dynamics with Project No. 740479, Natural Science Foundation of China under
grant Nos. 11988102 and 91852202 and Netherlands Organisation for Scientific
Research (NWO) through the Multiscale Catalytic Energy Conversion (MCEC)
research center.

We thank G.W. Bruggert, M. Bos, and T. Zijlstra for technical support building
the setup. Valuable discussions with R.A. Lopez de la Cruz, L. Thayyil Raju,
H. Tan, and Y. Li are also appreciated.

## Declaration of Interests

The authors report no conflict of interest.

## Appendix A: Details of the optimization process

In the optimization process, the loss function is set to the square of the
difference of left-hand side and right-hand side of Eqs. (3) and Eq. (4). A
basinhopping algorithm is used to search for global minimum. We gradually
increase the resolution, namely we decrease the size of the discretization
unit $dr$ as in Fig. 4(c), and finally obtain the results with $dr=2$ pixels.
In this way we get a rough, low-order, low-resolution estimation for the
oversaturation, which we then refine by decreasing $dr$, where we upscale the
low-resolution oversaturation profile to a higher resolution, which is our new
starting point for the minimization. This incremental procedure allows us to
find global minima for our high-dimensional system in an efficient manner. The
calculation procedure is repeated for every height independently, leading to
the oversaturation fields as in Fig. 7.

To check the validity of the results, the final oversaturation field is fed
back into Eq. (2) to reconstruct the intensity field, see Figs. 15a and 15e.
The difference between such reconstructed field and the original intensity
field (Figs. 15b and 15f) can then be used to check the deviation, see Figs.
15c, 15d, 15g, and 15h. Note that for direct comparison, the original
intensity field is down-sampled to match that of the reconstructed field. Fig.
15 clearly shows that the results are not perfect, with most deviation located
in the near field. We believe that there are multiple factors contributing to
the deviation, mainly including the down-sampling of the original intensity
field, the resolution limit in the near field, and the vicinity of the region
with $\widetilde{C}_{\text{oil,oversat}}>1$. Considering the aforementioned
factors, and the low deviation in most of the domain, we believe the results
obtained by the optimization process are valid for the semi-quantitative
analysis in this study.

## Appendix B: Comparison between far and zoomed-in recordings

Fig. 16 compares the profiles obtained from far view and zoomed-in recordings
in streamwise and radial directions. Figs. 16a and 16b presents the centerline
evolution of the results in Fig. 7, showing nice collapse between the two
fields except for the region with $\widetilde{C}_{\text{oil,oversat}}>1$. We
also compare the centerline evolution obtained from the curve-fitting in Figs.
16c and 16d, with similar features to those in Figs. 16a and 16b. Note that
for the intense nucleation region, the error bar is extremely large because
the data points for fitting are scarce, demonstrating the uncertainty of our
results in the area. Also, one might notice the fluctuation for each case in
the ramping-up stage, which can be attributed to the fluctuating position of
laminar/turbulent transition, causing the nucleation there not so smooth. To
complete the comparison, the jet width evolutions are presented in Figs. 16e
and 16f, showing consistent radial profiles between the two recordings. Note
that we do not subtract the abscissa with laminar length here (Hassanzadeh et
al., 2021), unlike the analysis in §4, so that the height $z$ here is aligned
with that in Fig. 7. By the rigorous comparison between the two recordings,
the consistency of our method is further demonstrated for the domain out of
the initial shear layer.

Figure 15: Evaluation of the oversaturation fields calculated by optimization
procedures. (a-d) $Re_{0}$=555, $w_{e}/w_{o}$=100, (e-h) $Re_{0}$=1387,
$2_{e}/w_{o}$=100. (a,e) are the reconstructed intensity fields by the
oversaturation fields. (b,f) are the original intensity fields recorded by the
camera. (c,g) present the absolute difference between the reconstructed and
original intensity fields. (d,h) show the intensity difference in fraction,
namely the results in (c,g) divided by those in (b,f). Note that the colorbars
for (c,g) are inverted.

Figure 16: Comparison of oversaturation profiles between far view and zoomed-
in recordings. (a,b) show the streamwise profiles, namely the centerline
evolution directly derived from the calculated fields in Fig. 7 , and (c,d)
are the centerline evolution obtained by curve-fitting. (e,f) display the
radial profile, namely the the jet width evolution, obtained by curve-fitting.

## Appendix C: Light attenuation in a scattering medium

As briefly mentioned in §.2.3, the superposition of level of light attenuation
is questionable in a scattering medium. Dahm (2013); Dahm & Dahm (2013)
discuss the way to theoretically address the light propagation in such medium.
The light attenuation technique we implemented is based on the Lambert-Beer
law, see eq. (1). For a non-scattering medium, the fraction of light
propagating through a medium is called transmittance, while the remaining
fraction of light not going through is absorbance. However, with oil droplets
induced by solvent exchange, the solution becomes a scattering medium, which
introduces an extra factor called remittance in Dahm (2013); Dahm & Dahm
(2013), that is,

$T+A+R=1,$ (14)

where $T=I/I_{ref}$ is transmittance, A is absorbance, and R is remittance.
The remittance represents the part of light that neither penetrates through
the medium nor absorbed by the medium, which results from its scattering
nature. Instead of using a linear superposition, Dahm & Dahm (2013) introduces
the so-called equations of Benford to deal with the overall transmittance,
absorbance, and remittance through layers of the scattering medium. These
equations are written as:

$\displaystyle T_{x+y}=\frac{T_{x}T_{y}}{1-R_{x}T_{y}},$ $\displaystyle
R_{x+y}=R_{x}+\frac{T^{2}_{x}R_{y}}{1-R_{x}R_{y}},$ $\displaystyle
A_{x+y}=1-T_{x+y}-R_{x+y},$ (15)

where x, y denote the two adjacent layers of the scattering medium with their
own thickness and concentration.

While the remittance R is not possible to measure in our setting, we measure
the transmittance T as a function of oil oversaturation with two calibration
cells of difference thickness (10mm and 2mm), namely $T_{x}(C)$ and
$T_{y}(C)$. And from $T_{x}(C)$ and $T_{y}(C)$ we can obtain $R_{x}(C)$ and
$R_{y}(C)$ from the equations of Benford. At this point we have the
transmittance and remittance curves as functions of oil oversaturation at the
thickness of 2mm and 10mm. Using either of these two curve sets and the
equations of Benford, we can obtain the transmittance and remittance curves
T(C) and R(C) for a very thin layer of the medium for further calculation.
Here we choose 1 pixel as the thickness for such base layer. Applying the
thickness matrix $dy_{mn}$ in eq. 4, we can estimate the amount of the base
layers in each item of $dy_{mn}$, from which we can construct the overall
transmittance $T_{sum}$ and remittance $R_{sum}$ using the equations of
Benford. $T_{sum}$ needs to match the value from the averaged jet in Fig.
4(b). Note that the order/trajectory of light propagation matters when
applying the equations of Benford, which is not the case for the Lambert-Beer
law. To elaborate the procedure further, we now take a look at Fig. 4(c). To
estimate $T_{sum}$, we have to start from the discretization unit closest to
the light source. Knowing the amount of base layers (1 px) in this unit, the
overall transmittance till the end of this unit can be obtained, and then we
enter the next unit. When applying the equations of Benford, the value $T_{x}$
keeps getting updated with the results obtained from the previous step, while
$T_{y}$ is the value of the following base layer, which could be in the same
or the next unit. We repeat such algorithm following the direction of light
propagation, reaching $T_{sum}$ right in front of the camera.

We then implement such theoretical optical framework into the optimization
procedures in Appendix A, replacing the superposition described by eq. 4.
Unfortunately, with the extra complexity, the optimization procedure is unable
to converge to a sensible concentration field.

Considering the discussion in this appendix, we must remind the readers that
the preliminary analysis and results in this work is not on firm and water-
proof footing, and will require further examination once the updated algorithm
can successfully capture the optics involved in this complex phenomena.

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