# Mixing and solvent exchange near the turbulent/non-turbulent interface in a
quasi-2D jet

You-An Lee  Sander G. Huisman  Detlef Lohse

###### Abstract

We inject with jet mixtures of ethanol and dissolved anise oil upward into
quiescent water with jet Reynolds numbers, $500<Re_{0}<810$. Nucleation of oil
droplets, also known as the ouzo effect, follows from the entrainment and
mixing with the ambient water, in which the oil has much lower oil solubility
than the initial jet fluid. We experimentally investigate the local
concentration during solvent exchange near the turbulent/non-turbulent
interface (TNTI) in the quasi-two-dimensional turbulent jet. Using a light
attenuation technique, we measure the concentration fields for the solvent
exchange case (the nucleated oil droplets) and the reference dye case (passive
scalar).

Combining conventional and conditional mean profiles, we show that the
nucleation of oil droplets is initiated near the TNTI upstream, and penetrates
into the turbulent region as the jet travels downstream. Persistent nucleation
not only sustains the scalar dissipation rate, but also leads to enhanced
temporal fluctuations of the concentrations till the downstream (high)
position. The probability density functions (PDFs) of the concentration
exhibit pronounced bimodal shapes near the TNTI and positively-skewed curves
toward the centerline, which also suggest that the oil droplets nucleate
across the jet upon mixing.

In addition to the qualitative investigation, we also describe the process
more quantitatively. Based on the idea that the concentration field of the
nucleated oil micro-droplets is linked to the ethanol concentration field
running in the background, we estimate the nucleation rate near the TNTI using
a control volume approach. We model the concentration field of the nucleated
oil using that of the passive scalar and the ternary phase diagram of a water,
ethanol, and anise oil mixture.

This study extends our previous work on the mean field, revealing more details
of the turbulent statistics induced by solvent exchange. The findings shed new
light on the interplay between mixing and nucleation in a quasi-2D turbulent
jet. We also provide the first modelling of solvent exchange in turbulent
flows with a simple model based on ethanol concentration field and the phase
diagram.

###### keywords:

Solvent exchange , Nucleation , Turbulent jet , TNTI

††journal: International Journal of Multiphase Flow

[inst1]organization=Physics 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,addressline=, city=Enschede,
postcode=7500 AE, state=, country=Netherlands

[inst2]organization=Max Planck Institute for Dynamics and Self-
Organization,addressline=Am Faßberg 17, city=Göttingen, postcode=37077,
state=, country=Germany

Mixing near the turbulent/non-turbulent interface induces solvent exchange and
initiates droplet nucleation.

Concentration field in the opaque flow is measured using a light attenuation
technique.

Continuous solvent exchange leads to prolonged temporal and spatial
concentration fluctuations across the entire turbulent jet.

##  1 Introduction

Turbulent mixing has been of central interest in the turbulence community for
decades; it is a fundamental and essential topic for turbulence research
(Rothstein et al., 1999; Dimotakis, 2005; Villermaux, 2019). Built upon the
knowledge of passive scalar mixing, more complicated processes related to
mixing also attract tremendous attention due to their relevance to various
industrial applications and natural phenomena. We refer to these processes as
physicochemical hydrodynamics, which can be broadly categorized into chemical
reactions, in particular combustion (Peters, 2000; Pitsch, 2006; Bisetti et
al., 2012; Attili et al., 2014a), and thermodynamic phase transitions
(Lesniewski and Friedlander, 1998; Murfield et al., 2013; de Rivas and
Villermaux, 2016; Lohse and Zhang, 2020). While the mechanisms behind these
physicochemical hydrodynamical processes are very different, they all involve
the conversion from one substance or phase to another due to turbulent mixing.

For reactive flows, the competition between the timescale of mixing at small
scales and that of the chemical reaction determines the rate-limit of the
process and the resulting velocity and scalar fields, which is characterized
by the Damk$\ddot{\text{o}}$hler number, the ratio between the hydrodynamics
and reaction timescales. Karasso and Mungal (1996) and Mingotti and Cardoso
(2019) studied chemical reactions in turbulent mixing layers and turbulent
plumes, respectively. They both indicated that mixing is the rate-limiting
process for fast reactions, i.e., those with high Damk$\ddot{\text{o}}$hler
number. On the other hand, for slow reactions, the reactant and the product
scalar field depend mainly on the chemical kinetics of the involved reaction.
To characterize turbulent mixing, in addition to the scalar field itself, the
scalar dissipation rate, the spatial gradient of the scalar field, gives
additional insight. This quantity is key to the precise modelling of turbulent
combustion. Bilger (2004) suggested that the scalar dissipation rate is
proportional to the reaction rate. Furthermore, Mellado et al. (2009) and
Sutton and Driscoll (2013) used the dissipation rate conditioned on the
mixture fraction, where the mixture fraction measures the mass fraction of the
fuel stream in the fuel/air mixture. They showed a non-monotonic dependence
and a local optimum of the dissipation rate in the mixture fraction parameter
space.

While the aforementioned studies focused on the scalar fields in a continuous
phase, physicochemical hydrodynamics following turbulent mixing can also lead
to precipitation of a dissolved component into a dispersed phase, which
undergoes oversaturation, droplet nucleation, droplet growth, and sometimes
also shrinkage due to evaporation (Marié et al., 2014; Li et al., 2023) in the
turbulent flow. Such precipitation processes include aerosol formation through
condensation (Lesniewski and Friedlander, 1998; Zhou et al., 2014; Ng et al.,
2021; Li et al., 2023), soot formation in turbulent combustion (Bisetti et
al., 2012; Attili et al., 2014a), particle formation by chemical reaction
(Schikarski et al., 2022; Tang et al., 2022), and droplet formation upon
solvent exchange (Lee et al., 2022). Lesniewski and Friedlander (1998)
experimentally studied dibutyl-phthalate (DBP) aerosol formation in a
turbulent jet flow, indicating that with high DBP vapor concentration, the
droplets nucleate both within the initial shear layer and in the downstream
turbulent region. Zhou et al. (2014) also investigated the formation of the
DBP particles, in a mixing layer using a numerical approach. They showed that
nucleation and growth of the particles are preferentially located at different
sides of the mixing layer. Murfield et al. (2013) focused on the modelling of
nucleation in general turbulent flows. They confirmed the experimental finding
of Lesniewski and Friedlander (1998) that nucleation of the DBP particles was
initiated both in the initial shear layer and by the large-scale eddies.
Combining experiments and numerical modelling, Schikarski et al. (2022)
investigated the precipitation of ibuprofen particles in a T-mixer upon mixing
and reaction, aiming at pharmaceutical applications. They showed the effect of
the Damk$\ddot{\text{o}}$hler number on the particle size distribution, and
demonstrated that their numerical framework correctly predict the ibuprofen
particle size distribution.

Solvent exchange with subsequent precipitation, also known as the ouzo effect,
is a substance-extraction process utilizing a solubility constraint in a
multi-component solution. The solution consists of a good solvent, a solute,
and a poor solvent. Mixing the good solvent with a sufficient amount of the
poor solvent decreases the solubility of the solute, leading to saturation and
nucleation, which will either be collected or dispersed in the flow. While the
physicochemical hydrodynamics of the ouzo effect has been extensively studied
in the laminar regime in microfluidic setups (Hajian and Hardt, 2015; Zhang et
al., 2015; Li et al., 2021), it remains almost unexplored in the turbulent
regime. In our previous work (Lee et al., 2022), we studied the concentration
field of nucleated oil in an axisymmetric turbulent jet. Implementing a light
attenuation technique, we successfully measured the mean structure of the
field. The results revealed the effects of continuous nucleation in the jet,
including reduced centerline decay rate, larger jet width, and monotonically
increasing mass flow rate with streamwise distance.

Due to the opaque nature of the ouzo effect, our optimization algorithm (Lee
et al., 2022) constrains the findings to the time-averaged axisymmetric
concentration field. Although the mean field unveils the centerline scaling
and the spreading of the profiles, we did not learn much about the temporal
dynamics. Moreover, the averaging operation smears out the meandering of the
jet, which is vital to identify nucleation near the turbulent/non-turbulent
interface (TNTI). TNTI is the interface separating the turbulent region from
the non-turbulent one, which has been extensively studied in various kinds of
turbulent shear flows, such as in axisymmetric jets (Westerweel et al., 2009;
Mistry et al., 2019), planar jets (Watanabe et al., 2015; Silva and da Silva,
2017), mixing layers (Attili et al., 2014b), and gravity currents (Krug et
al., 2015). Conditioning the flow fields on the distance to the TNTI reveals a
sharp gradient of vorticity and concentration across the interface in all the
aforementioned studies. Westerweel et al. (2005, 2009) suggested that the
small-scale eddy motion (‘nibbling’) dominates over inviscid action of the
large-scale eddies (‘engulfment’) as the main entrainment mechanism across the
TNTI for an axisymmetric jet. Watanabe et al. (2015) focused on the TNTI of a
reactive planar jet, revealing a significant effect of the interface
orientation on the concentration profile. Such an effect is more pronounced
for the case with fast reactions. Lee et al. (2022) argued that the measured
sub-Gaussian mean concentration profiles result from the meandering TNTI and
the accompanying mixing and nucleation.

The opaque nature of the axisymmetric jet of Lee et al. (2022) makes it very
difficult, if not impossible, to further investigate the effect of turbulence
on the solvent exchange near the TNTI experimentally. We therefore opt for a
quasi-two-dimensional jet (Landel et al., 2012a, b) by confining the jet
laterally. Eliminating the depth-integrated information in a 3D jet (Lee et
al., 2022), we then can extract information on the turbulent statistics of the
scalars. As a reference case, we also injected dyed ethanol to induce quasi-2D
turbulent buoyant jets. Utilizing a light attenuation technique, we
reconstruct the scalar fields for both dyed ethanol and nucleated oil. We
perform conventional and conditional statistics, and compare the spatial
profiles and the statistics of the ouzo case and the passive scalar case,
revealing the effect of nucleation. Moreover, we develop a very simple model
to be able to predict the concentration of the nucleated oil from the passive
scalar field. We provide qualitative explanations of the deviation between the
model and the measurement.

The paper is structured as follow: In §2 we detail the experimental setup and
the methods to measure the concentration fields. Also, the detection of the
TNTI is discussed in §2. In §3 we show the results of the conventional and
conditional statistics, with supporting discussion and qualitative analysis in
§4. Conclusions and an outlook on future work are presented in §5.

##  2 Experimental method

###  2.1 Set-up

Figure 1: Experimental set-up. The syringe pump injects the ethanol/oil
mixture upwards to the quasi-2D geometry filled with water.

We have conducted the experiments 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. To confine
the flow within a quasi-2D geometry, a Hele-Shaw-like flow cell consisting of
two pieces of glass is placed within the tank. The flow cell has a gap of
$2\text{\,}\mathrm{mm}$, and it is open to the ambient on all four sides to
prevent internal circulation. We refill the tank with decalcified water before
every run of the experiments. We injected a solution upwards through a round
blunt tip needle with an inner diameter d = $0.84\text{\,}\mathrm{mm}$ and an
length l = $25.4\text{\,}\mathrm{mm}$. The injected solution consisted of
mixtures of ethanol and trans-anethole (Sigma Aldrich, $\geq$99%) with mass
ratio $w_{e}:w_{o}=100:1$. As ethanol is the dominant component in the ouzo
solution, we also injected dyed ethanol as the reference case, offering a
comparison between the common passive scalar jet and the ouzo jet. The fluid
was injected by a Harvard 2000 syringe pump at three flow rates, forming
quasi-2D turbulent buoyant jets with initial Reynolds numbers ranging from 500
to 800, where buoyancy comes from the approximate 22% density difference
between ethanol and water. The initial Reynolds number is defined as
$Re_{0}=u_{0}d/\nu$. To minimize the effect of the temperature-dependence of
the solubility in the ternary liquids system, the water in the tank and the
injected solution were kept at room temperature
($$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$\pm$1\text{\,}\mathrm{K}$$) before
all the experiments.

We measure the local concentrations using a light attenuation technique. For
the purpose we use a LED backlighting with a diffuser and recorded the degree
of light attenuation using two Photron FASTCAM Mini AX200 high-speed cameras
with 1024 $\times$ 1024 pixels resolution at 50 fps. The cameras were equipped
with Zeiss $100\text{\,}\mathrm{mm}$ objectives, and installed at two
different axial positions to capture the local concentration fields, see e.g.
Fig. 4. The cameras were carefully arranged so that the centerline and the
TNTI of the jet can both be captured. The injection and the recording lasted
for 40–50 s, providing 2000–2500 frames for each experiment. For each
condition, the experiment is repeated 3 times for reliable statistical
results.

###  2.2 Oversaturation

Figure 2: Schematic ternary phase diagram and estimation of oversaturation.
(a)Ternary phase diagram. The metastable regime in (a) is where the ouzo
effect occurs. (b) Binodal curve and the theoretical diffusion path for ouzo
mixture ${w_{e}/w_{o}=100}$. The dashed line in (a) corresponds to the black
diffusion path in (b). Note that the binodal curve in (b) is determined partly
by titration experiments (green solid line) and partly by direct linear
approximation (green dashed line). The length of the yellow line segment in
(b) measures the oversaturation. The pink dashed line in (b) extends from the
yellow line segment to the pure oil phase, which can be approximated with a
vertical line considering the tiny amount of oil. (c) Non-monotonic variation
of oversaturation as a function of ethanol weight fraction, which is fitted by
a $3^{rd}$-order polynomials for $C_{\text{ethanol}}>0.3$, and by a piecewise
linear function for $C_{\text{ethanol}}\leq 0.3$.

To estimate the concentration of the nucleated oil, the oversaturation of the
oil needs to be calculated. Fig. 2(a) shows the ternary phase diagram for our
system. Before the injection, the initial solution lies on the ethanol-oil
boundary, close to the ethanol phase. Once mixed with water, the local fluid
parcel follows the trajectory indicated by the dashed arrow in Fig. 2(a). As
the trajectory reaches the green binodal curve, the solution is saturated with
oil. Mixing with more water, the local fluid parcel enters the metastable
regime and oil droplets start to nucleate. Converting the three-component
phase diagram in Fig. 2(a) to two components, we obtain Fig. 2(b). The green
solid line is determined by titration following the procedures detailed in Lee
et al. (2022). The green dashed line represents the part of the binodal curve
approximated by a direct linear extrapolation. We do not perform titration
here since it is very close to the pure water phase and the titration gets
very difficult. The black diffusion path in Fig. 2(b) is identical to the
black dashed arrow in Fig. 2(a), which we assume to be straight. A straight
diffusion path suggests that the ethanol/oil ratio remains unchanged during
the whole mixing process, which has been discussed in Li et al. (2021) and
proved to be a valid assumption. The difference between the diffusion path and
the binodal curve, that is, the length of the yellow line segment in 2(b), is
the amount of oversaturation. The oversaturation first increases with the
water fraction in the local fluid parcel, reaching the peak, and followed by a
decline if the water fraction keeps increasing. The decrease in the
oversaturation signals the weakening of nucleation, such that dilution
overcomes nucleation beyond the optimum mixture composition. Fig. 2(c) shows
the obtained oversaturation as a function of the local ethanol fraction, where
a theoretical maximum oversaturation can be clearly seen around
$C_{\text{ethanol}}=0.3$. For $C_{\text{ethanol}}<0.3$ in Fig. 2(c), the
oversaturation is calculated by the approximated dashed line in Fig. 2(b).

###  2.3 Calibration

With the definition of the oversaturation, we can apply a light attenuation
technique. An in-situ calibration is carried out in a separate closed cell
with a $2\text{\,}\mathrm{mm}$-gap immersed in the same tank as in Fig. 1. The
cell is first filled with a specific amount of ethanol/oil solution
($w_{e}:w_{o}=100:1$), which is transparent and thus serves as the background
image. We gradually add doses of water and mix the solution in the cell until
it is visually homogeneous, followed by taking 200 frames for each dose. For
the reference dye case, the calibration procedure is the same, but the cell is
then first filled with pure ethanol followed by adding doses of red food dye.
Averaging over 200 frames and dividing by the background image, the average
light attenuation level for every pixel is obtained. To reduce the noise, we
then average the light attenuation level for every block of 5 $\times$ 5
pixels and take it as one image unit. Relating the logarithmic light
attenuation level to the estimated concentration (oversaturation), we obtain
the in-situ calibration data for every image unit, see the data points in Fig.
3.

Figure 3: Calibration data and the fitted curves. (a) For the reference dye
case. (b) For the ouzo case. $\Phi$ is the degree of light attenuation across
the 2mm gap, $\Phi=\log(I_{\text{{ref}}}/I)$. The red data points are obtained
by the calibration doses, and the black curves are the fitted curve, which is
linear in (a) and an empirical function in fitted by Eq. 1 in (b). The
abscissa in (b) $C_{\text{{oil,sat}}}$ is the oversaturation of the oil.
$C_{\text{thres}}$ denotes the threshold oversaturation used to normalized oil
oversaturation.

The red line in Fig. 3(a) shows that a linear calibration curve is established
below $4000\text{\,}\mathrm{p}$pm, which results in our choice of
$3500\text{\,}\mathrm{p}$pm as the initial concentration for the dye case. On
the other hand, the calibration curve for the ouzo case in Fig. 3(b) seems
non-linear and thus we fit it by the empirical function:

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

where $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, and $a_{5}$ are the fitting
parameters, and $C$ is the oil oversaturation. In Fig. 3(b) we can see that
there is a threshold oversaturation, $C_{\text{thres}}$, for the light
attenuation to set in. Also, we found in the calibration that the light
attenuation level does not monotonically increase with added water, and the
peak attenuation level occurs at an oversaturation value, $C_{\text{max}}$,
very close to that indicated in Fig. 2(c). For a better interpretation of the
results, we normalize the oil oversaturation using

$\widetilde{C}_{\text{oil}}=\frac{(C_{\text{oil}}-C_{\text{thres}})}{(C_{\text{max}}-C_{\text{thres}})}$
(2)

For the dye case we normalize the concentrations with the initial
concentration, written as

$\widetilde{C}_{\text{dye}}=\frac{C_{\text{dye}}}{C_{0}}$ (3)

Using the in-situ calibration curves, the concentration fields displayed in
Figs. 4(a,b,d,e) are calculated from the recorded images.

###  2.4 TNTI detection

The detection of the TNTI is critical for conditional statistics, and is thus
always an essential part of the relevant studies. Following the method
presented in Prasad and Sreenivasan (1989), we plot the histogram of the bit
value across the recorded frames, and identify the first local minimum as the
threshold for the TNTI detection, see Figs. 4(c,f). Figs. 4(a,b,d,e) show that
the detected TNTIs fit reasonably well to the contour of the jets for both the
dye and the ouzo case, at both low and height positions. For the histogram and
the following results on conventional and conditional statistics, we only
collect the data when the flow is in a steady state, and we exclude the
transients at the start of the injection.

Figure 4: Instantaneous scalar fields and TNTI detection for $Re_{0}=606$. The
color code in (a,b,d,e) represents the normalized concentrations. The white
contours in (a,b,d,e) indicate the detected TNTI. (a) For the dye case at the
low position, $40\leq z/h\leq 60$. (b) For the dye case at the high position,
$125\leq z/h\leq 145$. (c) The threshold value for the TNTI detection in case
(a). (d) For the ouzo case at the low position, $40\leq z/h\leq 60$. (e) For
the ouzo case at the high position, $125\leq z/h\leq 145$. (f) The threshold
value for the TNTI detection in case (d).

##  3 Results

###  3.1 Conventional statistics

Fig. 5 displays the streamwise evolution of the normalized mean concentration
profiles. Fig. 5(a) shows a graphical guide for the location of the profiles
shown in Figs. 5(b,c). For the dye case, we see in Fig. 5(b) evolving Gaussian
profiles following continuous mixing and dilution in the streamwise direction.
On the other hand, for the ouzo case, the effect of nucleation sets in at
$20\leq z/h\leq 40$, exhibiting increasing concentration levels and off-center
peaks in Figs. 5(a,c). The off-center peaks indicate that nucleation is
initiated near the TNTI. As the jet travels downstream to $40\leq z/h\leq 60$,
the concentrations further increase. Although the off-center peaks are no
longer present, we can more or less see off-center humps, suggesting ongoing
nucleation events near the TNTI. The concentration profiles from $40\leq
z/h\leq 60$ to $125\leq z/h\leq 145$ in Fig. 5(c) only marginally change,
showing the effect of nucleation on reducing dilution.

Figure 5: Conventional mean concentration profiles for $Re_{0}=606$. (a) shows
the mean concentration fields at three recording views for ouzo case, which
serves as a graphical legend for (b,c). (b) is the profiles for dye and (c)
for ouzo. Note that the recordings for $20\leq z/h\leq 40$ are not discussed
in §2.4 since these were recorded separately, and we do not perform
conditional statistics on these images. The profiles for $125\leq z/h\leq 145$
are cut off on the left-hand side due to the limited recording view, see Figs.
4(b,e).

From the concentration profiles as those in Fig. 5, we extract the centerline
concentrations for all three $Re_{0}$. However, we think the unproportionally-
high centerline concentrations in Figs. 5(b) are partly caused by oil attached
to the glass wall. We therefore extract the centerline value fitted by the
generalized normal distribution function,

$\widetilde{C}(x)=\widetilde{C}_{m}e^{-\frac{|x|^{\kappa}}{\kappa\sigma^{\kappa}}},$
(4)

where $x$ is the horizontal distance to the centerline, $\widetilde{C}_{m}$
the centerline concentration, $\sigma$ is the jet width, $\kappa$ a shape
parameter, and deviation from $\kappa=2$ indicates non-Gaussianity. The high
concentrations near the centerline are excluded when performing the curve-
fitting, see the example in Fig. 6(a).

Figure 6: Fitted centerline evolution for the concentration profiles in Fig.
5. (a) demonstrates the deviation between the measured data and the fitted
curves near the centerline, which is caused by oil attachment. (b) For the dye
cases. (c) For the ouzo cases.

Figs. 6(b,c) present the fitted centerline evolution for $40\leq z/h\leq 60$
and $125\leq z/h\leq 145$. Note that the data from the lowest position
($20\leq z/h\leq 40$) is not included as they were recorded separately, and
the off-center peaks do not fit the desired generalized normal distribution
profile. At first glance, the effect of $Re_{0}$ in Fig. 6(b) is counter-
intuitive. For the lowest $Re_{0}$, the mixing is weakest within $40\leq
z/h\leq 60$, and therefore the concentration is the highest for $Re_{0}=505$.
However, what is surprising is that the decay rate of the concentration is
fastest for $Re_{0}=505$ and slowest for $Re_{0}=808$. We attribute this
phenomenon to the limited mixing and the wall friction due to our confining
quasi-2D geometry. With the fastest transition toward turbulence and the
strongest mixing, the centerline concentration for $Re_{0}=808$ decays to the
lowest value before $z/h=40$. However, the reduced entrainment due to lateral
confinement restricts the mixing and the decay of concentration downstream.
This is also why the concentrations within $125\leq z/h\leq 145$ turn out to
be highest for $Re_{0}=808$.

Comparing Fig. 6(c) to Fig. 6(b), the effect of nucleation can be clearly
identified by the increasing concentration at $40\leq z/h\leq 60$. Also, the
level of nucleation increases with $Re_{0}$, as the concentration is highest
for $Re_{0}=808$. The oil droplets nucleate strongest upstream for the highest
$Re_{0}$, and the value almost does not decay from the low to the high
position, which results from the restricted dilution and the extended
nucleation.

In addition to the mean concentration profiles displayed in Fig. 5, we show
the r.m.s. profiles for $Re_{0}=606$ in Fig. 7 to complete the analysis on the
conventional statistics. The typical off-center peaks for the r.m.s.
concentration profiles are present for the dye and the ouzo cases at all three
locations. Figs. 7(b), however, show that there is a secondary or even the
strongest peak near the centerline. We believe that this is also a consequence
of the gradual build-up of nucleated oil on the wall, similar to what we see
in Figs. 5(b).

Figure 7: Conventional r.m.s. concentration profiles for $Re_{0}=606$ for (a)
dye and (b) ouzo. The profiles for $125\leq z/h\leq 145$ are cut off on the
left-hand side due to the limited recording view, see Figs. 4(b,e). Figure 8:
Evolution of the ratio of the peak-to-centerline for the r.m.s. profiles in
Fig. 7. (a) For the dye cases, and (b) for the ouzo cases.
$C_{\text{ratio}}=\widetilde{C}_{\text{peak}}/\widetilde{C}_{m}$ denotes the
ratio between the off-center peak shown in Fig. 7 and the centerline value.
Note that for (b) we take the minimum close to the centerline instead of the
centerline value itself, as the distinct peak in the centerline shown by the
blue curves in Fig. 7(b) is not a typical feature of r.m.s. profiles, which
will lead to an inappropriate comparison.

To have a deeper look into the off-center peaks featured in Fig. 7, we extract
the ratio between the peak and the centerline concentrations in Fig. 8.
Comparing Fig. 8(b) to Fig. 8(a), the concentration ratio for the ouzo case is
slightly higher than that for the dye case for $Re_{0}=505$ and 606,
suggesting enhanced fluctuations near the TNTI due to nucleation. As for
$Re_{0}=808$, the most intense nucleation might have occurred before $z/h=40$,
causing the off-center peak ratio very close to that as in the dye case.

###  3.2 Conditional statistics

While the conventional statistics have already provided us with insights into
the solvent exchange process of a turbulent jet, the information near the TNTI
is smeared due to temporal averaging. Using the detected TNTI in Fig. 4, we
utilize a distance transform on the TNTI for the conditional statistics, see
the examples in Fig. 9. We define $\tilde{x}_{n}$ as the distance to the TNTI,
with $\tilde{x}_{n}>0$ in the turbulent region and $\tilde{x}_{n}<0$ in the
non-turbulent region.

Figure 9: Conditional distance to the TNTI for the ouzo case with
$Re_{0}=606$. (a) $40\leq z/h\leq 60$ and (b) $125\leq z/h\leq 145$. (c,d) The
concentration fields corresponding to (a,b), respectively. The green contours
mark the TNTI.

Since the conditional profiles look similar for all three $Re_{0}$, in the
following analysis, we only show the results for $Re_{0}=606$. As for the
biased centerline concentrations due to nucleation on the wall we discussed in
§.3.1, we exclude those data points with the help of a joint histogram of the
concentrations and the scalar dissipation rate. The troublesome data are
almost separate from most of the data points on the 2D histogram, and thus we
can cut them off as they are clear outliers.

Figure 10: Mean concentration profiles conditioned on the distance to the TNTI
for $Re_{0}=606$. (a) dye and (b) ouzo.

For the conditional mean concentration profiles, the existence of a sharp
concentration gradient across the TNTI has been widely confirmed. The profiles
in Fig. 10 show that we successfully capture this feature for both the dye
case and the ouzo case. Similar to what we observe in the conventional mean
profiles in Fig. 5, nucleation of the oil serves as a source term, competing
with the dilution. Fig. 10(b) illustrates not only the almost unchanged
concentration levels for low positions, but also the reduced dilution from low
to high positions, as compared to the dye case in Fig. 10(a). Focusing on the
turbulent region, $\tilde{x}_{n}>0$, most of the profiles in Fig. 10 extend
beyond the peak concentration, reaching a plateau or starting to decline. This
is caused by the frames with part of the TNTI out of the scope, which leads to
the large $\tilde{x}_{n}>$ value marked by the dark blue color in Fig. 9(b).

Figure 11: Normalized mean concentration profiles conditioned on the distance
to TNTI for $Re_{0}=606$. (a) dye and (b) ouzo. The concentrations are
normalized by the centerline value $\widetilde{C}_{m}$, and the distance is
normalized by the distance from the centerline to the TNTI $x_{\text{max}}$.

To facilitate a fair comparison for the dye and the ouzo case, we further
normalize the mean profiles by the corresponding centerline concentrations,
$\widetilde{C}_{m}$, and the distance from the TNTI to the centerline,
$x_{\text{max}}$, see the axes of Fig. 11. Note that the normalization with
$x_{\text{max}}$ automatically excludes the beyond-the-peak plateau mentioned
in the previous paragraph. Also, the centerline concentrations used here
originate from the fitting results presented in Fig. 6. Fig. 11 shows that the
profiles collapse well across the height for $40\leq z/h\leq 145$ for both the
dye and the ouzo case. Focusing on the gradient across the TNTI, we
intuitively expected to see a larger gradient for the ouzo case due to
nucleation. Surprisingly, Fig. 11 shows that the magnitude for the ouzo case
is smaller than that for the dye case, which results from the higher
centerline concentrations we use to normalize the profiles.

Fig. 11(b) also shows that the concentration gradient for the ouzo case is
larger than that for the dye case within the turbulent region
($\tilde{x}_{n}>0$). Together with what we just discussed near the TNTI, it is
revealed that the nucleation events are not restricted to the region near the
TNTI. Instead, it gets stronger as we move towards the centerline within the
turbulent region.

Figure 12: Normalized concentration fluctuation profiles conditioned on the
distance to TNTI for $Re_{0}=606$. (a) dye and (b) ouzo. The concentration
r.m.s. are normalized by the mean values shown in Fig. 10, and the distance is
normalized by the distance from the centerline to the TNTI. Note that we force
the value in the non-turbulent region ($\tilde{x}_{n}<0$) to be zero as the
fluctuation there is only noise.

While the mean concentration profile exhibits the outcome of the solvent
exchange process upon mixing, the fluctuations of the concentration in time
and space are also crucial to understand the process. Fig. 12 presents the
r.m.s. profiles normalized by the mean profiles in Fig. 10, that is, the
relative fluctuation of concentrations in time. For the dye case in Fig.
12(a), the fluctuation decreases monotonically from the TNTI to the
centerline, resulting from the relatively intense mixing across the TNTI. On
the other hand, the non-monotonic variation at the low positions in Fig. 12(b)
reveals that the peak fluctuations are shifted slightly toward the turbulent
region, where nucleation might be the strongest. The blue curves in Fig. 12(b)
also show that the relative fluctuations increase significantly as the jet
travels downstream, especially near the TNTI and the centerline. We attribute
the enhanced fluctuations near the centerline and near the TNTI to the
continuous nucleation of the oil droplets.

Figure 13: Mean scalar dissipation rate profiles conditioned on the distance
to TNTI for $Re_{0}=606$. (a) dye and (b) ouzo.

To study the spatial fluctuation of the concentration fields, we calculate the
conditional (on distance from the TNTI) mean scalar dissipation rate

$\chi=2D\sum\limits_{i=1,2}\left(\frac{\partial C}{\partial x_{i}}\right)^{2}$
(5)

where $C$ is the local concentration, $D$ is the diffusion coefficient of
ethanol, and $x_{1}$ and $x_{2}$ denote the $x$ and $z$ directions in the
recorded view. Fig. 13 shows that we capture the peak dissipation rate near
the TNTI, which has been reported in the literature (Attili et al., 2014b). As
the jet travels downstream and mixes with the ambient water, the scalar
dissipation rate naturally decreases, see the decay from the red to the blue
curves in Fig. 13(a). However, for the ouzo case, $\chi$ remains almost
unchanged from the low to the high position, suggesting that nucleation
sustains the spatial gradient of the concentrations in the entire span of the
jet, especially on the two ends, namely the TNTI and the centerline. This
finding further supports our argument that mixing and nucleation occur within
the entire turbulent region and last till a very downstream position.

Figure 14: PDFs for the concentrations conditioned on the distance to TNTI for
$Re_{0}=606$. The color code given in (b) holds for all four plots. (a,c) dye
and (b,d) ouzo. (a,b) at $40\leq z/h\leq 60$. (c,d) at $125\leq z/h\leq 145$.
$\tilde{x}_{n}/x_{\text{max}}$ denotes the distance to the TNTI normalized by
the local jet width. The value increases from 0 near the TNTI to 1 near the
centerline. The concentrations are normalized by the mean centerline values in
the entire domain (20h $\times$ 20h), $\langle\widetilde{C}_{m}\rangle$.

To evaluate the interaction between turbulent mixing and nucleation, the
probability density functions (PDFs) of the concentration field can provide
further insights. In Fig. 14 we present the histogram of the conditional
concentrations for the dye and the ouzo cases, at low and high positions.
While the data is conditioned on the distance to the TNTI for each case, the
height dependence is ignored within the domain. Fig. 14(a) shows bimodal
distribution profiles close to the TNTI (red and orange curves) for the dye
case at the low position, which indicates that the entrained water is not
fully mixed. On the other hand, for the same position in the ouzo case, the
red curves in Fig. 14(b) exhibit more pronounced bimodal profiles close to the
TNTI. We attribute the shape to nucleation in the diluted area near the TNTI,
which enhances the bimodal curves for the dye case. The left peak of the
bimodal shape for the red curves in Fig. 14(b) shows the presence of the
unmixed water, consistent with the red curves in Fig. 14(a). The local minimum
and the right peak indicate the effect of nucleation on the concentration
distributions.

Moving to the high position, Figs. 14(c, d) show that nucleation positively
skews the distribution across the jet, especially close to the centerline. One
more noticeable thing is that the spread between the profiles for the ouzo
case in Figs. 14(b,d) is significantly larger than the dye case in Figs.
14(a,c). Such a larger spread again proves that the nucleation events
penetrate deeply into the centerline, aligned with the large concentration
gradient within the turbulent region shown in Fig. 11(b).

Up to this point, we have been discussing the effect of the solvent exchange
by comparing the concentration field of the ouzo case to the passive scalar
case. However, we would like to emphasize that the concentration field of the
ouzo case is in fact linked to the passive scalar case (ethanol
concentrations). In other words, the turbulent mixing between the (dyed)
ethanol in the jet fluid and the ambient water leads to the nucleation of the
oil droplets for the ouzo case. To relate the nucleated oil concentration
field to the passive scalar one, we condition the scalar dissipation rate on
the local concentrations. Mellado et al. (2009) and Sutton and Driscoll (2013)
presented a non-monotonic variation and a local optimum of the dissipation
rate in the mixture fraction parameter space, where the mixture fraction
measures the mass fraction of the fuel stream in the fuel/air mixture. As the
optimum can be interpreted as the concentration for a fast reaction rate
(Bilger, 2004), we argue that we can replace the reaction rate with the
nucleation rate in our study, at least qualitatively, as the ouzo effect is
generally considered as a spontaneous emulsification process (Ruschak and
Miller, 1972; Sitnikova et al., 2005).

Figure 15: Scalar dissipation rates conditioned on the concentration and the
distance to TNTI for $Re_{0}=606$. (a,c) dye and (b,d) ouzo. (a,b) at $40\leq
z/h\leq 60$. (c,d) at $125\leq z/h\leq 145$. The concentrations are normalized
the same way as that in Fig. 14, and the dissipation rates are normalized by
the mean value in the entire domain (20h x 20h), $\langle\chi_{m}\rangle$.

In Fig. 15(a) we identify a local maximum of $\chi$ at low concentration,
which indicates intense mixing between the jet fluid and the ambient water for
the passive scalar. The scalar dissipation rate of the passive scalar
eventually creates the concentration field for the ouzo case, leading to all
the results for the ouzo case in Fig. 10 – Fig. 14. The solvent exchange
process then in turn builds up the spatial gradient, delivering the results in
Fig. 15(b). The pronounced local maximum near the TNTI in Fig. 15(b) indicates
nucleation of a modest amount of the oil following entrainment and mixing,
corresponding to the local minimum in Fig. 14(b). At the downstream position,
Fig. 15(c) shows smeared spatial gradient with sufficient mixing, with almost
collapsing curves across the jet. Fig. 15(d), on the other hand, exhibits a
large spread of the scalar dissipation rate across the jet. Approaching the
centerline, the dissipation rate increases almost monotonically with the
concentration, proving the continuous nucleation at the downstream position.

##  4 Discussion

###  4.1 Scalar flux and nucleation rate near TNTI

In §3.2 and Fig. 11, we present a qualitative description of the conditional
mean concentration profiles. To gain quantitative insights from the
concentration gradient, we apply a theoretical framework adapted from
Westerweel et al. (2009). Westerweel et al. (2009) estimated the momentum flux
and the scalar flux transported across the TNTI using the conditional
profiles, see Figs. 2 & 13 and §2.1 & §4.5 in their paper for the details.
Using a control volume analysis, the momentum and scalar transport can be
written as

$\displaystyle E_{b}\Delta
U\approx\Delta\tilde{\tau}\approx-\nu_{e}\frac{U_{s}}{\sigma_{x}},$ (6)
$\displaystyle E_{b}\Delta C\approx\Delta\tilde{F}\approx-
D_{e}\frac{C_{s}}{\sigma_{x}},$ (7)

where $E_{b}$ is the velocity of the interface toward the non-turbulent
region, $\Delta U$ and $\Delta C$ denotes the jump or the sharp velocity and
concentration difference across the TNTI, $\tilde{\tau}$ and $\tilde{F}$ is
the momentum flux (stress) and the scalar flux, $\nu_{e}$ and $D_{e}$ is the
eddy viscosity and the eddy diffusivity, $U_{s}$ and $C_{s}$ is the velocity
and concentration difference within the turbulent region, and $\sigma_{x}$ is
the width of the turbulent region.

Using Eq. 7 and the approximated relation between the eddy viscosity and eddy
diffusivity derived from rapid distortion theory, they suggested

$\left(\frac{\Delta U}{U_{s}}\right)\left(\frac{\Delta
C}{C_{s}}\right)^{-1}\approx\frac{\nu_{e}}{D_{e}}\approx 0.5$ (8)

Substituting Eq. 8 into Eq. 7, we get

$\frac{\Delta U}{U_{s}}=\frac{\Delta
C}{2C_{s}}=\frac{-D_{e}}{2E_{b}\sigma_{x}}$ (9)

For the dye case, Eq .9 can be written as

$\frac{-D_{e}}{E_{b}}=\sigma_{x}\frac{\Delta C_{dye}}{C_{s,dye}}$ (10)

For the ouzo case, to estimate the concentration flux across the TNTI due to
nucleation, we introduce a source term $\dot{P}$ into Eq. 7

$E_{b}\Delta C=-\left(D_{e}\frac{C_{s}}{\sigma_{x}}+\dot{P}\right)$ (11)

Substituting $D_{e}$ using Eq. 10 , we get

$\dot{P}=-E_{b}\left(\Delta C_{oil}+C_{s,oil}\frac{\Delta
C_{dye}\sigma_{x,dye}}{C_{s,dye}\sigma_{x,oil}}\right)$ (12)

Figure 16: Normalized concentration profiles conditioned on the distance to
TNTI for $Re_{0}=606$. The green and red dashed lines mark the range where we
estimate the scalar flux, and the blue lines indicate the associated
concentration gradient. The (a,c) dye and (b,d) ouzo. (a,b) At $z/h=45$. (c,d)
At $z/h=130$.

All the parameters in Eq. 12 can be obtained by the normalized conditional
profiles shown in Fig. 16, except for $E_{b}$. The source term divided by
$E_{b}$ can then be calculated at different heights, $z/h$, delivering the
estimated nucleation rate divided by $-E_{b}$ shown in Fig. 17. Although we
are not able to accurately measure $E_{b}$, as the time-averaged jet width
doesn’t increase at the high position, it is fair to estimate that $-E_{b}$
decreases with height. Therefore, it is certain that the nucleation rate
$\dot{P}$ at the low ($40\leq z/h\leq 60$) is larger than that at the high
position ($125\leq z/h\leq 145$), which indicates that nucleation near the
TNTI is significantly more intense upstream. The analysis here provides
quantitative information about the nucleation rate near the TNTI.

Figure 17: Estimated nucleation rate divided by the propagating velocity of
the TNTI for $Re_{0}=606$.

###  4.2 Modelling concentration of the nucleated oil

As mentioned in §3.2, we believe that the concentration field of the ouzo case
is linked to the concentration field of the passive scalar case, which is in
fact the ethanol concentration. Therefore, we take one step further,
estimating the concentration field of the nucleated oil using the passive
scalar field and the red curve in the phase diagram presented in Fig. 2(c).
This modelling approach converts the ethanol concentration histogram in Fig.
18(a) to the oil concentration histogram in Fig. 18(b). Compared to the
measured result in Fig. 18(c), the modelled oil concentration in Fig. 18(b) is
obviously lower.

Figure 18: Joint histogram of the concentration distribution and the mean
concentration profiles conditioned on the distance to TNTI for $Re_{0}=606$ at
$z/h=50$. The yellow and red profiles are the averaged values of the
corresponding joint histogram. (a) For dye. (b) Oversaturated oil scalar field
modelled from (a) and the phase diagram in Fig. 2(b). (c) Measured
oversaturated oil scalar field.

We attempt to identify the possible causes for such a deviation. First, the
converting curve in Fig. 2(c) cannot capture the threshold oversaturation to
induce nucleation, see the discussion for the calibration curve in Fig. 3(b).
As a result, the sharp concentration gradient across the TNTI significantly
reduces in Fig. 18(b).

Second, $\widetilde{C}_{oil}$, the rescaled oversaturation from the
measurements exceeds the theoretical maximum, 1, which is certainly not
predicted by the model. The over-estimated oversaturation reveals the
limitation of the light attenuation technique, which was also observed in our
previous work on the round jet (Lee et al., 2022). We attribute such a
limitation to the effect of droplet size on the light attenuation level, which
is neglected in the measurement. With the in-situ calibration approach, we
automatically assume the droplet size is monodisperse and remains the same in
the calibration and the turbulent jet. While we can’t resolve the oil
droplets, we argue that the oil droplets nucleated in the jet might be smaller
than those in the calibration, especially in the region with intense mixing
and nucleation. The size inconsistency compromises the accuracy of the
measurement. Bearing this in mind, the oversaturation we measure here is
essentially an apparent concentration of the nucleated oil. The apparent
concentration includes the effect of the droplet size, which requires future
investigation to disentangle.

Last but not least, the modelling framework relies on a phenomenological
assumption. The nucleated oil droplets must stay in the same local fluid
parcel, such that the phase diagram and the ethanol concentrations can be
exploited to calculate the local oil concentrations. As long as the oil
droplets separate from the local fluid parcel, the oil concentration can no
longer be estimated by the ethanol concentration, which compromises the
phenomenological assumption.

##  5 Conclusions

We have experimentally investigated the concentration field induced by the
solvent exchange in a quasi-2D turbulent buoyant jet. We adapt the light
attenuation technique applied in Lee et al. (2022) to estimate the local
concentration of the nucleated oil droplets. The quasi-2D geometry not only
simplifies the data processing algorithm in Lee et al. (2022), but also
overcomes the limitation of the opaque flow. The geometry enables us to obtain
all the instantaneous concentrations and thus the turbulent statistics, which
is a step further from the mean field in our previous work (Lee et al., 2022).
We have further obtained the concentration profiles and the statistics
conditioned on the distance to the TNTI, where the turbulent entrainment and
mixing are initiated (Westerweel et al., 2009; Mistry et al., 2019). The
conditional profiles provide physical insights about the spatial distribution
of the solvent exchange process in the turbulent jet flows.

In the streamwise direction, the increasing concentrations upstream and the
reduced dilution downstream clearly reveal the effect of nucleation. Using a
control volume analysis, we quantitatively show that the nucleation rate near
the TNTI is higher at the low position, and decreases as the jet travels
downstream to the high position. However, the nucleation events are not
restricted to the TNTI. Instead, they penetrate deeply into the turbulent
region, evidenced by the steep conditional mean profiles in Fig. 11(b), the
enhanced fluctuation in Fig. 12(b), the sustained dissipation rate in Fig.
13(b), and the wide-spread PDFs in Figs. 14(b,d).

Although the modelling attempt does not deliver satisfying results, we
identify the possible causes for the deviation between the measurements and
the model, including the limitations of the measurement technique and the
phenomenological assumptions. While the concentration of the nucleated oil is
closely connected to the passive scalar field (ethanol concentrations), the
spatial and temporal trajectories of the nucleated droplets play a vital role
in determining their concentrations.

This experimental investigation unveils the complexity of the solvent exchange
in a quasi-2D turbulent jet. Despite being mostly qualitative, the results
still serve as stepping stones for the future modelling efforts and the
droplet-resolved experimental improvement.

## Acknowledgement

The authors acknowledge the funding by ERC Advanced Grant Diffusive Droplet
Dynamics (DDD) with Project No.740479 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 D. Krug on the TNTI is also appreciated.

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