# Intermittency in turbulent emulsions

1<EMAIL_ADDRESS>H.-C. Smith1  J.Q. Long2  M. Crialesi-Esposito1
G. Boffetta2  L. Brandt3,4  S. Chibbaro5  S. Musacchio2  1 INFN, Sezione di
Torino, via Pietro Giuria 1, 10125, Torino, Italy 2 Dipartimento di Fisica and
INFN, Università degli Studi di Torino, via P. Giuria 1, 10125 Torino, Italy.
3 FLOW Centre, KTH Royal Institute of Technology, Stockholm, Sweden 4
Department of Energy and Process Engineering, Norwegian University of Science
and Technology(NTNU), Trondheim, Norway 5 Université Paris-Saclay, CNRS, LISN,
91400 Orsay, France

###### Abstract

We investigate the statistics of turbulence in emulsions of two-immiscible
fluids of same density. We compute for the first time velocity increments
between points conditioned to be located in the same phase or in different
phases and examine their probability density functions (PDF) and the
associated structure functions (SF). This enables us to demonstrate that the
the presence of the interface reduces the skewness of the PDF at scales below
the Kolmogorov-Hinze scale and therefore the magnitude of the energy flux
towards the dissipative scales, which is quantified by the third-order SF. The
analysis of the higher order SFs shows that multiphase turbulence is more
intermittent than single-phase turbulence. In particular, the local scaling
exponents of the SFs display a saturation about the Kolmogorov-Hinze scale and
below, which indicates the presence of large velocity gradients across the
interface. Interestingly, the statistics approach of classic homogeneous
isotropic turbulence when significantly increasing the viscosity of the
dispersed phase.

###### keywords:

Emulsions, multiphase turbulence, intermittency, structure functions.

MSC Codes  (Optional) Please enter your MSC Codes here

##  1 Introduction

Emulsions, _i.e._ mixtures composed of two immiscible (totally or partially)
liquids with similar densities, are extremely common in industrial
applications environment such as pharmaceuticals (Nielloud, 2000; Spernath &
Aserin, 2006), food processing (McClements, 2015) and oil production (Kokal &
Others, 2005; Mandal et al., 2010; Kilpatrick, 2012). Emulsions are also
important in geophysical applications: as example, when oil or industrial
wastes spill into water streams (from rivers to oceans), the oil droplet
distribution becomes fundamental for quantifying the environmental damage (Li
& Garrett, 1998; French-McCay, 2004; Gopalan & Katz, 2010).

At very low-volume-fraction, turbulent emulsions are mainly characterized by
the breakup of droplets. The droplet size distribution is produced by the
turbulent stresses and the feedback of the dispersed phase on the carrier flow
is small, and often neglected. The dynamics of droplet breakup, for a dilute
emulsion in a homogeneous and isotropic turbulent flow was initially
investigated by Kolmogorov (1949) and Hinze (1955), who derived an expression
for the maximum size of droplets resisting breakup as a function of the flow
characteristics and the fluid properties. This is usually referred to as the
Kolmogorov-Hinze (KH) scale. Recent numerical investigations of droplets,
bubbles and emulsions confirmed the general validity of the KH theory, both in
isotropic and homogeneous turbulence (Perlekar et al., 2014; Mukherjee et al.,
2019; Rivière et al., 2021; Crialesi-Esposito et al., 2022b; Girotto et al.,
2022; Begemann et al., 2022), and in anisotropic flows (Soligo et al., 2019;
Rosti et al., 2020; Pandey et al., 2022), while some theoretical corrections
were lately proposed to account for scale-local nature of the process
(Crialesi-Esposito et al., 2022a; Qi et al., 2022).

At finite volume fraction of the dispersed phase, the distribution of the
droplet sizes results from the interplay between breakup and coalescence. In
this regime, the presence of droplets also modulates the underlying
turbulence, affecting the flow statistics both at large (Yi et al., 2021; Wang
et al., 2022) and small scales (Mukherjee et al., 2019; Freund & Ferrante,
2019; Vela-Martín & Avila, 2021; Crialesi-Esposito et al., 2022b). In
particular, the presence of a dispersed phase alters significantly the
statistics at the small scales, producing large deviations from the average
values of dissipation and vorticity (Crialesi-Esposito et al., 2022b).

In this work, we address the effects of the dispersed phase on the velocity
increments of the turbulent flow at moderate ($10\%$) and high ($50\%$) volume
fractions. We study the role of the interface separating the two phases by
examining the statistics of velocity increments between two points which are
conditioned to be either in the same phase or in different phases. We find
that the most important deviations from the statistics of single-phase flows,
quantified by the PDFs of the velocity increments, are concentrated in regions
around the interface, i.e. when the two points belongs to different phases.
Moreover, we show that the amplitude of the third-order structure function
(SF) is reduced because of the contribution of the points located around the
interface. This is associated with a reduction of the flux of kinetic energy
in the turbulent cascade, which, in combination with the surface tension term,
alters significantly the energy transport across scales. Finally, we discuss
the effects of the droplets on the local scaling exponents of the high-order
structure functions, which display a striking saturation at small scales.

The remaining of this paper is organized as follows. In Section 2 we introduce
the numerical method adopted for the simulations, Section 3 is devoted to the
presentation of the results and Section 4 summarises the main conclusions.

##  2 Methodology

We consider the velocity field ${\bm{u}}({\bm{x}},t)$ obeying the Navier-
Stokes equations

$\rho\left(\partial_{t}{\bm{u}}+{\bm{u}}\cdot{\bm{\nabla}u}\right)=-{\bm{\nabla}}p+{\bm{\nabla}}\cdot\left[\mu\left({\bm{\nabla}u}+{\bm{\nabla}u}^{T}\right)\right]+{\bm{f}}^{\sigma}+{\bm{f}}$
(1)

and the incompressibility condition ${\bm{\nabla}}\cdot{\bm{u}}=0$. In
Equation 1 $p$ is the pressure and $\rho$ is the density and $\mu({\bm{x}},t)$
is the local viscosity. The surface tension force is represented by the term
${\bm{f}}^{\sigma}=\sigma\xi\delta_{S}{\bm{n}}$ where $\sigma$ is the surface
tension coefficient, $\xi$ is the local interface curvature, ${\bm{n}}$ the
surface normal unit vector and $\delta_{S}$ represents a delta function which
ensures that the surface force is applied at the interface only (Tryggvason et
al., 2011). The last term ${\bm{f}}$ is a constant in time body force which
sustains turbulence by injecting energy at large scales. Here, we adopt the
so-called ABC forcing (Mininni et al., 2006) which reads
${\bm{f}}=(A\sin(k_{f}z)+C\cos(k_{f}y),B\sin(k_{f}x)+A\cos(k_{f}z),C\sin(k_{f}y)+B\cos(k_{f}x))$.
The forcing scale is given by $L_{f}=2\pi/k_{f}$.

We solve Equation 1 in a triply-periodic, cubic domanin of size $L=2\pi$,
discertized on a staggered uniform Cartesian grid. Spatial derivative are
discretised with a second-order centered finite difference scheme and time
integration performed by means of a second-order Adam-Bashford scheme. To
reconstruct the interface, we use the algebraic Volume of Fluid method,
MTHINC, introduced by Ii et al. (2012). A constant-coefficient Poisson
equation is obtained using the pressure splitting method (Dodd & Ferrante,
2014), which is solved using a Fast Fourier Transform direct solver. All the
simulations have been performed with the code FluTAS, described in Crialesi-
Esposito et al. (2023), where further details on the numerical methods
employed in this study can be found.

We consider four different cases, all using a fixed ABC forcing with $A=B=C=1$
and $k_{f}=2\pi/L_{f}=2$. The reference single phase (SP) simulation assumes
viscosity $\mu=0.006$, corresponding to a Taylor-scale Reynolds number
$Re_{\lambda}=15k^{2}/(\nu\varepsilon)^{1/2}\approx 137$, with
$\varepsilon=\nu\langle(\nabla{\bm{u}})^{2}\rangle$ the energy dissipation
rate, $k$ the turbulent kinetic energy and $\nu$ the kinematic viscosity. We
vary the volume fraction $\alpha=V_{d}/V$, defined as the ratio between the
volume of the dispersed phase $V_{d}$ and the total volume $V=L^{3}$, and the
viscosity ratio $\gamma=\mu_{d}/\mu_{c}$. As regards different volume
fractions, we analyze the two cases with $\alpha=0.1$ (hereafter MP10) and
$\alpha=0.5$ (hereafter MP50), while keeping $\gamma=1$ for both cases.
Finally, we study the case $\alpha=0.1$ and $\gamma=100$ (hereafter MPM). For
all multiphase (MP) simulations the density ratio among the two phases is kept
equal to $1$ and the Weber number $We=\rho L_{f}u_{rms}^{2}/\sigma$=42.6.

All the simulations are performed at a resolution $N=512$ which is sufficient
to resolve all the scales (see Crialesi-Esposito et al., 2022b); statistics
are accumulated over several large eddy turnover times $T=L_{f}/u_{rms}$ once
statistical stationary conditions have been reached. For further details on
the simulation setup, we refer the reader to Crialesi-Esposito et al. (2022b).

##  3 Statistics of the multiphase flow

In turbulent multiphase flows, part of the kinetic energy of the carrier phase
is absorbed at large scales by the deformation and breakup of the interface of
the dispersed phase, while the coalescence of small droplets, their surface
oscillations and relaxation from high local curvature re-inject energy in the
carrier phase at scales smaller than the Kolmogorov-Hinze scale (Crialesi-
Esposito et al., 2022a). The consequences of this complex exchange of energy
between the two phases are evident in the kinetic energy spectrum shown in
Figure 1. Comparing the spectra of a multiphase flow with that of a single-
phase flow sustained by the same forcing, we observe a suppression of energy
at low wavenumbers (i.e. large scales) and an enhancement at high wavenumbers.
This effect increases with the volume fraction $\alpha$ of the dispersed phase
(Mukherjee et al., 2019; Crialesi-Esposito et al., 2022b).

Figure 1: Kinetic energy spectra of SP flow (black continuous line), and MP
flows at $\alpha=0.1$ (red dashed line) and $\alpha=0.5$ (blue dash-dotted
line)

Because of the injection of energy at small scales, due to the droplet
dynamics, we expect higher intermittency of the velocity fluctuations in the
MP flow than in the SP flow at fixed amplitude of the external forcing. In
order to quantify this effect we compute the probability density functions
(PDF) of the longitudinal velocity increments
$\delta_{\ell}u=({\bm{u}}({\bm{x}}_{2})-{\bm{u}}({\bm{x}}_{1}))\cdot({\bm{x}}_{2}-{\bm{x}}_{1})/\ell$
at distance $\ell=|{\bm{x}}_{2}-{\bm{x}}_{1}|$. The comparison of the PDFs at
two scales within the inertial range, shown in Figure 2, confirms that the
velocity increments have larger fluctuations in the case of MP flows, in
particular at smaller values of $\ell$. This effect increases with the
concentration $\alpha$ of the dispersed phase. We also observe that in the
case $\gamma=100$ (i.e. when the dispersed phase is much more viscous than the
carrier phase) the effect of the droplets on the velocity increments vanishes
due to the damping of fluctuations in the dispersed phase, and we recover the
statistics of the SP flow.

(a)(b)

Figure 2: PDF of velocity increments at distance $\ell=0.03L_{f}$ (left panel)
and $\ell=0.12L_{f}$ (right panel), normalized by the standard deviation of
the SP case. The single-phase Kolmogorov scale is at $\ell\approx 0.008L_{f}$

Note that the PDF shown in Figure 2 are computed over the full simulation
domain, i.e. the velocity increments are computed among points
${\bm{x}}_{1,2}$ which can belong to both phases unconditionally. To
understand the role of the interface in the turbulent statistics, we therefore
compute the PDF of the velocity increments conditioned to points belonging to
the same or to different phases. Hence, we introduce three different PDFs of
the velocity increments depending on which phase the two points ${\bm{x}}_{1}$
and ${\bm{x}}_{2}$ belong to. We denote by $P_{cc}$, $P_{dd}$ and $P_{cd}$ the
PDFs relative to points belonging only to the carrier phase $c$, only to the
dilute phase $d$ and to both phases, respectively. We remark that, for small
values of $\alpha$, the statistics in the two phases are different. For
$\alpha=0.5$ and $\gamma=1$ the two phases are equivalent and therefore
$P_{dd}=P_{cc}$.

(a)(b)

(c)(d)

Figure 3: PDFs of velocity increments conditioned to the phases on which the
two velocities are measured: $cc$ (both points in the carrier phase, red
line), $dd$ (both points in the dispersed phase, orange line), $cd$ (one point
in each phase, blue line). Upper panel: Simulation MP10 with $\alpha=0.1$ and
$\gamma=1$. Lower panel: Simulation MP50 with $\alpha=0.5$ and $\gamma=1$.
Black line: PDF of velocity increments for a single-phase simulation with the
same parameters of the MP simulation. Dashed black line: Gaussian
distribution. All the PDFs are rescaled with the variance of the SP case.

Figure 3 (panels a, b) shows the conditional PDF (normalized with the
corresponding variance of the SP case) pertaining the simulation with volume
fraction $\alpha=0.1$. First, we note that the PDF of the carrier phase is not
too far from that of the SP case (and this is the case also for the variance).
In the dispersed phase, on the contrary, velocity increments develop
relatively larger tails at small separations. Remarkably, the PDF $P_{cd}$
develops the largest tails at small scale (panel a), a clear indication of the
role of the interface for small-scale intermittency in MP flows.

Similar observations can be made for the emulsion with $\alpha=0.5$, shown in
Figure 3 (panels c, d). As expected $P_{cc}=P_{dd}$, and also in this case the
data show that the leading contribution to the increased intermittency at
small scales comes from velocity increments across the interface, $P_{cd}$.
Note that, although the shapes of $P_{cd}$ are similar for MP10 and MP50,
their contribution to the overall flow statistics is different because of the
different statistical weight (i.e. the different extension of the total
interface).

A remarkable feature shown in Figure 3 is that the skewness of $P_{cd}$ at
small scales is opposite (i.e. positive) to that of $P_{cc}$. We remind that
the sign of the skweness is linked to the direction of the turbulent energy
cascade via the third-order velocity structure function (SF) defined as
$S_{3}(\ell)=\langle(\delta_{\ell}u)^{3}\rangle$. In the case of SP flows,
under the assumption of statistical stationariety, homogeneity and isotropy,
the Kolmogorov $4/5$ law gives $S_{3}(\ell)=-(4/5)\varepsilon\ell$, where the
viscous energy dissipation rate $\varepsilon$ is equal to the flux of the
turbulent cascade (Frisch, 1995). The negative skewness of the PDF of the
longitudinal velocity increments is therefore related to the direction of the
energy transfer and the negative amplitude of $S_{3}(\ell)$ is proportional to
the energy flux.

In MP flows, because of the opposite sign of the skewness of $P_{cd}$ with
respect to $P_{cc}$, we expect that the presence of the interface reduces the
energy flux associated to the turbulent cascade. This can be quantified by
looking at the third-order velocity structure function
$S_{3}(\ell)=\langle(\delta_{\ell}u)^{3}\rangle$, whose average can be
unconditioned, or conditioned to two points belonging either to the carrier
phase $S_{3}^{cc}(\ell)$, or to the dispersed phase $S_{3}^{dd}(\ell)$, or
points located on different sides of the interface $S_{3}^{dc}(\ell)$. These
are shown in Figure 4.

(a)(b)

Figure 4: Third-order stucture function $S_{3}(\ell)$ averaged on the whole
domain (violet line) or conditioned on the two phases of the flows. Simulation
with $\alpha=0.1$ (left panel) $\alpha=0.5$ (right panel) and $\gamma=1$. The
black line represents the SP case.

We see in the figure that the third-order SF of the MP turbulent flows is
qualitatively similar to the SP flow when averaged over the whole domain, yet
with a smaller amplitude. This is due the fact that part of the turbulence
energy is used to break the interface and the direct transfer of energy to
small scales is reduced (Crialesi-Esposito et al., 2022b). If we consider the
same quantity averaged over one of the two phases only: $S_{3}^{cc}(\ell)$ and
$S_{3}^{dd}(\ell)$, which are equivalent in a binary flow, the magnitude of
the flux increases and approaches the SP limit, indicating that the turbulent
cascade is not significantly affected when considering flow structures living
in one of the two phases. On the contrary, the flux across two points
belonging to different phases is strongly suppressed: the associated
$S_{3}^{dc}(\ell)$ is closer to zero and even changes sign at intermediate
scales for $\alpha=0.5$ (consistently with what suggested in Figure 3). The
physical interpretation is that the interface “decouples” the velocity fields
in the two phases which become less correlated and therefore with a reduced
energy flux, signaled by the reduction of $S_{3}(\ell)$. The precise behavior
of $S_{3}^{dc}(\ell)$ depends on the value of $\alpha$, as shown by the
comparison with the case $\alpha=0.1$ (see Figure 4, left panel). Positive
values of $S_{3}^{dc}(\ell)$, hint however to the possibility of scale-local
backscatter, which becomes more relevant at increasing volume fractions.
Nevertheless, the reduction of the energy flux at intermediate scales is a
general feature, independent on $\alpha$.

(a)(b)

(c)(d)

Figure 5: Local scaling exponents $\zeta_{p}$ of the structure functions, for
the single phase (Upper left panel), case MP10 at $\alpha=0.1$ (Upper right
panel), case MP50 at $\alpha=0.5$ (Lower left panel), case MPM at $\gamma=100$
(Lower right panel). In each figure, the exponents of different order $p$
assume increasing values. Vertical black dashed lines represent the
Kolmogorov-Hinze scale, computed in Crialesi-Esposito et al. (2022a). Curves
for $p=3$ is omitted.

The effects of the presence of a dispersed phase on the statistics of the
velocity fluctuations affects also the scaling behavior of the structure
functions of the absolute values of the longitudinal velocity increments
defined as $S^{a}_{p}(\ell)=\langle|\delta_{\ell}u|^{p}\rangle$. It is well
known that in SP flows the SFs display a power-law behavior
$S^{a}_{p}(\ell)\sim\ell^{\zeta^{p}}$ at scales $\ell$ in the inertial range
(Frisch, 1995). In this context, intermittency manifests in the non-linear
behavior of the scaling exponents: $\zeta^{p}\neq p/3$. In the MP flows,
because of the different physical processes which occur at scales larger and
smaller than the Kolmogorov-Hinze scale (dominated by brak-up and coalescence
respectively), we expect to observe a more complex scaling behavior. To
address this issue, we compute the local scaling exponents defined as the
logarithmic derivative of the SFs
$\zeta^{p}_{\ell}={\text{d}\log(S^{a}_{p}(\ell))}/{\text{d}\log(\ell)}$ and
here applied to multiphase flows for the first time.

The local scaling exponents $\zeta^{p}_{\ell}$ are displayed for $p\leq 8$ in
figure 5, where they are divided by the reference scaling exponent
$\zeta^{3}_{\ell}$ of the third order SF. In the SP case, panel a, we find
that the ratios $\zeta^{p}_{\ell}/\zeta^{3}_{\ell}$ are almost constant in the
inertial range $0.09\leq\ell/L_{f}\leq 0.32$. In the MP flows, the value of
the exponents are a little smaller but comparable to that of the SP case only
at large scales; we observe a dramatic decrease of the scaling exponents at
scales $\ell$ smaller than the KH scale $L_{KH}\approx 0.14L_{f}$ for the case
MP10 (panel b) and $L_{KH}\approx 0.19L_{f}$ for the case MP50 (panel b) (see
vertical line in figure). In particular, we observe a striking saturation of
the scaling exponents of the high-order SF with $p\geq 5$ at scales
$\ell\simeq 0.02L_{f}$ for the MP50 case and $\ell\simeq 0.04L_{f}$ for the
MP10 case. The saturation of the scaling exponents of the high order SFs
reveals the presence of strong velocity differences across the interface
between the two phases, which originates from the pressure jump at the
interface caused by the surface tension forces.

Note also that the saturation of the exponents is not observed when the
dispersed phase presents higher viscosity, case MPM in panel d. This is
consistent with the previous observations in Figure 2, showing that when
velocity gradients across the interface are significantly reduced (_e.g._ by
higher viscosity) no exponent saturation is observed.

##  4 Conclusions

We have discussed intermittency and scaling exponent obtained from direct
numerical simulations (DNS) of turbulent emulsions at moderate ($10\%$) and
high ($50\%$) volume fractions and two different values of the viscosity
contrast. As observed in previous works (Perlekar, 2019; Pandey et al., 2020;
Crialesi-Esposito et al., 2022b), the presence of a deformable interface
increases the intermittency in the flow and the energy content at small
scales, when the surface tension offers an alternative path for energy
transport across scales.

By investigating the statistics of the velocity increments conditioned to
points belonging to a single phase or to different phases, we demonstrate that
the increased intermittency is mostly due to the presence of strong velocity
differences across the interface between the carrier and the dispersed phase.

We also show that the presence of the dispersed phase causes a decrease of the
negative skewness of the PDF of the longitudinal velocity increments. This is
associated with a reduction of the flux of the kinetic energy from the forcing
scale to the viscous scales. In other words, the presence of a deformable
interface affects the vortex stretching and tilting associated to the classic
turbulent energy cascade of single-phase flows.

This effect becomes remarkable at the highest volume fraction considered here,
when the flux related to points lying on either side of the interface gives a
positive contribution to the distribution skewness. This suggests a not-
negligible backscatter in multiphase flows, expected in proximity of the
interface separating the two fluids. We interpret this reduced flux as due to
the absorption and dissipation of part of the kinetic energy of the turbulent
flow by the deformation and break-up of drops of the dispersed phase.

Finally, to understand the local properties of turbulence, we have analysed
the longitudinal Structure Functions at higher orders. Interestingly, at
scales larger than the Kolmogorov-Hinze, the exponents are only slightly
smaller than in the single-phase flow, which implies increased intermittency,
yet a similar anomalous scaling. More importantly, we report a neat saturation
of the exponents for structure functions higher than 3 at scales smaller than
the Kolmogorov-Hinze length. This is typically related to a strongly
intermittent dynamics and to the presence of jumps, here due to the pressure
differences across the interface induced by the surface tension.

A further demonstration that the interface is responsible of the increased
intermittency is given by the results for the flow at viscosity ratio
$\gamma=100$. In this case, small-scale fluctuations are damped, especially in
the more viscous dispersed phase, and the statistics approach those of the
single-phase turbulence with no exponent saturation.

These observations may prove fundamental for understanding small scale
dynamics in multiphase flows and for their future sub-grid modelling. Indeed,
our results indicate that a correct model would need to account for the
reduction of the energy fluxes near an interface. Moreover, we have shown that
the turbulence statistics approach those of the single-phase flow when the
droplets consist of a highly viscous fluid. This suggests that, despite
several global measures seem to indicate a similar dynamics (Olivieri et al.,
2022; Yousefi, 2022), the turbulence modulation is significantly different in
the case of rigid particles and deformable intrusions.

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[Acknowledgements]MCE, GB and SM acknowledge the support from the Departments
of Excellence grant (MIUR) and INFN22-FieldTurb. The authors acknowledge
computer time provided by the National Infrastructure for High Performance
Computing and Data Storage in Norway (Sigma2, project no. NN9561K) and by SNIC
(Swedish National Infrastructure for Computing).

[Funding]LB acknowledge the support from the Swedish Research Council via the
multidisciplinary research environment INTERFACE, Hybrid multiscalemodelling
of transport phenomena for energy efficient processes, Grant no. 2016-06119.

[Declaration of interests]The authors report no conflict of interest.

[Data availability statement]Data are available from the corresponding author
upon reasonable request.