# Elastic turbulence in two-dimensional cross-slot viscoelastic flows

D. O. Canossi  G. Mompean  S. Berti  Univ. Lille, ULR 7512 - Unité de
Mécanique de Lille Joseph Boussinesq (UML), F-59000 Lille, France

###### Abstract

We report evidence of irregular unsteady flow of two-dimensional polymer
solutions in the absence of inertia in cross-slot geometry using numerical
simulations of Oldroyd-B model. By exploring the transition to time-dependent
flow versus both the fluid elasticity and the polymer concentration, we find
periodic behaviour close to the instability threshold and more complex flows
at larger elasticity, in agreement with experimental findings. For high enough
elasticity we obtain dynamics pointing to elastic turbulence, with temporal
spectra of velocity fluctuations showing a power-law decay, of exponent in
between $-3$ and $-2$, and probability density functions of velocity
fluctuations that weakly deviate from Gaussianity while high non-Gaussian
tails characterise those of local accelerations.

## Introduction

The rheological behaviour of viscoelastic flows at vanishingly small inertia
can be related to strongly non-linear phenomena and includes an association of
viscous and elastic effects, with the latter being typically due to the
presence of flexible long-chain polymers in the solution. The elasticity of
the flow can give rise to complex dynamics that are relevant for both
fundamental studies and industrial applications, as e.g. efficient mixing and
heat transfer in microdevices Traore  _et al._ (2015), or painting and coating
processes Larson (1992); Shaqfeh (1996); Li  _et al._ (2012).

The purely elastic instabilities marking the transitions between different
flow regimes have been documented in a variety of geometrical configurations
Muller  _et al._ (1989); Shaqfeh (1996); Groisman and Steinberg (2000); Berti
_et al._ (2008), including complex ones, such as the abrupt axisymmetric
contraction McKinley  _et al._ (1991) and the lid-driven cavity Pakdel and
McKinley (1996). The cross-slot setup, made of two perpendicularly
intersecting channels with two inlets and two outlets, is, in this sense, no
exception. Due to its relevance for mixing and rheology, it has been the
subject of extensive studies. Indeed, experimental Scrivener  _et al._ (1979);
Arratia  _et al._ (2006); Sousa  _et al._ (2015), theoretical Harlen  _et al._
(1990); Becherer  _et al._ (2008) and numerical Remmelgas  _et al._ (1999);
Poole  _et al._ (2007); Rocha  _et al._ (2009) investigations have reported
about the existence of instabilities solely driven by elasticity in this
setup. It is now known that low-Reynolds-number polymeric flows in this
geometry can display two types of instabilities: a first one, corresponding to
a supercritical pitchfork bifurcation, to steady asymmetric flow Poole  _et
al._ (2007); Rocha  _et al._ (2009), and a second one leading to unsteady
oscillatory behaviour Arratia  _et al._ (2006); Poole  _et al._ (2007); Haward
_et al._ (2012). Concerning the latter, it is interesting to recall that it
has been provided numerical evidence, in a two-dimensional (2D) flow, that it
occurs via a supercritical Hopf bifurcation Xi and Graham (2009); a mechanism
relying on the role of stress gradients and the existence of a stagnation
point at the centre of the setup was also proposed Xi and Graham (2009).

Above a critical Weissenberg number ($Wi$), meaning for elasticity larger than
a threshold, purely elastic instabilities can lead to the appearance of
disordered flows corresponding to the dynamical regime known as elastic
turbulence Vinogradov and Manin (1965); Groisman and Steinberg (2000). As
shown in the seminal work Groisman and Steinberg (2000), where a swirling flow
between counter-rotating parallel disks was considered, and in subsequent ones
also employing different geometries Groisman and Steinberg (2001, 2004), such
flows are reminiscent of the turbulent ones occurring in Newtonian fluids. In
particular, they are characterised by a whole range of active scales,
irregular temporal behaviour, growth of flow resistance and enhanced mixing
properties Groisman and Steinberg (2001). Interestingly, however, the spectrum
of velocity fluctuations displays power-law behaviours, in both the temporal
($E(f)\sim f^{-\delta}$) and spatial ($E(k)\sim k^{-\delta}$) domains, with an
exponent (in absolute value) $\delta\approx 3.5>3$, corresponding to a smooth
flow essentially dominated by the largest spatial scales. It is worth to
remark that such experimental findings are supported by theoretical
predictions based on a simplified uniaxial model of viscoelastic fluid
dynamics in the absence of walls and in homogeneous isotropic conditions
Fouxon and Lebedev (2003). At the same time, it was recently pointed out in
Gupta and Vincenzi (2019) that numerical simulations based on standard
constitutive models may be dramatically affected by the polymer-stress
diffusivity typically added to the evolution equations to ensure numerical
stability, and that this particularly applies to flows characterised by
regions of pure strain. Notably, using a cellular forcing in two dimensions,
it was shown that kinetic energy spectra are considerably flatter in the
absence of artificial polymeric diffusion and scale as $k^{-2.5}$ Gupta and
Vincenzi (2019).

The elasticity-driven transition to turbulent-like states was experimentally
investigated in cross-slot devices of different aspect ratio (vertical size
over channel width), for more and less concentrated polymer solutions Sousa
_et al._ (2018). Independently of the aspect ratio, it was found that the more
concentrated solution undergoes a transition to unsteady flows that become
progressively more irregular when the Weissenberg number is increased. The
power spectra of velocity fluctuations, obtained from single-point time series
of the streamwise component measured in the outlet channel at midway from the
lateral walls (both in the horizontal and vertical directions), were
characterised by the presence of marked peaks (a fundamental frequency plus
some harmonics), and by a power-law behaviour of exponent smaller than $-3$,
at small and large $Wi$ values, respectively. In particular, for the smaller
aspect ratio, continuous spectra and features typical of elastic turbulence
were observed when $Wi\gtrsim 25$. For the more dilute solution, although the
phenomenology of the transitions was similar, the chaotic flow observed at
high $Wi$ did not show similar spectral properties.

In this letter we explore the unsteady viscoelastic flow regime occurring in a
2D cross-slot geometry at high elasticities and vanishing Reynolds number
($Re$) by means of extensive numerical simulations, for different polymer
concentrations. For this purpose we adopt Oldroyd-B model, i.e. the simplest
possible one, to describe the dynamics of the viscoelastic fluid. As in van
Buel  _et al._ (2018), where elastic turbulence was simulated in a 2D Taylor-
Couette system, we integrate the model evolution equations using the open-
source code OpenFOAM® Weller  _et al._ (1998); Pimenta and Alves (2017), which
allows control of the numerical instabilities associated with large
Weissenberg numbers Sureshkumar and Beris (1995). We provide numerical
evidence of the emergence of turbulent-like features for quite concentrated
solutions when $Wi$ is large enough. We analyse the transition to irregular
dynamics and we characterise the statistical properties of the high-$Wi$
flows, discussing the similarities and differences with experimental results.

## Model and methods

We consider an isothermal, incompressible, inertialess, 2D viscoelastic fluid
flow in a cross-slot geometry. The latter consists of two perpendicular and
bisecting channels of identical width $d$, with opposing inlets (here, along
the $x$ direction) and outlets (along the $y$ direction), as shown
schematically in fig. 1. The velocity field
$\bm{u}(\bm{x},t)=(u_{x}(\bm{x},t),u_{y}(\bm{x},t))$ at position $\bm{x}$ and
time $t$ evolves according to the momentum conservation equation

$\rho\left[\frac{\partial\bm{u}}{\partial
t}+\left(\bm{u}\cdot\bm{\nabla}\right)\bm{u}\right]=\bm{\nabla}\cdot\bm{T}-\bm{\nabla}p$
(1)

and the incompressibility condition $\bm{\nabla}\cdot\bm{u}=0$, where $\bm{T}$
is the total (viscous plus elastic) stress tensor, $p$ the pressure and $\rho$
the density.

Figure 1: (Colour online) Schematic of the cross-slot geometry. The dotted
square is the area where the analyses were conducted, with the two dots
indicating the positions where time series were recorded: probe 1 (entrance,
red), probe 2 (exit, blue). Inset: zoom of the central area and typical mesh
refining towards the centre of the setup; note that simulations were performed
with at least twice finer meshes.

In the framework of Oldroyd-B model Oldroyd (1950); Bird  _et al._ (1987), the
stress tensor $\bm{T}$ is the sum of a viscous component
$\bm{\sigma}=\eta_{s}\,\dot{\bm{\gamma}}$, with $\eta_{s}$ the zero-shear
dynamic viscosity of the solvent and
$\dot{\bm{\gamma}}=\bm{\nabla}\bm{u}+\left(\bm{\nabla}\bm{u}\right)^{T}$ the
strain-rate tensor, and an elastic one $\bm{\tau}$ due to polymers. The
constitutive equation for the extra-stress tensor $\bm{\tau}$ reads:

$\bm{\tau}+\lambda\left[\frac{\partial\bm{\tau}}{\partial
t}+\bm{\nabla}\cdot\left(\bm{u}\bm{\tau}\right)-\left(\bm{\nabla}\bm{u}\right)^{T}\cdot\bm{\tau}-\bm{\tau}\cdot\bm{\nabla}\bm{u}\right]=\eta_{p}\,\dot{\bm{\gamma}}\,,$
(2)

where $\lambda$ represents the largest polymer relaxation time and $\eta_{p}$
the polymer contribution to viscosity. An important parameter is the viscosity
ratio $\beta=\eta_{s}/(\eta_{s}+\eta_{p})$, which is inversely proportional to
the polymer concentration. Let us remark that in the limit $\beta\to 0$, one
recovers the upper-convected Maxwell (UCM) model Bird  _et al._ (1987),
accounting for the dynamics of very concentrated solutions. At fixed $\beta$,
the control parameters of the dynamics specified by eqs. (1) and (2) are the
Reynolds $Re=\rho U_{b}d/(\eta_{s}+\eta_{p})$ and Weissenberg $Wi=\lambda
U_{b}/d$ numbers, where $U_{b}$ is the (uniform) velocity at the inlet.

In spite of important limitations, such as the infinite extensibility of
polymers - and the consequent unbounded nature of extensional viscosity at
strain rates $\geq 1/(2\lambda)$ \- or the absence of shear-dependent
viscosity, Oldroyd-B model corresponds to the simplest differential
constitutive equation for viscoelastic fluids, and it exhibits normal stress
differences. Furthermore, it has been successfully employed to numerically
reproduce the basic phenomenology of elastic turbulence in different 2D
configurations Berti  _et al._ (2008); Plan  _et al._ (2017); van Buel  _et
al._ (2018).

### Numerical simulations

Equations (1) and (2) are integrated by means of the open-source numerical
solver rheoTool®, which was developed in the framework of the OpenFOAM®
simulation code Pimenta and Alves (2017). This solver is based on a finite-
volume discretisation and makes use of the log-conformation technique Fattal
and Kupferman (2004) to control the numerical instabilities appearing at high
$Wi$ values. We remark that no polymer-stress diffusion is included.

The cross-slot configuration has recently been proposed as a benchmark problem
Cruz  _et al._ (2014), for its geometrical characteristics and the existence
of the instability leading to asymmetric flow at appropriate $\beta$, $Re$ and
$Wi$ values. Similarly to the reference studies with this setup, here we set a
length to width ratio of $10:1$ for each of the four “arms”, which was
previously shown to be enough to ensure a fully developed flow away from the
inlet in a channel Durst  _et al._ (2005). The global mesh adopted for the
numerical integration is composed of four blocks, each of which corresponds to
an arm, with increased density of grid points when approaching the centre of
the system, plus a central square block with the smallest (uniform) grid size.
The results presented in the following were obtained with a total of $12801$
computational cells, corresponding to $51\times 51$ cells and a minimal grid
spacing $\Delta x_{min}=\Delta y_{min}\approx 0.02\,d$ in the central region.
The mesh refinement towards the centre in each arm is realised via a geometric
progression relation with a stretching factor $f_{s}=0.931$. In order to
verify the robustness of our results, some calculations, and particularly
those related to the instability thresholds, were repeated with a mesh twice
as refined. The results were qualitatively independent of the mesh size and
only slight differences in the values of the critical parameters were found.
The general phenomenology in the developed regime is also found to hold
similar using the more refined mesh.

A uniform velocity profile of amplitude $U_{b}$ is applied at both inlets,
where a homogeneous Neumann (zero gradient) boundary condition is specified
for the pressure field, whereas polymeric extra-stresses are set to zero. At
the outlets, a homogeneous Dirichlet (zero value) boundary condition is
imposed for pressure, as well as zero-gradient ones for velocity and extra-
stress fields. At the walls, no-slip conditions ($\bm{u}=0$) are applied to
the velocity field and a linear extrapolation technique is adopted for the
extra-stresses Pimenta and Alves (2017). The velocity and stress initial
condition corresponds to no flow.

The Weissenberg number was varied by changing the polymer relaxation time
$\lambda$ only; the polymer concentration was set by choosing $\eta_{s}$ and
$\eta_{p}$ such that their sum is constant. The Reynolds number, accounting
for the relative strength of the non-linear inertial term to the viscous one
in eq. (1), was kept fixed at $Re=0$ by neglecting the term
$(\bm{u}\cdot\bm{\nabla})\bm{u}$ in eq. (1) Rocha  _et al._ (2009), but we
checked that including the latter (and setting $Re=0.1$) did not strongly
affect the results on the instability critical parameters. Further, the
dynamics appear not to be very sensitive to the presence of the term
$\rho\partial_{t}\bm{u}$ in eq. (1).

## Results

Figure 2: (Colour online) Snapshots of the magnitude of the velocity field
(colour) and flow streamlines (black lines) for $\beta=1/9$ and $Re=0$.
Increasing $Wi$, different regimes are observed: steady symmetric (a), steady
asymmetric (b), unsteady disordered flow (c, d).

When increasing the elasticity of the solution, while keeping $\beta$ fixed,
in our numerical integrations, we observe a destabilisation of the flow, in
agreement with previous studies Arratia  _et al._ (2006); Poole  _et al._
(2007). The sequence of flow states that are selected depends on the polymer
concentration, however, and here we provide a full picture of the stability
portrait of the system as a function of both $\beta$ and $Wi$. Let us
preliminarily remark that below the onset of purely elastic instabilities the
flow coming from each of the inlets splits into two streams of equal flow
rate, a symmetric state, at the outlets (see fig. 2a). For concentrated
solutions ($\beta\lesssim 0.56$), the flow first transitions to a steady
asymmetric state (fig. 2b, where $\beta=1/9$). By measuring the degree of
asymmetry, expressed in terms of the excess flow rate in a stream, as a
function of $Wi$ we verified (results not shown) that this transition is a
supercritical pitchfork bifurcation. Our values of the critical Weissenberg
number are in good agreement with those reported in previous benchmark studies
Cruz  _et al._ (2014) (relative difference of less than $0.05$) both for
$\beta=1/9$ and the UCM case $\beta=0$. In this range of low $\beta$ values, a
second instability manifests when $Wi$ is further increased beyond a second
threshold value close to $1$, leading to time-dependent behaviour in the form
of regular oscillations of the asymmetric flow pattern (which stays similar to
that of fig. 2b). The situation changes for more diluted solutions (i.e. when
$\beta\gtrsim 0.56$). Indeed, in this case, the steady asymmetric flow regime
does not set in and a direct change from steady symmetric to unsteady flow is
observed. Remarkably, the same qualitative phenomenology is also found in
experiments in micro-scale devices Sousa  _et al._ (2015). In the time-
dependent regime, and particularly for low $\beta$, an increase of $Wi$
eventually gives rise to spatially and temporally more complex flows akin to
elastic turbulence ones. Two illustrative examples at fixed time are shown in
fig. 2c,d for $\beta=1/9$ and two different values of $Wi$.

Figure 3: (Colour online) Stability diagram in the $(\beta,Wi)$ plane at
$Re=0$. The green squares, blue diamonds and red dots respectively correspond
to steady symmetric, steady asymmetric and unsteady flow. The dashed
($Wi_{c}^{(\mathrm{I})}$) and continuous ($Wi_{c}^{(\mathrm{II})}$) lines are
fits using eq. (3); here $a_{0}^{(\mathrm{I})}\simeq 2.75$,
$a_{-1}^{(\mathrm{I})}\simeq-3.94$, $a_{0}^{(\mathrm{II})}\simeq 0.85$,
$a_{-1}^{(\mathrm{II})}\simeq 0.05$. Inset: amplitude and frequency of
$|\bm{u}(\bm{x}_{*}^{(2)},t)|$ vs $Wi$ at the onset of unsteady flow, for
$\beta=1/9$.

The complete stability portrait, obtained by spanning the ($\beta,Wi$) plane
with a large number of simulations, is shown in fig. 3, where the different
point types correspond to the different dynamical regimes observed; here we
only show a limited subset of the results from the simulations performed. By
measuring the amplitude and frequency of the time series of
$|\bm{u}(\bm{x}_{*}^{(2)},t)|$ at the fixed location $\bm{x}_{*}^{(2)}$
(corresponding to point $2$ in fig. 1) for $Wi$ close to the onset of the
unsteady regime and for different concentration values, we could assess that
the second instability is a supercritical Hopf bifurcation (see inset of fig.
3 for $\beta=1/9$), as also suggested by Xi and Graham (2009) using a FENE-P
model at non-zero $Re$ and large $\beta$. Indeed, the velocity signal displays
a growth of its amplitude that is fairly well described by $(Wi-
Wi_{c}^{(\mathrm{II})})^{1/2}$, with $Wi_{c}^{(\mathrm{II})}$ the critical
Weissenberg number, and an approximately linear decrease of its frequency with
$Wi$. For both the first and the second instability, the critical Weissenberg
number, $Wi_{c}^{(\mathrm{I})}$ and $Wi_{c}^{(\mathrm{II})}$ respectively,
grows with growing $\beta$, which is reasonable since increasing $\beta$
corresponds to decreasing polymer concentration. The faster growth of
$Wi_{c}^{(\mathrm{I})}(\beta)$ causes the shrinking of the region of steady
asymmetric flow. Determining the functional dependencies $Wi_{c}^{(i)}(\beta)$
(with $i=\mathrm{I},\mathrm{II}$) from stability analysis is not an easy task,
due to the formation of a birefringent strand and a diverging base state
associated with the infinite extensibility of polymers Becherer  _et al._
(2008). Since here we are mainly interested in characterising the boundaries,
in the $(\beta,Wi)$ plane, of the regions where elastic turbulence could be
excited, we proceed heuristically, especially focusing on
$Wi_{c}^{(\mathrm{II})}(\beta)$. In order to account for non-zero $\beta$
effects, we conjecture that
$Wi_{c}^{(\mathrm{II})}(\beta)=Wi_{c}^{(\mathrm{II})}(0)f(\beta)$, where
$f(\beta)$ is a positive analytic function, except for $\beta\to 1$ where a
divergence is expected, since the fluid becomes Newtonian and no purely
elastic instability should occur; clearly $f(0)=1$. Our numerical results
suggest that the data are compatible with a Laurent expansion at second order
around the point $\beta=1$. Somehow more surprisingly, we find that the same
functional shape can also be used to fit the $Wi_{c}^{(\mathrm{I})}(\beta)$
data, indicating that:

$Wi_{c}^{(i)}=Wi_{c}^{(i)}(0)\left[a_{0}^{(i)}+\frac{a_{-1}^{(i)}}{1-\beta}+\frac{a_{-2}^{(i)}}{(1-\beta)^{2}}\right]\,,$
(3)

where $a_{-2}^{(i)}=1-a_{0}^{(i)}-a_{-1}^{(i)}$ using the constraint $f(0)=1$,
and $i=\mathrm{I},\mathrm{II}$. In fig. 3 we report a comparison between a fit
with function (3) (dashed and continuous lines for $i=\mathrm{I},\mathrm{II}$,
respectively) and the numerical data; the agreement is rather good for both
instability types, confirming our conjecture.

To conclude this discussion, we mention that in our calculations with a more
refined grid or at $Re=0.1$ (see the previous section for the details about
simulations) we did not observe any qualitative difference in the dynamical
regimes occurring for different values of $\beta$ and $Wi$.

We now consider the transition to turbulent-like flow. In the following we
will present the results of the analysis performed for increasing $Wi$ at
$\beta=1/9$. Notwithstanding some quantitative differences, the phenomenology
holds similar in the whole range ($\beta\lesssim 0.56$) of concentrated
solutions, including for UCM ($\beta=0$). In the case of more diluted
solutions, while we observed some hints of the onset of irregular flow, we
could not reach a fully developed regime and we cannot conclude about the
emergence of elastic turbulence. Notice that for such large values of $\beta$,
the critical Weissenberg number $Wi_{c}^{(\mathrm{II})}$ grows very rapidly,
making the simulations more and more delicate.

Figure 4: (Colour online) Temporal evolution (subset of the total data set,
see text) of the $y$-component of velocity at the outlet (probe $2$),
normalised by its time average over the whole time series, after the
transient, for $Wi=1.55,3,6,12$ (from top to bottom), $Re=0$ and $\beta=1/9$.

Our analysis is based on the measurement of time series of the velocity
components at two different positions marked as probe $1$ ($\bm{x}_{*}^{(1)}$,
entrance) and probe $2$ ($\bm{x}_{*}^{(2)}$, exit) (see fig. 1), over long
durations corresponding to at least $800\lambda$, and up to $1000\lambda$. As
for the experiments reported in Sousa  _et al._ (2018), we choose to focus on
the axial component $u_{y}(\bm{x}_{*}^{(2)},t)$ at the exit probe, whose
behaviour is presented in fig. 4 for several values of $Wi$. Remark that in
this figure the initial transient was removed and only a subset of the data
record is shown.

Figure 5: (Colour online) Temporal spectra of fluctuations of the axial
velocity at the outlet $u_{y}(\bm{x}_{*}^{(2)},t)$, normalised by their
integral $E_{y}^{tot}$ in the elastic turbulence regime for $Re=0$ and
$\beta=1/9$; the curves have been vertically shifted to ease readability. The
dashed black curves stand for $E_{y}(f)\sim f^{-\delta}$, the fitted values of
$\delta$ are $\delta\simeq(2.8,2.5,2.2,2.1)\pm 0.4$ for $Wi=6,12,20,25$,
respectively. Inset: similar spectra at lower elasticity. For $Wi=1.55\gtrsim
Wi_{c}^{(\mathrm{II})}$, a single frequency peak is found; at larger $Wi=3$
more discrete frequencies are present.

The spectra of $u_{y}(\bm{x}_{*}^{(2)},t)$ are shown in fig. 5. All those
corresponding to the developed regime are averages over ten spectra computed
from consecutive subintervals of the velocity time series obtained for a given
value of $Wi$ (after the transient). For $Wi\gtrsim Wi_{c}^{(\mathrm{II})}$,
time dependency manifests in the form of regular oscillations with a single
frequency close to $0.4/\lambda$ (see inset of fig. 5). At slightly higher
Weissenberg number ($Wi=3$ in fig. 4) the flow is still periodic but it is now
characterised by more discrete frequencies; correspondingly, the spectrum
shows several distinct peaks associated with a fundamental frequency and some
harmonics (inset of fig. 5). The occurrence of a transitional periodic regime
was also found in different setups Schiamberg  _et al._ (2006); Berti and
Boffetta (2010). Above $Wi\approx 5$, the flow loses periodicity and the
velocity spectra become continuous. Indeed, starting from $5\lesssim
Wi\lesssim 10$ they result to be quite well described by a power-law function
(fig. 5). When elasticity is increased in the range $Wi>10$, the faster
fluctuating behaviour of the flow is accompanied by quite wide and irregular
oscillations, over longer durations. The flow now loses its spatial asymmetry
to alternatively select the outlet in the positive/negative $y$-direction.
Such a phenomenon has a strong impact on the statistics of the transversal
velocity component $u_{x}(\bm{x}_{*}^{(2)},t)$ at the outlet (and similarly on
$u_{y}(\bm{x}_{*}^{(1)},t)$ at the inlet), whose fluctuations are accompanied
by irregular jumps between two mean values of opposite sign (see fig. 6), thus
complicating their analysis. A detailed investigation of the behaviour of such
a two-state system goes beyond the scope of the present work.

Figure 6: (Colour online) Temporal evolution (subset of the total record) of
$u_{x}(\bm{x}_{*}^{(2)},t)$, normalised by its time average over the whole
time series, after the transient, for $Wi=11,12,15,20$ (from top to bottom),
$Re=0$ and $\beta=1/9$.

In the turbulent-like regime ($Wi>5$), the spectrum of velocity fluctuations
displays a power-law behaviour $E_{y}(f)\sim f^{-\delta}$ beyond a frequency
that, as in experimental studies Sousa  _et al._ (2018), slightly increases
with $Wi$. The absolute value of the exponent is found to be in the range
$2\lesssim\delta\lesssim 3$ and shows some tendency to decrease at higher
$Wi$; the latter feature is also detected in experiments Sousa  _et al._
(2018); Varshney  _et al._ (2016). In particular, we find
$\delta\simeq(2.8,2.5,2.2,2.1)\pm 0.4$ for $Wi=6,12,20,25$, respectively. The
spectra are thus overall less steep than those previously found in experiments
Groisman and Steinberg (2000); Sousa  _et al._ (2018) and those theoretically
predicted assuming homogeneity and isotropy Fouxon and Lebedev (2003),
pointing to more energetic small scales, as e.g. the quite localised ones
(fig. 2c,d) stemming from intense polymer stretching, and less smooth flow.
However, they bear an interesting similarity with those obtained in 2D
numerical simulations, without artificial polymer-stress diffusion, of
Oldroyd-B model in the presence of a cellular forcing generating distinct
regions of strain and vorticity Gupta and Vincenzi (2019). A possible reason
for the difference with the prediction of Fouxon and Lebedev (2003) is the
lack of the statistical symmetries assumed by the theory in the present case.
Indeed, our flow is neither homogeneous (due to the presence of the walls, but
also of the high-strain region close to the centre of the setup), nor fully
isotropic, as we typically observe that $u_{y}^{rms}>u_{x}^{rms}$ for the
root-mean-square (rms) velocity components. Moreover, the turbulent intensity
$u^{rms}/\overline{u}$, here defined as the ratio of the rms to the mean value
of the full velocity modulus $u\equiv|\bm{u}|$ (with the overbar denoting a
temporal average), can quite easily exceed $0.5$, and be as high as $\approx
0.8$ in conjunction with the temporal oscillations of the spatial asymmetry of
the flow. Therefore, the validity of Taylor’s hypothesis Taylor (1938); Frisch
(1995), allowing to convert spectra from the frequency to the wavenumber
domain, appears questionable. It might be the case that its refined version
could be applied, as in Burghelea  _et al._ (2005), but addressing this
question requires further investigations. Finally, although previous numerical
studies in two dimensions have revealed that the spectral exponent of elastic
turbulence seems to be quite insensitive to the space dimensionality Berti
_et al._ (2008); Plan  _et al._ (2017); van Buel  _et al._ (2018), we cannot
exclude that the 2D nature of our flow has an impact. Note, too, that values
of $\delta$ for $Wi\geq 25$ should be taken with caution, as they may also
likely depend on the length of the inlet/outlet channels.

Figure 7: (Colour online) Probability density functions of normalised velocity
fluctuations $u_{y}^{\prime}=(u_{y}-\overline{u_{y}})/\sigma_{u_{y}}$ (a) and
temporal increments
$w_{y}=(\partial_{t}u_{y}-\overline{\partial_{t}u_{y}})/\sigma_{\partial_{t}u_{y}}$
(b), where $u_{y}\equiv u_{y}(\bm{x}_{*}^{(2)},t)$, the overbar denotes the
temporal average and $\sigma$ the standard deviation, for different values of
$Wi$, $Re=0$ and $\beta=1/9$. The insets show the pdf’s of the same quantities
along $x-$direction (velocity and temporal-increment fluctuations, in (a) and
(b), respectively). In all panels the solid black lines are standard Gaussian
pdf’s.

To further characterise the statistical properties of our elastic turbulent
flows, we computed the probability density functions (pdf’s) of the
fluctuations of the velocities $u_{x,y}(\bm{x}_{*}^{(2)},t)$, as well as of
the local accelerations $\partial_{t}u_{x,y}(\bm{x}_{*}^{(2)},t)$, obtained
from the temporal signals at probe $2$. The results are presented in fig. 7,
where all variables are rescaled with the corresponding standard deviation
$\sigma$. The statistics of $u_{y}$ fluctuations are not far from Gaussian,
but negatively skewed for large enough $Wi$ (fig. 7a), probably due to the
establishment of a transversal flow component via intermittent bursts
Burghelea  _et al._ (2007). Those of $u_{x}$ are less so (inset of fig. 7a),
instead, and show a bimodal shape for $10\lesssim Wi\lesssim 20$, which
reflects the importance of the flow-asymmetry alternation events in this range
of elasticities. A qualitatively similar phenomenology is found at the
entrance probe $1$, provided $x$ and $y$ indices are exchanged. The statistics
of fluctuations of the accelerations are remarkably less dependent on the
Weissenberg number (and the probe location), suggesting a faster (with $Wi$)
onset of scaling properties at small scales. As it can be seen in fig. 7b, the
corresponding pdf’s display high tails that are indicative of non-Gaussian
statistics, as is typical in turbulent flows and as observed in elastic
turbulence experiments Burghelea  _et al._ (2007). This finding highlights the
intermittent behaviour of local accelerations, likely due to the passage
through the system of transient intense filamentary structures (fig. 2c,d, but
see also Berti and Boffetta (2010); Grilli  _et al._ (2013); Varshney and
Steinberg (2019) about the role of elastic propagating wavy patterns).

## Conclusions

We investigated numerically the dynamics of Oldroyd-B fluids in a 2D cross-
slot geometry for broad ranges of the Weissenberg number and the polymer
concentration, focusing on the possibility to obtain elastic turbulence. We
detected two instabilities: the first one, present only for rather
concentrated solutions (see also Sousa  _et al._ (2015)), leads to steady
asymmetric flow; the second one, less documented, manifests for all viscosity
ratios $\beta<1$ and corresponds to a supercritical Hopf bifurcation. By
characterising the dependence of the critical Weissenberg number
$Wi_{c}^{(\mathrm{II})}$ on the viscosity ratio, we found a heuristic
expression that allows to quantitatively delimit the regions
$Wi>Wi_{c}^{(\mathrm{II})}(\beta)$ where elastic turbulence may be excited.

Close to the onset of the second instability, the flow of quite concentrated
solutions displays regular oscillations in time, while at larger elasticities
its dynamics appear more irregular. The frequency spectra measured in one of
the outlets and far from the walls show distinct peaks for $Wi\gtrsim
Wi_{c}^{(\mathrm{II})}$, while for $Wi\gtrsim 5$ they are well described by
continuous power-law functions, of exponent $-\delta$, pointing to elastic
turbulence. As in experiments Sousa  _et al._ (2018); Varshney  _et al._
(2016), the scaling range occurs beyond a frequency that moderately increases
with $Wi$, and $\delta$ decreases with $Wi$. However, we obtain values
$2<\delta<3$, somehow smaller than the experimental ones and the theoretical
prediction for the homogeneous isotropic case Fouxon and Lebedev (2003). While
we cannot exclude an impact of the 2D nature of our flow here, and we recall
the influence of the inlets/outlets’ length on the results for $Wi\geq 25$, we
remark that the symmetries assumed in the theory clearly do not hold for our
setup. Similarly energetic spectra have been recently found in simulations of
2D Oldroyd-B cellular flows without polymer-stress diffusion Gupta and
Vincenzi (2019).

Further, the statistics of axial velocity components are found to be weakly
non-Gaussian in the developed regime, while those of transversal ones also
exhibit a bimodal pdf for $10<Wi<20$ due to the alternations of the spatial
flow asymmetry occurring in this range of $Wi$. The pdf’s of both components
of the local accelerations, instead, present high non-Gaussian tails
indicative of intermittency. Such a phenomenology agrees with that observed in
experiments (see e.g. Burghelea  _et al._ (2007)).

In summary, we reproduced the different dynamical regimes experimentally
observed in cross-slot devices, and we obtained turbulent-like states bearing
good statistical resemblance with elastic turbulence. The quantitative
differences highlighted call for further theoretical and numerical
developments. In the future it would be interesting to explore such dynamics
in three-dimensional flows.

###### Acknowledgements.

D. O. Canossi acknowledges financial support from a PhD grant funded by the
Brazilian agency CNPq (Conselho Nacional de Desenvolvimento Científico e
Tecnológico). We are grateful to N. Ouarzazi and E. Calzavarini for
interesting discussions about instabilities and turbulent-like features,
respectively.

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# Benchmarking performance of object detection under image distortions in an
uncontrolled environment

###### Abstract

The robustness of object detection algorithms plays a prominent role in real-
world applications, especially in uncontrolled environments due to distortions
during image acquisition. It has been proven that the performance of object
detection methods suffers from in-capture distortions. In this study, we
present a performance evaluation framework for the state-of-the-art object
detection methods using a dedicated dataset containing images with various
distortions at different levels of severity. Furthermore, we propose an
original strategy of image distortion generation applied to the MS-COCO
dataset that combines some local and global distortions to reach much better
performances. We have shown that training using the proposed dataset improves
the robustness of object detection by 31.5%. Finally, we provide a custom
dataset including natural images distorted from MS-COCO to perform a more
reliable evaluation of the robustness against common distortions. The database
and the generation source codes of the different distortions are made publicly
available111https://github.com/Aymanbegh/Benchmarking-
performance,222https://github.com/Aymanbegh/Distorted-Natural-COCO.

Index Terms—  Deep learning, Object detection, Distortion, Robustness,
Benchmarking

##  1 Introduction

Object detection (OD) and recognition is one of the most challenging and
widely studied problems in computer vision and especially in uncontrolled
environment. Deep Neural Networks (DNN) based approaches [1, 2, 3] have been
proven very promising paradigms to solve such complex problems in various
real-world applications. However, generally, the proposed DNN-based
architectures do not consider the sensitivity of the learning models to common
data distortions and corruption effects. Indeed, it has been shown by Szegedy
et al.[4] that a small perturbation invisible to the human visual system can
have significant repercussions on the performance of object recognition
models. It is therefore important to conduct ad hoc studies to understand the
effects of such perturbations and to be able to generate them at different
levels in order to develop more robust OD networks. The limitations of object
detection methods, in terms of robustness against some common signal
degradation, data corruptions, or adversarial examples, have been highlighted
in several works [5, 6, 7, 8]. A solution would be to perform pre-processing
before the high-level phases. But this implies being able to identify the
distortion automatically in order to apply the most appropriate solution. A
more elegant and efficient solution present in many research [9, 10, 11, 12]
is to enrich the database used for learning by simulating the most relevant
distortions. It is in this last framework that the proposal described in this
work is made. Thereby, we propose a full framework evaluation of robustness
for a set of object detection methods [13, 14, 15] through several distortions
applied on the MS-COCO dataset [16]. This data augmentation is performed
through some common global distortions (noise, motion blur, defocus blur,
haze, rain, contrast change, and compression artefacts) in the whole image and
some local distortions (local blur motion, BackLight illumination and local
defocus blur) in specific areas that include the possible dynamic objects or
scene conditions. To the best of our knowledge, our work is the first to
consider the local distortions to assess the robustness of OD models. Finally,
we proceeded to a manual selection of the distorted images present in the
validation set of the MS-COCO dataset and retained only the annotations of the
distorted objects to perform a more relevant evaluation of the impact of
distortions on models. The main contributions of our study are summarized as
follows:

  * •

A comprehensive evaluation of the robustness of the state-of-the-art OD
methods against global and local distortions at 10 levels of distortion is
conducted (see section 4.1).

  * •

A dataset dedicated to the study of the impact of local and global distortions
on the robustness of OD is built from MS-COCO dataset [17].

  * •

It is shown that the robustness of OD process is improved by enriching the
training process using data augmentation based on generated distorted images
(see table 3).

  * •

An additional new benchmark dataset constructed for natural in-capture
distortions from MS-COCO dataset that allows to better assess the reliability
of the OD models in case of real distortions is provided (see tables 2 and 4)
[18].

The remainder of the paper is organized as follows. Section 2 summarizes
previous related literature. Section 3 is devoted to detail the methods of
evaluation and the distortion generations. Then, section 4 is dedicated to
show and discuss results. Finally, concluding remarks and perspectives are
provided in section 5.

(a) Original

(b) Local defocus blur

(c) BackLight illumination (BL)

(d) Local motion blur

Fig. 1: Illustration of some local distortions.

##  2 Related Work

The evaluation of OD methods from images acquired in an uncontrolled
environment has been the subject of a few studies [19, 20]. But most of the
time, the used databases contain images with limited number of degradation
types and therefore cannot be generalized to practical cases where one has to
face several distortions at the same time. Some papers deal with the impact of
various distortions on the detection performance, described in [21], that
attempts to evaluate the robustness of some backbones of CNN architectures
against geometric transformations. A comparative study between DNN-based
architectures (ResNet-152, VGG-19, GoogLeNet) and human observer performance
was conducted in [22]. In this study twelve common distortions are considered
in the performance evaluation of object recognition methods. It has been shown
that these DNN-based architectures perform remarkably well in the case of the
distortions on which the training was performed. However, this performance
decreases, compared to that of the human observer, for unseen distortions.
Another interesting study on the evaluation of the robustness of
classification models of images from the imagenet database and subjected to
fifteen artificial distortions at five levels of severity, revealed the effect
of image quality degradation on the stability of the training process for
various DNN-based architectures [6]. Recently, a benchmark study on the
robustness of many OD models against 15 types of global distortions and
stylized images through data augmentation process has been done in [5].

##  3 Proposed benchmarking framework

In order to evaluate the robustness of the considered models against image
distortions in an uncontrolled environment, we propose 10 types of distortions
(global and local) at 10 linearly increasing levels of severity [17]. In
figures 1 (b), (c), and (d), distortion levels are 8, 7, and 10 respectively.

###  3.1 Image distortions

The distortions of the image signal can be classified into two main categories
described below. Global distortions affect the image as a whole and come from
different sources related in general to the acquisition conditions. Some are
directly dependent on the physical characteristics of the camera and are of
photometric or geometric origin. Among the most common distortions that affect
the quality of the signal are defocus blur (Defocus-Blur), photon noise,
geometric or chromatic aberrations, and blur (Motion-Blur) due to the movement
of the camera. The other types of degradation are related to the environment
and, more particularly, the lighting and atmospheric disturbances in the case
of outdoor scenes. Compression and image transmission artifacts are another
source of degradation that is difficult to control. These common distortions
have been already considered in benchmarking the performance of some models
[5, 6, 8]. Local distortions are undesirable signals affecting one or more
localized areas in the image (see figure 1). A typical case is the Motion Blur
(Loc. MBlur) due to the movement of an object of relatively high speed.
Another photometric distortion is the appearance of a halo around the object
contours due to the limited sensitivity of the sensors or backlight
illumination (BackLight). The artistic blur (Loc. Defoc.) affecting a
particular part of the targeted scene, the object to be highlighted by the
pro-shooter, is another type of local distortion. Thus, integrating the local
distortions in the database increases its size and makes it richer and more
representative of scenarios close to real applications. It is worth noticing
that these local distortions are made possible thanks to the annotations
provided in the MS-COCO database and, in particular, the details of the shape
and location of the object of interest.

###  3.2 Description of the considered models

In this study we focus on few models of state-of-the-art object detection
methods which have the best performance or notoriety, namely YOLOv4 [13], Mask
R-CNN [15]

Table 1: State-of-the-art object detection models.

Methods |  Architecture |  Size |  Add-blocks
---|---|---|---
Mask R-CNN [15] |  Resnet-101 |  512 |  FPN, RPN [23, 24]
YOLOv4 [13] |  CSPDarknet53 |  512 |  PAN [25]
YOLOv4-tiny |  416
EfficientDet [14] |  EfficientNet-B0 |  512 |  Bi-FPN [14]
EfficientNet-B1 |  640
EfficientNet-B2 |  768
EfficientNet-B3 |  896
EfficientNet-B4 |  1024

FPN: Feature Pyramid Network, RPN: Region Proposal Network, PAN: Path
Aggregation Network, Bi-FPN: Bi-directional FPN.

and EfficientDet [14] with distinct backbone architectures that highlights the
correlation between the backbone and their robustness. The table 1 summarizes
the different models and their components which are used in this benchmarking.

###  3.3 Robustness Evaluation against Natural Distortions

We manually identified the natural distortions present in the COCO validation
set to better assess the robustness in real scenarios. Indeed, in the existing
methods, robustness evaluations were performed on synthetic distortions or
specific datasets, thus limiting those studies’ extension in real-world
applications. It is worth noticing that the selected sub-sets from COCO
contain images with global distortions and the associated object annotations.
In contrast, only affected objects are considered in the case of local
distortions (see figure 2).

(a) Original annotations

(b) Sorted annotations

Fig. 2: Natural distortions selection in the MS-COCO dataset.

This benchmarking framework for robustness evaluation uses one of the most
widespread databases for OD, restricted to a limited set of seven natural
distortions types (see table 2). In this way, we aim to lay the foundations
for a future common evaluation method for evaluating OD robustness against
real-world distortions.

Table 2: Features of distorted natural sub-sets.

|  Noise |  Contrast |  Blur |  Defoc. |  Rain |  O.Blur |  BackL.
---|---|---|---|---|---|---|---
Images |  44 |  42 |  32 |  201 |  21 |  127 |  128
Objects |  289 |  312 |  224 |  1299 |  110 |  464 |  934
Ratio |  1.0 |  0.83 |  0.97 |  0.72 |  0.95 |  0.34 |  0.68

Ratio: Number of retained objects/annotated objects in images.

It is worth noting that some natural distortions such as Defocus, Object Blur,
and BackLight have low ratios (see table 2). This results from the nature of
local distortions, which only affect specific objects in the images (see
figure 2).

##  4 Results and discussions

###  4.1 Evaluation of the robustness

We evaluated the robustness of models against the distortions on the MS-COCO
2017 validation set,

Fig. 3: Robustness of object detection models against global(left)/local
(right) distortions.

which contains 5K annotated images and corrupted by the 10 types of
distortions we generated [17]. it is worth noting that level 0 of distortion
corresponds to non-distorted images. We assessed the performance of the ODn
algorithms using the AP (Average Precision) metric.

Fig. 4: Robustness of object detection models against distortions.

Figures 3 and 4 illustrate the evolution of the AP score of the considered OD
models according to each distortion type and level. These results show that
the distortions reduce object detection performance of the OD methods by 20%
to 89.2%. Moreover, they highlight the weaknesses of YOLOv4 method against
distortions (strongly decreasing curves in figures 4 and 3), which needs a
deeper study. The evaluations in figure 3 indicates that the global
distortions have more impact than local distortions on OD performance. While
local and atmospheric disturbances have a roughly uniform impact on OD methods
despite their different architectures, the other distortions produce effects
separate and independent. Indeed, overall models performance are greatly
affected by distorted images but not uniformly. Each type of distortion has a
different rate of impact on models performance depending on the distortion
level and OD method. In order to better evaluate the impact of distortions on
OD models, we computed the robustness rate that represents the ratio between
mAP scores of the models related to each distortion type and level, and non-
distorted images (level 0). This robustness rate is illustrated in (see figure
5) using violin plot where the most relevant statistical features are
highlighted. Results from figures 3, 4 and 5 indicate that the EfficientDet
method’s robustness decreases despite its network depth (D0 to D3).

*/-: Mean/Median of mAP for all distortion type and level, Light/dark surrounding shape: probability density/percentiles.

Fig. 5: Average robustness rate of object detection models against the
distortions.

Furthermore, the YOLOv4-tiny model was the least robust to distortions with an
average decrease with AP score of 36.7% between the distortion levels 0 and 10
(lowest average robustness rate at 63.3%, see mAP score in figure 5). This
observation lead us to choose this model to evaluate the improvement in
robustness using a data augmentation as proposed in our dataset.

###  4.2 Training on distorted dataset

This section presents experimental results on MS-COCO 2017 evaluation set with
5K images. YOLOv4-tiny model is trained following the protocol in MS-COCO
challenge, i.e., using the trainval35k set for training that includes 80k
images from the train set and 35k images from the validation set.

Table 3: Object detection percentage performance of the YOLOv4-tiny trained model on our distorted dataset. Distortion | Distortion level (%) |  mAP
---|---|---
Type |  2 |  4 |  6 |  8 |  10 |  Average
Noise |  9.58 |  22.3 |  37.9 |  80.3 |  114 |  47.2%
Contrast |  -0.54 |  1.8 |  2.47 |  3.23 |  5.71 |  1.99%
Compression |  -2.36 |  -0.48 |  -0.5 |  4.84 |  35.4 |  4.26%
Rain |  12.7 |  40.4 |  71.4 |  101.8 |  114.3 |  61.7%
Haze |  -0.49 |  3.14 |  10.5 |  29.8 |  51.7 |  15.8%
Motion-Blur |  6.11 |  39.5 |  95.3 |  151.4 |  183.3 |  86.3%
Defoc-Blur |  1.02 |  14.2 |  22.4 |  26.3 |  26.7 |  16.5%
Loc. MBlur |  10.3 |  31.6 |  46.9 |  59.3 |  67.0 |  39.8%
Loc. Defoc. |  3.08 |  16.0 |  23.3 |  25.5 |  25.9 |  17.2%
BackLight |  1.45 |  5.85 |  19.1 |  40.3 |  76.8 |  24.1%
Average |  4.11% |  17.9% |  32.9% |  52.3% |  70.1% |  31.5%
per level

We first performed this training with the original MS-COCO trainval35k set to
obtain reference values, then training with the distorted dataset, which
allowed us to evaluate the contribution of the data augmentation on the
robustness of the model. Each distortion type represents 5% of the database
used for the training process, i.e., 5.9K images for each distortion. In our
case, the hyper-parameters are taken from the YOLOv4 method but with some
tuning. The training step is 500,050, with the initial learning rate
multiplied by 0.1 at 400,040 step then 450,045 step. The evaluation of the
training improvement is achieved by comparing the evaluation scores of both
models trained on original and distorted train sets, respectively, on the
distorted MS-COCO validation sets for each distortion level. These results
shown in table 3 (Columns 2-6) are obtained by computing the ratio between AP
COCO scores of the distorted model on the natural model. We observed that the
training has a significant impact on the robustness improvement with an
average increase of 31.5% of performance (see last column in table 3).
Furthermore, the study highlighted that the training improves robustness for
high-level distortions (see last row in table 3).

Table 4: Object detection performance of the YOLOv4-tiny trained model on our
distorted natural sub-sets.

Natural |  AP |  rel. AP |  mIoU |  rel.mIoU
---|---|---|---|---
distortions
Noise |  0.003 |  0.99% |  0.005 |  0.65%
Contrast |  0.011 |  5.47% |  0.007 |  0.95%
Motion-Blur |  -0.012 |  -5.17% |  -0.004 |  -0.54%
Defocus |  0.015 |  12.9% |  0.0 |  0.0%
Rain |  -0.009 |  -2.81% |  0.006 |  0.79%
Loc. MBlur |  -0.006 |  -3.13% |  0.0 |  0.0%
BackLight |  0.011 |  8.94% |  0.005 |  0.68%

rel.: ratio between scores from distorted and original images, mIoU: mean
Intersection over Union.

It can be observed from the results of Table 4, that training has a weak
impact on robustness against natural distortions with an average improvement
of 2.5% for the mAP score. However, the interpretation of these results
deserves further investigations, taking into account the fact that the
distortions present in the original database are of low amplitude [18].
Indeed, the proposed data augmentation makes use of images with higher
amplitude distortion (see average per level of column 2 in table 3).

##  5 Conclusions and Future Work

This study revealed the impact of the quality of the acquired images in the
development of OD methods. Indeed, through this study we have shown that
increasing the training database with complex scenarios containing different
distortions improves the performance of OD models. We have also shown that
this is true even for the best state-of- the-art methods. We, particularly,
propose an original strategy of image distortion generation applied to the MS-
COCO dataset that combines some local and global distortions to reach a much
better performance of OD process. We have shown that training with this
distorted dataset improves the robustness of models by up to an average of
31,5%. We, also, provide a custom dataset including the natural images
distorted from MS-COCO dataset to perform a more relevant evaluation of the
robustness concerning distortions. The database and the generation source
codes of the different distortions is made publicly available on github
plateform. Considering dataset with local and global distortions is an
interesting way for future research.

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