# Nested traveling wave structures in elastoinertial turbulence

Manish Kumar1  Michael D. Graham1<EMAIL_ADDRESS>1Department of Chemical
and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering
Dr, Madison, WI 53706, USA

###### Abstract

Elastoinertial turbulence (EIT) is a chaotic flow resulting from the interplay
between inertia and viscoelasticity in wall-bounded shear flows. Understanding
EIT is important because it is thought to set a limit on the effectiveness of
turbulent drag reduction in polymer solutions. In the present study, we
analyze simulations of two-dimensional EIT in channel flow using Spectral
Proper Orthogonal Decomposition (SPOD), discovering a family of traveling wave
structures that capture the sheetlike stress fluctuations that characterize
EIT. The frequency-dependence of the leading SPOD mode contains several
distinct peaks and the mode structures corresponding to these peaks exhibit
well-defined traveling structures. The structure of the dominant traveling
mode exhibits shift-reflect symmetry similar to the viscoelasticity-modified
Tollmien–Schlichting (TS) wave, where the velocity fluctuation in the
traveling mode is characterized by the formation of large-scale regular
structures spanning the channel and the polymeric stress field is
characterized by the formation of thin, inclined sheets of high polymeric
stress localized at the critical layers near the channel walls. The structures
corresponding to the higher-frequency modes have a very similar structure, but
nested in a region roughly bounded by the critical layer positions of the
next-lower frequency mode. A simple theory based on the idea that the critical
layers of mode $\kappa$ form the “walls” for the structure of mode $\kappa+1$
yields quantitative agreement with the observed wave speeds and critical layer
positions. The physical idea behind this theory is that the sheetlike
localized stress fluctuations in the critical layer prevent velocity
fluctuations from penetrating them.

##  1 Introduction

Adding a tiny amount of high molecular weight polymer to a fluid dramatically
reduces turbulent drag (Toms, 1949). Therefore, the polymeric additives are
used to reduce pumping costs in pipeline transport of crude oil and home
heating and cooling systems, and to reduce fuel transfer time in airplane tank
filling (Brostow, 2008). Polymer additives also have been envisioned for flood
remediation and enhancement of the drainage capacity of sewer systems (Kumar &
Graham, 2023; Bouchenafa et al., 2021; Sellin, 1978). The turbulent flow
contains streamwise vortices close to walls, which dominate the near-wall
momentum transport and thus the drag. During drag reduction, the polymeric
chains get stretched due to turbulence and wrap around the streamwise
vortices, weakening them to lead to lower turbulent drag (Li & Graham, 2007;
Kim et al., 2007; Graham & Floryan, 2021). However, this suppression of near-
wall vortices does not generally lead to full relaminarization, but rather to
a limiting state called the maximum drag reduction (MDR) asymptote. Some
understanding of this observation has come from the discovery of
elastoinertial turbulence (EIT), a complex chaotic flow that is sustained,
rather than suppressed by viscoelasticity, and thus helps explain the absence
of relaminarization (Samanta et al., 2013; Dubief et al., 2023; Shekar et al.,
2019). Nevertheless, the structure and mechanism underlying EIT remain poorly
understood and are the topic of the present work.

Figure 1: Snapshots of instantaneous (a) wall-normal component of the velocity
field ($u_{y}$) and (b) the xx-component of conformation tensor
($\alpha_{xx}$) in EIT at $\Rey=3000$ and $\textit{Wi}=35$.

EIT arises in parameter regimes of Reynolds number $\Rey$ and Weissenberg
number Wi (product of polymer relaxation time and nominal strain rate) where
the laminar flow is linearly stable and is thus a nonlinearly self-sustaining
flow. The basic structure in both channel and pipe flows (Samanta et al.,
2013; Lopez et al., 2019) is two-dimensional (2D) (Sid et al., 2018),
characterized by vorticity fluctuations localized in narrow regions near the
walls with tilted sheets of highly stretched polymers emanating from these
regions. Figure 1 shows a snapshot of EIT structure in channel flow.

Despite the absence of an obvious linear instability mechanism for EIT, it has
been hypothesized that EIT is related to the nonlinear excitation of either a
“wall mode” or a “center mode” structure arising in the linear stability
problem for the laminar state (Drazin & Reid, 1981; Datta et al., 2022). A
wall mode has a wave speed much less than the centerline velocity, and
critical layer positions, i.e. where the wave speed equals the local laminar
velocity, near the walls. In contrast, a center mode travels at nearly the
centerline velocity and accordingly the critical layer position is near the
centerline. The Tollmien-Schlichting (TS) mode of classical linear analysis of
plane Poiseiulle flow is a wall mode, and there is a strong structural
resemblance of the viscoelastic extension of the TS wave to EIT (Shekar et
al., 2019, 2020, 2021). Figure 2 shows the viscoelastic linear TS mode at the
same conditions as Figure 1. Sheets of highly stretched polymer are generated
in the TS wave due to the presence of hyperbolic stagnation points (in the
frame traveling with the wave) in the Kelvin cat’s-eye structure of the
velocity field in the critical layer (Shekar et al., 2019). The possibility of
a center mode structure is of interest in part because there is a linear
center mode instability at low Reynolds number $\Rey$ that may organize
“elastic turbulence” at very small Reynolds number $\Rey$ (Garg et al., 2018;
Khalid et al., 2021; Choueiri et al., 2021; Morozov, 2022). Nevertheless, in
the elastoinertial regime considered here, while center mode structures can
exist (Dubief et al., 2022), they do not appear to play an active role in the
structure and self-sustenance of EIT (Beneitez et al., 2024). The present work
is consistent with this picture, and indeed deepens the connection between EIT
and wall modes, showing in particular the existence of a nested family of such
structures.

Figure 2: Structures of (a) wall-normal velocity ($u_{y}$) and (b) the
perturbation of the xx-component of conformation tensor from mean
($\alpha_{xx}^{\prime}$) of viscoelastic linear TS wave at $Re=3000$ and
$Wi=35$. This mode ultimately vanishes as viscoelastic channel flow is
linearly stable at this parameter regime.

In the present study, we investigate the structure and dynamics of EIT in
channel flow using a modal decomposition technique known as spectral proper
orthogonal decomposition (SPOD) (Towne et al., 2018). SPOD characterizes
coherent structures in complex flows that have well-defined structures and
persist both in space and time via a frequency-domain variant of Proper
Orthogonal Decomposition (POD) (Lumley, 1967). At a given frequency, SPOD
generates an energetically ordered and spatially orthogonal set of modes
characterizing the flow. The modes and eigenvalues obtained using SPOD
analysis can be interpreted as physical structures and the energy associated
with those structures (Schmidt & Towne, 2019). SPOD has been successfully used
in inertial turbulence to understand coherent structures and develop a low-
dimensional model for turbulence (Schmidt et al., 2017; Araya et al., 2017;
Tutkun & George, 2017; Braud et al., 2004; Hellström & Smits, 2014; Nekkanti &
Schmidt, 2021). Here, we use it to investigate traveling coherent structures
underlying the chaotic dynamics of EIT. We will focus on the frequency-
dependence of the most energetic SPOD mode, as it reveals important coherent
features of the flow.

##  2 Formulation and governing equations

Because the self-sustaining dynamics of elastoinertial turbulence are
fundamentally 2D (Sid et al., 2018), we consider two-dimensional (2D)
viscoelastic channel flow with nondimensional equations of mass and momentum
conservation:

$\nabla\cdot\bm{u}=0,\ \ \ \ \ \ \frac{\partial\bm{u}}{\partial
t}+\bm{u}\cdot\nabla\bm{u}=-\nabla
p+\frac{\beta}{Re}\nabla^{2}\bm{u}+\frac{1-\beta}{Re}\nabla\cdot\bm{\tau_{p}}+f(t)\bm{e}_{x},$
(1)

where $\bm{u}$ and $p$ are non-dimensional velocity field and pressure field,
respectively. Laminar centerline velocity ($U_{c}$) and channel half width
($H$) have been used as characteristic velocity scale and length scale,
respectively. The ratio between solvent viscosity ($\eta_{s}$) to zero shear
rate solution viscosity ($\eta$) has been denoted by $\beta=\eta_{s}/\eta$.
The Reynolds number has been defined as $Re=\rho U_{c}H/\eta$, where $\rho$
represents fluid density. We use no-slip boundary conditions for the velocity
field at the channel wall. Periodic boundary conditions have been used at the
inlet and outlet of the channel. Flow is driven by an external forcing term
$f(t)\bm{e}_{x}$, where $\bm{e}_{x}$ denotes the streamwise direction. The
forcing term varies with time to keep the volumetric flow rate at its laminar
value. The polymer stress tensor is denoted $\bm{\tau_{p}}$ and we choose the
FENE-P constitutive model with an artificial diffusion term to model its
evolution:

$\frac{\partial\bm{\alpha}}{\partial
t}+\bm{u}\cdot\nabla\bm{\alpha}-\bm{\alpha}\cdot\nabla\bm{u}-(\bm{\alpha}\cdot\nabla\bm{u})^{T}=-\bm{\tau_{p}}+\frac{1}{ReSc}\nabla^{2}\bm{\alpha},$
(2)
$\bm{\tau_{p}}=\frac{1}{\textit{Wi}}\left(\frac{\bm{\alpha}}{1-tr(\bm{\alpha})/b}-\bm{I}\right),$
(3)

where $\bm{\alpha}$ is the conformation tensor and parameter $b$ characterizes
the maximum extensibility of the polymeric chains. The Weissenberg number
$\textit{Wi}=\lambda U_{c}/H$, where $\lambda$ is the polymeric relaxation
time. The Schmidt number $Sc=\eta/\rho D$, where $D$ is the diffusion
coefficient, represents the ratio of momentum diffusivity to mass diffusivity.
The artificial diffusion term is used to stabilize the numerical scheme during
the integration of Eq. 2. The presence of this term leads to the requirement
of boundary conditions for the conformation tensor. At the channel walls, we
determine $\bm{\alpha}$ by solving the governing equations considering
$Sc\to\infty$.

We solve the governing equations with a spectral method using the Dedalus
framework (Burns et al., 2020). The computational domain has a length $L=5$.
The governing equations have been discretized using 256 Fourier basis
functions and 1024 Chebyshev basis functions in the streamwise ($x$) and wall-
normal ($y$) directions, respectively. We use $\beta=0.97$, $b=6400$,
$Re=3000-6000$, and $\textit{Wi}=35-70$, which are relevant to turbulent drag
reduction and are in the range of values used in previous studies, e.g.
(Shekar et al., 2019; Dubief et al., 2022). In reality, the Schmidt number for
a polymer solution is very large ($Sc\sim 10^{6}$). Numerical simulation with
such a large value of $Sc$ would require an extremely fine mesh and small
time-step, making numerical simulations computationally very expensive. At the
same time, small $Sc$ (i.e., $Sc<10$) smears out small-scale dynamics and
suppresses EIT (Sid et al., 2018; Dubief et al., 2023). Therefore, numerical
simulations of EIT based on artificial diffusion generally use $Sc\sim O(100)$
(Sid et al., 2018; Buza et al., 2022). In the present study, we use $Sc=100$
which is sufficient to sustain EIT and also numerically tractable.
Viscoelastic channel flow in the parameter regime considered here is linearly
stable. Therefore, we provide random perturbations to the conformation tensor
to initiate EIT. In computing statistics, initial transients are dropped so
that we consider only statistically stationary results.

For the SPOD analysis, we use the MATLAB tool developed by Schmidt (2022).
Details of the method and its numerical implementation can be found in
literature (Towne et al., 2018; Schmidt & Colonius, 2020). We use $300$ time
units of data generated using EIT simulation to perform SPOD analysis, which
is sufficient for the convergence of SPOD. The data set consists of
$N_{t}=4000$ snapshots, which are sampled at the interval of $\Delta
t_{s}=0.075$ time units. To obtain converged spectral densities, multiple
realizations of the flow are used (Bendat & Piersol, 2000). This can be
achieved by dividing the data set into multiple overlapping blocks (Welch,
1967). In the present study, we use $N_{f}=500$ snapshots in each block with
$50\%$ overlap, which leads to a total of $N_{b}=15$ blocks. The number of
modes obtained in SPOD is the same as the number of blocks, where the first
mode has the highest energy and the last mode has the lowest energy. The
number of non-negative frequencies is given by $N_{f}/2+1$ and the interval
between them is $\Delta f=1/(\Delta t_{s}N_{f})$.

##  3 Results and discussion

This study focuses on the case $\Rey=3000,\textit{Wi}=35$, for which a
snapshot of the wall-normal velocity $u_{y}$ and $xx$ polymer stretch
$\alpha_{xx}$ are shown in Figure 1. Results obtained for
$\Rey=3000,\textit{Wi}=70$ and $\Rey=6000,\textit{Wi}=35$ display nearly
identical features. The dynamics of $u_{y}$ in EIT are dominated by the
downstream advection of large-scale structures spanning the channel
(Supplementary video: Movie1), whereas the dynamics of the polymeric stress
field are dominated by the flapping of thin inclined sheets of polymeric
stress in the vicinity of the channel walls (Supplementary video: Movie2). We
note that the velocity fluctuations at EIT are generally quite small (e.g.
$u_{y}\sim 10^{-2}$), so the velocity profile does not greatly differ from
laminar. Since $u_{y}$ is identically zero in the laminar state, it is the
most straightforward velocity component to analyze with SPOD. Additionally,
most of the polymer stretching in EIT is in the $x$-direction, so our SPOD
analyses will consider $u_{y}$ and $\alpha_{xx}^{\prime}$, where $(^{\prime})$
denotes deviation from the mean.

SPOD energy spectra as a function of frequency for the first several modes in
$u_{y}$ and $\alpha_{xx}^{\prime}$ are shown in Fig. 3(a) and Fig. 3(b),
respectively, along with the energy of the sum of all modes. The SPOD analyses
of $u_{y}$ and $\alpha_{xx}^{\prime}$ have been done separately. For wall-
normal velocity, the leading mode contains most of the energy ($\approx 75\%$)
and hence dominates the flow structure (Fig. 3(a)). In the energy spectrum of
$\alpha_{xx}^{\prime}$, the leading mode has a relatively smaller contribution
to the total energy ($\approx 32\%$). The leading mode of wall-normal velocity
contains distinct peaks at specific frequencies indicated with red symbols,
and the energy of these peaks decreases as the frequency increases. The
higher-order modes do not have such distinct peaks.

(a)

(b)

Figure 3: SPOD energy spectra of (a) wall-normal velocity ($u_{y}$) and (b)
the perturbation of the xx-component of conformation tensor
($\alpha_{xx}^{\prime}$) at $\Rey=3000$ and $\textit{Wi}=35$. Red symbols
indicate the first few peaks in the leading mode of the energy spectrum of
$u_{y}$. Inset of (a): SPOD energy spectrum of the leading SPOD mode for
$u_{y}$ on a linear scale.  Figure 4: Structures of SPOD modes of (a-e)
$u_{y}$ and (f-j) $\alpha_{xx}^{\prime}$ corresponding to the frequencies
denoted by different symbols in the energy spectra (Figs. 3(a) and 3(b)).

The local peaks in the energy spectrum of wall-normal velocity indicate that
the structures corresponding to these frequencies have distinct features in
the dynamics of EIT. These peaks are _not_ at integer multiples of the lowest-
frequency peak, so are not simply harmonics; the relationship between them is
elucidated below. The SPOD mode structures of $u_{y}$ and
$\alpha_{xx}^{\prime}$ corresponding to the peak frequencies in the leading
mode of the wall-normal velocity have been shown in Fig. 4. Each mode has a
distinct wavenumber $\kappa$, which we measure in wavelengths per domain
length. These modes are all traveling waves with wave speed $C_{w}=fL/\kappa$,
as further discussed below.

The most dominant SPOD structure ($f=0.08$) has unit wavenumber ($\kappa=1$).
The velocity field for this mode consists of large-scale structures spanning
the channel (Fig. 4a), and the polymeric stress field displays thin layers
close to the walls having inclined alternating sheets of positive and negative
stress fluctuations (Fig. 4f). The structures approximately obey a shift-
reflect symmetry: i.e. $u_{y}(x,y)\approx-u_{y}(x+L/2,-y)$ and
$\alpha_{xx}^{\prime}(x,y)\approx\alpha_{xx}^{\prime}(x+L/2,-y)$. This is the
symmetry obeyed by the TS mode (Drazin & Reid, 1981), and comparison to Figure
2 indicates a strong similarity in structure. From here onward, we refer to
the regions having positive and negative $u_{y}$ as “positive lobe” and
“negative lobe”, respectively. The mode structures corresponding to other
peaks have similar structures, where the wavenumber of structures increases
with frequency (Fig. 4). The spanwise extent of the lobes in the velocity
field decreases as the wavenumber increases. Relatedly, the layers of strong
$\alpha_{xx}^{\prime}$ move away from the wall as the frequency increases.

The wave speeds of the traveling structures in the leading SPOD mode are shown
in Figure 5(a). They initially increase with the wavenumber (and frequency)
and ultimately saturate to a value close to the centerline velocity of the
channel ($C_{w}\to 0.94$). The wave speed of the dominant mode
($f=0.08,\kappa=1$) is very close to that of the Newtonian nonlinear TS (NNTS)
wave at the corresponding Re ($C_{TS}=0.42$), shown in red on Figure 5(a),
further strengthening the evidence connecting EIT to the TS mode. By contrast,
a center mode would have a wave speed close to unity and thus a frequency
close to $\kappa/L$. For $L=5$ this would be multiples of $0.2$, and examining
Figure 3(a) shows no peaks at these positions. In fact, $f=0.2$ and its
multiples are close to local minima in energy for the dominant mode. In short,
we see no evidence of a center mode structure. We elaborate below on the
origin of the peak positions in Figure 3(a).

(a)

(b)

Figure 5: (a) Wave speed and location of the critical layer for the traveling
waves associated with the peaks in the leading SPOD mode along with the
Newtonian nonlinear Tollmien–Schlichting (NNTS) mode results and the
predictions of the scaling model Eqs. 4 and 5. (b) Streamwise averaged $L_{2}$
norm of $\alpha_{xx}^{\prime}$ (solid lines) and critical layers (dashed
lines) for traveling modes having wave speed $C_{w}=0.4$ (black), $C_{w}=0.73$
(green), $C_{w}=0.84$ (blue), and $C_{w}=0.9$ (red).

As discussed in the Introduction, corresponding to the wave speed of a
perturbation there is a critical layer position $y_{c}$. As this is the
position where the fluid and the perturbation are moving together, it is the
most favorable position for the two to exchange energy. The velocity
fluctuations in EIT are very weak so the local streamwise velocity
$u_{x}\approx 1-y^{2}$, and the location of critical layers can be given as
$y_{c}\approx\pm\sqrt{1-C_{w}}$. Figure 5(a) shows the critical layer
positions corresponding to the wave speeds of the traveling structures of the
leading SPOD mode, as well as for the NNTS mode. As with wave speed, the
critical layer position of the $f=0.08,\kappa=1$ SPOD mode is very close to
that of the NNTS mode. To explore the relation between the critical layer and
the location of the polymeric sheets in the SPOD structures, in Figure 5(b) we
plot the streamwise averaged $L_{2}$ norm of $\alpha_{xx}^{\prime}$
($||\alpha_{xx}^{\prime}||_{x}$) along with the positions of the critical
layers for the traveling structures of the dominant SPOD mode. The peak
regions represent the locations of the sheets of polymer stretching, and we
see that their locations correspond to the critical layers. A similar
observation has been made for viscoelasticity-modified TS waves and it has
been reported that thin sheets of high polymeric stress emanate from the
critical layers of TS waves (Shekar et al., 2019; Hameduddin et al., 2019).

We noted above that as the wavenumber increases, $u_{y}$ becomes more
localized toward the channel center, as do the critical layer positions where
the stress fluctuations are high. More specifically, consider the $u_{y}$
profile at the second peak ($f=0.29$, Figure 4b) and the
$\alpha_{xx}^{\prime}$ profile at the first peak ($f=0.08$, Figure 4f). It
appears that the “lobes” where $u_{y}$ is large in the former figure are
roughly bounded by the layers where $\alpha_{xx}^{\prime}$ is large in the
latter. Similar observations can be made about all of the succeeding modes. We
visualize this point in Figure 6, which replots the results of Figure 4 by
showing contour lines of $u_{y}$ from the SPOD modes at wavenumber $\kappa+1$
juxtaposed with color contours of $\alpha_{xx}^{\prime}$ at wavenumber
$\kappa$. From this figure, we see that the velocity lobes at wavenumber
$\kappa+1$ are “nested” within the stress fluctuations, or equivalently
between the upper and lower critical layer positions, at wavenumber $\kappa$.
In contrast, the regions between the critical layers and the channel walls
contain small-scale and irregular structures both in the velocity field as
well as stress field.

The nested nature of the structures revealed by SPOD suggests that the
locations of the polymeric sheets of a slow-moving (low wavenumber) traveling
wave act like “walls” for the immediately faster-moving (and higher
wavenumber) wave. Consider the existence of a “primary” mode with wave speed
$C_{w,1}$ and thus critical layer positions $y_{c,1}=\pm(1-C_{w,1})^{1/2}$. We
take the next higher mode to occupy the domain $|y|<|y_{c,1}|$; if its
critical layer position $y_{c,2}$ is at the same fractional position in this
new domain, it will thus be at $\pm|y_{c,1}|^{2}=\pm(1-C_{w,1})^{1}$.
Continuing in this way, and noting that successively higher-speed waves can be
labeled by their wavenumber $\kappa$, we have a simple scaling result

$y_{c,\kappa}=(1-C_{w,1})^{\kappa/2}.$ (4)

Relatedly, the successive wave speeds are then

$C_{w,\kappa}=1-(1-C_{w,1})^{\kappa}.$ (5)

Using the SPOD results for $y_{c,\kappa}$ to find a best-fit value of
$C_{w,1}$ yields predictions for $y_{c,\kappa}$ and $C_{w,\kappa}$ that agree
very closely with the data, as shown in Figure 5(a). Furthermore, the value of
$C_{w,1}=0.44$ is very close to the NNTS wave speed $C_{TS}=0.42$. These
observations indicate that the structure of EIT is dominated by nested
structures that closely resemble TS waves.

Finally, we briefly describe a possible mechanism for the appearance of this
nested structure. A highly stretched elastic sheet resists lateral
deformation. Similarly, flows in which polymer molecules are strongly
stretched along one direction resist deformations transverse to that
direction. A classical example of this mechanism is the suppression of shear-
layer instability in a viscoelastic fluid, where the strong stretching in the
shear layer mimics an elastic sheet (Azaiez & Homsy, 2006). Relatedly,
viscoelastic Taylor-Couette instability is suppressed by axial flow (Graham,
1998), and in porous media flows, sheetlike regions with high polymeric stress
resist the flow passing through them and hence act like flow barriers (Kumar
et al., 2023). A similar mechanism is likely at work here, in which the sheets
of high polymer stretch in the critical layers from the primary mode prevent
velocity fluctuations from the higher modes from passing through the critical
layer, acting as “walls” as noted above, and so on successively with the
higher modes.

Figure 6: Contours of velocity fluctuations ($u_{y}$) of a faster-traveling
wave on the top of the stress fluctuations of the immediate slower traveling
wave: (a) stress at $C_{w}=0.4$ and velocity at $C_{w}=0.73$, (b) stress at
$C_{w}=0.73$ and velocity at $C_{w}=0.84$, (c) stress at $C_{w}=0.84$ and
velocity at $C_{w}=0.9$, and (d) stress at $C_{w}=0.9$ and velocity at
$C_{w}=0.93$.

For completeness, we also report in Figure 7 the structure corresponding to
the second-most energetic mode from SPOD (blue curve on Figure 3(a)) at
$f=0.08$. Note that the energy of this mode is more than an order of magnitude
smaller than that of the leading mode and that the spectrum of this structure
does not contain distinct peaks. This mode also has $\kappa=1$ and thus the
same wave speed and critical layer position as the most energetic mode. It
again exhibits localized polymer stretch fluctuations in the critical layer,
but now displays simple reflection symmetry rather than the shift-reflect
symmetry of the dominant mode. We view this as a higher-order correction on
the dominant structure elucidated above.

Figure 7: Mode structures the second-most energetic mode at $f=0.08$; (a)
$u_{y}$, (b) $\alpha_{xx}^{\prime}$.

##  4 Conclusions

In the present study, we use Spectral Proper Orthogonal Decomposition (SPOD)
to elucidate the structure underlying the chaotic dynamics of elastoinertial
turbulence in channel flow. The SPOD energy spectrum of wall-normal velocity
for the most energetic structure has distinct peaks. The mode structures
corresponding to these peaks exhibit a family of well-defined traveling
structures, where the velocity field contains large-scale regular patterns and
the polymeric stress field contains the formation of thin inclined sheets of
high and low stress at the critical layers of the wave. The structure of the
most dominant traveling wave (first, highest peak) of this family exhibits
shift-reflect symmetry and resembles the structure of the Tollmien–Schlichting
wave indicating its origin in a nonlinearly self-sustained wall mode. The
traveling structures corresponding to the higher frequency peaks have very
similar structure and symmetry, however, their wavenumber increases, and the
size of large-scale structures decreases. We discover that the formation of
polymeric sheets at the critical layers acts like walls for the traveling
structure of the next higher wave number and hence leads to a nested
arrangement of the waves. Based on this observation, a simple theory
quantitatively captures the relationship between the wave speeds and the
locations of critical layers for different waves. From this analysis, a
picture emerges of EIT as a nested collection of nonlinearly self-sustaining
TS-wave-like structures.

## Acknowledgments

This research was supported under grant ONR N00014-18-1-2865 (Vannevar Bush
Faculty Fellowship). We are grateful to Aaron Towne for helpful discussions
and Oliver Schmidt for making available his SPOD code.

## References

  * Araya et al. (2017) Araya, Daniel B., Colonius, Tim & Dabiri, John O. 2017 Transition to bluff-body dynamics in the wake of vertical-axis wind turbines. Journal of Fluid Mechanics 813, 346–381.
  * Azaiez & Homsy (2006) Azaiez, J & Homsy, G M 2006 Linear stability of free shear flow of viscoelastic liquids. Journal of Fluid Mechanics 268 (-1), 37 69.
  * Bendat & Piersol (2000) Bendat, Julius S & Piersol, Allan G 2000 Random data analysis and measurement procedures. Measurement Science and Technology 11 (12), 1825–1826.
  * Beneitez et al. (2024) Beneitez, Miguel, Page, Jacob, Dubief, Yves & Kerswell, Rich R. 2024 Multistability of elasto-inertial two-dimensional channel flow. Journal of Fluid Mechanics 981, A30, arXiv: 2308.11554.
  * Bouchenafa et al. (2021) Bouchenafa, Walid, Dewals, Benjamin, Lefevre, Arnaud & Mignot, Emmanuel 2021 Water Soluble Polymers as a Means to Increase Flow Capacity: Field Experiment of Drag Reduction by Polymer Additives in an Irrigation Canal. Journal of Hydraulic Engineering 147 (8), 1–9.
  * Braud et al. (2004) Braud, Caroline, Heitz, Dominique, Arroyo, Georges, Perret, Laurent, Delville, Joël & Bonnet, Jean-Paul 2004 Low-dimensional analysis, using POD, for two mixing layer–wake interactions. International Journal of Heat and Fluid Flow 25 (3), 351–363.
  * Brostow (2008) Brostow, Witold 2008 Drag reduction in flow: Review of applications, mechanism and prediction. Journal of Industrial and Engineering Chemistry 14 (4), 409–416.
  * Burns et al. (2020) Burns, Keaton J., Vasil, Geoffrey M., Oishi, Jeffrey S., Lecoanet, Daniel & Brown, Benjamin P. 2020 Dedalus: A flexible framework for numerical simulations with spectral methods. Physical Review Research 2 (2), 023068, arXiv: 1905.10388.
  * Buza et al. (2022) Buza, Gergely, Beneitez, Miguel, Page, Jacob & Kerswell, Rich R. 2022 Finite-amplitude elastic waves in viscoelastic channel flow from large to zero Reynolds number. Journal of Fluid Mechanics 951, A3, arXiv: 2202.08047.
  * Choueiri et al. (2021) Choueiri, George H., Lopez, Jose M., Varshney, Atul, Sankar, Sarath & Hof, Björn 2021 Experimental observation of the origin and structure of elastoinertial turbulence. Proceedings of the National Academy of Sciences 118 (45), e2102350118, arXiv: 2103.00023.
  * Datta et al. (2022) Datta, Sujit S., Ardekani, Arezoo M., Arratia, Paulo E., Beris, Antony N., Bischofberger, Irmgard, McKinley, Gareth H., Eggers, Jens G., López-Aguilar, J. Esteban, Fielding, Suzanne M., Frishman, Anna, Graham, Michael D., Guasto, Jeffrey S., Haward, Simon J., Shen, Amy Q., Hormozi, Sarah, Morozov, Alexander, Poole, Robert J., Shankar, V., Shaqfeh, Eric S. G., Stark, Holger, Steinberg, Victor, Subramanian, Ganesh & Stone, Howard A. 2022 Perspectives on viscoelastic flow instabilities and elastic turbulence. Physical Review Fluids 7 (8), 080701, arXiv: 2108.09841.
  * Drazin & Reid (1981) Drazin, PG & Reid, WH 1981 Hydrodynamic stability (cambridge university .
  * Dubief et al. (2022) Dubief, Y, Page, J, Kerswell, R R, Terrapon, V E & Steinberg, V 2022 First coherent structure in elasto-inertial turbulence. Physical Review Fluids 7 (7), 073301.
  * Dubief et al. (2023) Dubief, Yves, Terrapon, Vincent E & Hof, Björn 2023 Elasto-Inertial Turbulence. Annual Review of Fluid Mechanics 55 (1), 675–705.
  * Garg et al. (2018) Garg, Piyush, Chaudhary, Indresh, Khalid, Mohammad, Shankar, V. & Subramanian, Ganesh 2018 Viscoelastic Pipe Flow is Linearly Unstable. Physical Review Letters 121 (2), 24502, arXiv: 1711.07991.
  * Graham (1998) Graham, Michael D 1998 Effect of axial flow on viscoelastic Taylor-Couette instability. Journal of Fluid Mechanics 360, 341 374.
  * Graham & Floryan (2021) Graham, Michael D. & Floryan, Daniel 2021 Exact Coherent States and the Nonlinear Dynamics of Wall-Bounded Turbulent Flows. Annual Review of Fluid Mechanics 53 (1), 227–253.
  * Hameduddin et al. (2019) Hameduddin, Ismail, Gayme, Dennice F. & Zaki, Tamer A. 2019 Perturbative expansions of the conformation tensor in viscoelastic flows. Journal of Fluid Mechanics 858, 377–406, arXiv: 1811.02501.
  * Hellström & Smits (2014) Hellström, Leo H. O. & Smits, Alexander J. 2014 The energetic motions in turbulent pipe flow. Physics of Fluids 26 (12).
  * Khalid et al. (2021) Khalid, Mohammad, Chaudhary, Indresh, Garg, Piyush, Shankar, V. & Subramanian, Ganesh 2021 The centre-mode instability of viscoelastic plane Poiseuille flow. Journal of Fluid Mechanics 915, A43, arXiv: 2008.00231.
  * Kim et al. (2007) Kim, Kyoungyoun, Li, Chang-F., Sureshkumar, R., Balachandar, S. & Adrian, Ronald J. 2007 Effects of polymer stresses on eddy structures in drag-reduced turbulent channel flow. Journal of Fluid Mechanics 584, 281–299.
  * Kumar & Graham (2023) Kumar, Manish & Graham, Michael D. 2023 Effect of polymer additives on dynamics of water level in an open channel. Journal of Non-Newtonian Fluid Mechanics 321 (June), 105129, arXiv: 2307.00626.
  * Kumar et al. (2023) Kumar, Manish, Guasto, Jeffrey S. & Ardekani, Arezoo M 2023 Lagrangian stretching reveals stress topology in viscoelastic flows. Proceedings of the National Academy of Sciences 120 (5).
  * Li & Graham (2007) Li, Wei & Graham, Michael D. 2007 Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Physics of Fluids 19 (8).
  * Lopez et al. (2019) Lopez, Jose M, Choueiri, George H & Hof, Björn 2019 Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit. Journal of Fluid Mechanics 874, 699 – 719.
  * Lumley (1967) Lumley, John Leask 1967 The structure of inhomogeneous turbulent flows. Atmospheric turbulence and radio wave propagation pp. 166–178.
  * Morozov (2022) Morozov, Alexander 2022 Coherent Structures in Plane Channel Flow of Dilute Polymer Solutions with Vanishing Inertia. Physical Review Letters 129 (1), 017801, arXiv: 2201.01274.
  * Nekkanti & Schmidt (2021) Nekkanti, Akhil & Schmidt, Oliver T. 2021 Frequency–time analysis, low-rank reconstruction and denoising of turbulent flows using SPOD. Journal of Fluid Mechanics 926, A26, arXiv: 2011.03644.
  * Samanta et al. (2013) Samanta, Devranjan, Dubief, Yves, Holzner, Markus, Schäfer, Christof, Morozov, Alexander N., Wagner, Christian & Hof, Björn 2013 Elasto-inertial turbulence. Proceedings of the National Academy of Sciences 110 (26), 10557–10562.
  * Schmidt (2022) Schmidt, Oliver T. 2022 Spectral proper orthogonal decomposition using multitaper estimates. Theoretical and Computational Fluid Dynamics 36 (5), 741–754, arXiv: 2112.10847.
  * Schmidt & Colonius (2020) Schmidt, Oliver T. & Colonius, Tim 2020 Guide to Spectral Proper Orthogonal Decomposition. AIAA Journal 58 (3), 1023–1033.
  * Schmidt & Towne (2019) Schmidt, Oliver T. & Towne, Aaron 2019 An efficient streaming algorithm for spectral proper orthogonal decomposition. Computer Physics Communications 237, 98–109, arXiv: 1711.04199.
  * Schmidt et al. (2017) Schmidt, Oliver T., Towne, Aaron, Colonius, Tim, Cavalieri, André V. G., Jordan, Peter & Brès, Guillaume A. 2017 Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability. Journal of Fluid Mechanics 825, 1153–1181.
  * Sellin (1978) Sellin, R. H. J. 1978 Drag reduction in sewers: First results from a permanent installation. Journal of Hydraulic Research 16 (4), 357–371.
  * Shekar et al. (2020) Shekar, Ashwin, McMullen, Ryan M., McKeon, Beverley J. & Graham, Michael D. 2020 Self-sustained elastoinertial Tollmien–Schlichting waves. Journal of Fluid Mechanics 897, A3, arXiv: 1910.11419.
  * Shekar et al. (2021) Shekar, Ashwin, McMullen, Ryan M., McKeon, Beverley J. & Graham, Michael D. 2021 Tollmien-Schlichting route to elastoinertial turbulence in channel flow. Physical Review Fluids 6 (9), 093301, arXiv: 2104.10257.
  * Shekar et al. (2019) Shekar, Ashwin, McMullen, Ryan M, Wang, Sung-Ning, McKeon, Beverley J & Graham, Michael D 2019 Critical-Layer Structures and Mechanisms in Elastoinertial Turbulence. Physical Review Letters 122 (12), 124503.
  * Sid et al. (2018) Sid, S., Terrapon, V. E. & Dubief, Y. 2018 Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Physical Review Fluids 3 (1), 011301, arXiv: 1710.01199.
  * Toms (1949) Toms, Be A. 1949 Some observations on the flow of linear polymersolutions through straight tubes at large reynolds numbers. In Proc. 1st Intl Congr. Rheol., , vol. 2, pp. 135–141.
  * Towne et al. (2018) Towne, Aaron, Schmidt, Oliver T. & Colonius, Tim 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. Journal of Fluid Mechanics 847, 821–867, arXiv: 1708.04393.
  * Tutkun & George (2017) Tutkun, Murat & George, William K. 2017 Lumley decomposition of turbulent boundary layer at high Reynolds numbers. Physics of Fluids 29 (2).
  * Welch (1967) Welch, P. 1967 The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Transactions on Audio and Electroacoustics 15 (2), 70–73.