12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

TOLLMIEN-SCHLICHTING ROUTE TO ELASTOINERTIAL TURBULENCE

Ashwin Shekar
Department of Chemical and Biological Engineering
University of Wisconsin-Madison
Madison, WI 53706 USA
shekar2@wisc.edu

Richard J. Hommel
Department of Chemical and Biological Engineering
University of Wisconsin-Madison
Madison, WI 53706 USA
hommel@wisc.edu

Ryan M. McMullen
Beverley J. McKeon
Graduate Aerospace Laboratories Graduate Aerospace Laboratories
CaltechCaltechPasadena, CA 91125 USA
Pasadena, CA 91125 USA
mcmullen@caltech.edu
mckeon@caltech.edu

Michael D. Graham
Department of Chemical and Biological Engineering
University of Wisconsin-Madison
Madison, WI 53706 USA
mdgraham@wisc.edu

ABSTRACTINTRODUCTION

The so-called elastoinertial regime of turbulence in polymer solutions is dominated by nearly two-dimensional structure. Direct simulations of two-dimensional channel flow ofa viscoelastic fluid have revealed the existence of a family of
Tollmien-Schlichting (TS) attractors that is nonlinearly selfsustained by viscoelasticity. Here, we describe the evolutionof this branch in parameter space and its connections to the
Newtonian TS wave and to elastoinertial turbulence (EIT). At
Reynolds number Re=3000, there is a solution branch with
TS-wave structure but which is not connected to the Newtonian solution branch. At fixed Weissenberg number, Wi andincreasing Reynolds number from 3000-10000, this attractor
goes from displaying a striation of weak polymer stretch localized at the critical layer to an extended sheet of very large
polymer stretch. This transition can be attributed to a coilstretch transition when the local Weissenberg number at thehyperbolic stagnation points of the Kelvin cat’s eye structureof the TS wave exceed 1/2. At Re=10000, unlike 3000, theNewtonian TS attractor evolves continuously into EIT as Wi
is increased. We describe how the structure of the flow andstress fields changes, highlighting a “sheet-shedding” process
by which the individual sheets associated with the critical layer
structure break up to form the layered multisheet structure
characteristic of EIT. We also find that at sufficiently high Wi,
this solution family extends down in Reynolds number to between 150 and 200, indicating that viscoelasticity can sustain
turbulence at Reynolds numbers well below Newtonian transition.

Addition of long chain polymer molecules to a fluids has
tremendous effects on wall-bounded turbulence, the most dramatic being the substantial reduction of the friction factor(Toms, 1949, 1977). This phenomenon has found wide use
in applications that seek energy efficiency in flow processes.
(Fink, 2012; Burger et al., 1982; King, 2012).
This work describes recent results that have led to an improved understanding of this phenomenon. The main focus
is channel flow of a dilute solution of high molecular weight
polymer, so the ratio between solvent and total viscosity, β satisfies 1 − β  1, and the ratio between extensional and shear
viscosities (Trouton ratio) Tr is  1. For the FENE-P constitutive model with chain length parameter b, this requires thatb(1 − β )  1. This is the regime of primary relevance for dragreduction, where as a practical matter it is desired to keep theshear viscosity of the fluid low (1−β  1), but the extensional
viscosity high (b(1 − β )  1). The Reynolds number regime
considered is Re ∼ 102 − 104 , i.e., near transition.
It is well-known that viscoelasticity suppresses the nearwall streamwise vortices that dominate Newtonian turbulence(Dubief et al., 2004; Kim et al., 2007; White & Mungal,
2008). A number of studies have captured this phenomena
by studying the effect of viscoelasticity on three-dimensional(3D) nonlinear traveling wave solutions of the Navier-Stokes
equations termed exact coherent states (ECS). (Stone et al.,
2002; Stone & Graham, 2003; Stone et al., 2004; Li et al.,2005, 2006; Li & Graham, 2007). These ECS contain the basic self-sustaining ingredients of transitional Newtonian turbulence i.e quasistreamwise vortices and streaks. A comprehensive review of ECS can be found in Graham & Floryan (2021).
Li and coworkers (Li et al., 2006; Li & Graham, 2007)
found that the ECS are so weakened by viscoelasticity that they1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)tions by direct numerical simulation (DNS):∂v
β 2(1 − β )+ v · ∇v = −∇p +∇ v+∇ · τp ,∂t
ReReWi∇ · v = 0,
α
− I,
τp =
tr(α)
1− b

(b)

∂α
−1
+ v · ∇α − α · ∇v − (α · ∇v)T =
τ p.
∂t
Wi

Figure 1. Snapshots of simulations of EIT in (a) channel flow
(Terrapon et al., 2014) and (b) pipe flow (Lopez et al., 2019).
In (a) color contours indicate polymer stretching and lines indicate vortex strength Q. In (a) isosurfaces indicate Q.

(1)
(2)
(3)
(4)

Here Re = ρUl/(ηs +ηp ), where ηs and η p are the solvent andpolymer contributions to the zero-shear rate viscosity. The viscosity ratio β = ηs /(ηs + ηp ). We fix β = 0.97 and b = 6400.
Since 1 − β is proportional to polymer concentration and b
to the number of monomer units, these parameters correspondto a dilute solution of a high molecular weight polymer. The
Weissenberg number Wi = λU/l, where λ is the polymer relaxation time.
For integrating the momentum equation, second-order
central differences were used for spatial discretization, andsecond-order Adams-Bashforth and Crank-Nicholson methods
were used for time-integration of the convection and diffusion
terms, respectively. The FENE-P equation was discretized using a high resolution central difference scheme.
Using computations in channel flow at Re = 1500, Shekar
et al. (2019) observed a narrow zone of Wi where the only attractor was the laminar base state. This zone separated dragreduced Newtonian turbulence at lower Wi and EIT at higherWi, corroborating the experimental observations of Choueiri
et al. (2018). The laminar flow remains linearly stable in
the EIT regime, but only very small (but finite) perturbations
are sufficient to drive the flow to EIT. EIT in this parameter
regime displays polymer stretch fluctuations localized near thewall. In particular, a clear resemblance was noted between the
EIT structure and the viscoelastic extension of the classicalTollmien-Schlichting (TS) mode, which at the chosen parameters is the slowest decaying mode from linear stability analysis. Similarly, resolvent analysis predicts strong amplificationof this structure in the presence of viscoelasticity. This strong
amplification implies, consistent with the fully nonlinear results, that even very weak disturbances may be sufficient totrigger EIT.
The viscoelastic TS mode displays polymer stretch fluctuations that are sharply localized to critical layers, i.e wallnormal positions near the top and bottom walls where thestreamwise velocity equals the real part of the wave speed.
Critical layers can be thought of as the most favorable positions for energy exchange between the mean and fluctuations,because they are the positions where both have the same speed.
These results indicate a role for TS-like critical layer mechanisms at EIT.
The key observations from Shekar et al. (2019) are summarized in Figure 2. Fig. 2a shows a snapshot from the3D DNS, illustrating the sharp localization of stretch fluctuations in a layer near the wall. Except in the layers where
the stress fluctuations are largest, this flow is dominated bystructures with a wavelength of half the domain size (i.e. 5
channel half-heights). Fig. 2b shows the structure filtered tokeep only this wavelength. Fig. 2c shows the slowest decaying structure from linear stability analysis at a wavelength of
5 – this is the viscoelastic continuation of the linear TollmienSchlichting mode. Finally, Fig. 2d shows the most amplified
resolvent mode under the same conditions. The resemblanceof the full nonlinear structure to the TS mode is apparent.

are no longer self-sustaining and lose existence. Recognizingthat in general, viscoelasticity is not experimentally observed
to drive relaminarization, these authors suggested the possibility of new viscoelastic mechanisms for turbulence coming into
existence and being unmasked as the Newtonian structure aresuppressed (Li et al., 2006).
Indeed, instead of complete relaminarization (except innarrow parameter ranges at transitional Re), recent studies
have unearthed a polymer-driven chaotic flow state dubbed
elastoinertial turbulence (EIT) dominating the flow at high levels of viscoelasticity (Samanta et al., 2013a). EIT (in thisparameter regime) displays multilayered sheets of polymer
stretch emanating from near the walls (see Figure 1a) andvery weak, spanwise-oriented vortices – a sharp contrast to
the 3D quasistreamwise vortex structures of Newtonian wall
turbulence. Sid et al. (2018) have found that the sheetlike
stress flucturations that dominate simulations of EIT in channel flow are still present in 2D simulations, indicating that, incontrast with Newtonian turbulence, EIT is not fundamentally
3D. Similarly, near-wall localized, nearly-axisymmetric vortex and stress structures (Figure 1b) have been reported in pipe
flow simulations of EIT (Lopez et al., 2019). The present work
aims to elucidate the origin of these structures.

FORMULATIONThis study focuses on two-dimensional pressure-driven
channel flow with constant mass flux. Two-dimensional flow
is chosen because of the strongly spanwise-oriented structures
observed in prior simulations in the EIT regime Sid et al.
(2018). We also report some results from 3D simulations toindicate how these affect the main 2D structure. The x andy axes are aligned with the streamwise and wall-normal directions, respectively. Lengths are scaled by the half channelheight l, so the dimensionless channel height Ly = 2. The domain is periodic in x with length Lx . Velocity v is scaled with
the Newtonian laminar centerline velocity U; time t with l/U,
and pressure p with ρU 2 , where ρ is the fluid density. The
polymer stress tensor τ p is related to the polymer conformation tensor α through the FENE-P constitutive relation, whichmodels each polymer molecule as a pair of beads connectedby a nonlinear spring with maximum extensibility b.
We solve the momentum, continuity and FENE-P equa-212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

(b)

(c)

Figure 3. (a) Snapshot of the finite amplitude TollmienSchlichting wave solution at Re = 3000, Wi = 10.White contours are wall-normal velocity, colors are deviations of xx polymer stretch from laminar.

(d)Figure 2. Summary of results at Re = 1500 (Shekar et al.,0 (filled2019).(a) Snapshot of v0 (line contours) and αxx
contours) from 3D nonlinear DNS at Re = 1500, Wi =20, where 0 denotes fluctuations. (b) Phase-matched average (kx Lx /2π, kz Lz /2π) = (2, 0) structures from 3D DNS.
(c) Structure of the TS mode at Re = 1500, Wi = 20, and thesame wavenumbers as in (b). (d) Structure of the most strongly
amplified resolvent mode at Re = 1500, Wi = 20, the same
wavenumbers as in (b), and c = 0.37. In all plots, contour levels are symmetric about zero. For v0 dashed - negative, solid
0 black - negative, red - zero and yellow - positive. For αxx
positive.

(a) Wi=0(b) Wi=4(c) Wi=8(d) Wi=13

Building on the above observations, we turn to elucidating the role of Tollmien-Schlichting wave structures in EIT. In
contrast to Re = 1500, at Re = 3000 a nonlinear self-sustaining
TS wave exists in the Newtonian limit. For Wi & 6, we finda viscoelastic family of attractors whose structure is virtually identical to the linear TS mode, and in particular exhibits
strongly localized stress fluctuations at the critical layer position of the TS mode. A snapshot of one of these solutions,at Wi = 10, Lx = 5, is shown in Figure 3; note the close resemblance to Figure 2c. At the parameter values chosen, thissolution branch is not connected to the nonlinear TS solution
branch found for Newtonian flow, and thus represents a solution family that is nonlinearly self-sustained by viscoelasticity (Shekar et al., 2020). (The laminar state remains linearly stable, though again, as in Shekar et al. (2019), only an
extremely small perturbation is required to drive the solution
away from laminar.) Evidence indicates that this branch is connected through an unstable solution branch to two-dimensionalelastoinertial turbulence (EIT).
Now moving up from Re = 3000 to Re = 10000, it isfound that the Newtonian nonlinear TS wave solutions attractor evolves continuously and without hysteresis into EIT as Wi
is increased from zero to about 13 – the two flows are part ofthe same solution family (Shekar et al., 2021). The snapshotsin Figure 4 illustrate the evolution of the flow and stress fields
as Wi increases. Figure 4a shows the self-sustained nonlinear
Tollmien-Schlicting wave in the Newtonian limit Wi = 0. As
Wi increases to 4, sheets of polymer stretch appear, driven by
the extensional flow at the hyperbolic stagnation points of the
Kelvin cat’s eye structure of the TS wave. As Wi increases
further, the simple sheet structures that originate with the TS
critical layer structure evolve into the multilayered structure of
EIT through a process that we call “sheet-shedding”: Individual sheets associated with the critical layer structure break up,
with the fragments further sheared as they travel downstream.
An intermediate stage of this process (Wi = 8) is shown in
Figure 4c. Finally as Wi increases further, the typical overlapping sheet structure becomes fully developed, as illustrated in

Figure 4. Snapshots of the finite amplitude TollmienSchlichting wave solution at Re = 10000 and (a) Wi = 0, (b)Wi = 4, (c) Wi = 8, (d) Wi = 13. White contours are wallnormal velocity, colors are deviations of xx polymer stretch
from laminar.

Figure 4d. Note the resemblance between this structure andFigure 1a.
An important question is the robustness of two dimensional result to three-dimensional perturbations. In particular,
given the tilted-sheet structure of the stress fluctuations, one
might expect that a 3D perturbation that changes the positionof these sheets will expose different parts of the sheet to different streamwise velocities – a portion of the sheet perturbed
toward the centerplane will be convected faster than an a portion perturbed toward the wall. This velocity differential might
be expected to tend to tear the sheets apart. Figure 5a-b show
snapshots of αxx at y = 0.8 at two instants from a 3D simulation at Re = 2000, Wi = 30 that starts from a 2D initial condition subjected to small random perturbations. Since the sheets
are tilted with respect to the flow direction, this view “slices”
obliquely through the sheets of stress. At short time (t = 36,
Fig. 5a), the sheets are coherent in the z direction, but as timeproceeds, (t = 180, Fig. 5), they become less so. Figure 5cshows a 3d view of the structure at t = 180, illustrating how
the sheets are perturbed in 3D. As the sheets lose coherence,
the overall structure becomes weaker and the drag closer to thelaminar value. Nevertheless the basic sheet structure remains.
We began this discussion with results at Re = 1500, well
below the regime where the Newtonian flow becomes linearly
unstable and also below the limiting Reynolds number forthe existence of self-sustaining nonlinear Tollmien-Schlichting
waves in that limit. The existence of nontrivial flows at finite
Wi in these cases illustrates the strongly subcritical nature of
the origin of these structures. Many studies (e.g. Samanta et al.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)Figure 6.

Norm of stress fluctuations vs. Re at Wi = 30.

(b)Figure 7. Snapshot of EIT solution at Re = 400, Wi = 30.
White contours are wall-normal velocity, colors are deviationsof xx polymer stretch from laminar.

(c)Figure 5. . Shapshots illustrating evolution of 3d αxx structure from 2d EIT at Re = 2000, Wi = 30, Lx = 6.2, Lz = 3.1.
Flow is left to right. (a) Slice at y = 0.8,t = 36. (b) As in (a)but at t = 180. (c) 3D view showing side, front and bottom.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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