Elastic Turbulence in Viscoelastic Flows

• Elastic turbulence is a disordered state in viscoelastic flows driven by elastic instabilities that emerge at low Reynolds numbers.
• It is characterized by key control parameters such as the Weissenberg and Deborah numbers, with spatial structures like elastic boundary layers and localized coherent streaks.
• Studies show a steep energy spectrum (e.g., kinetic energy ∼ k⁻⁴) and increased flow resistance, underscoring its impact on mixing and drag.

Elastic turbulence is a spatially and temporally disordered, turbulence-like state of viscoelastic flow in which the dominant nonlinearity is elastic rather than inertial. It arises when polymer stretching and relaxation generate stresses strong enough to destabilize a nominally laminar flow at very low Reynolds number, so that the relevant control parameters are typically the Weissenberg number and, in time-dependent forcing, the Deborah number. Although it was established experimentally in curved and swirling configurations, later work showed that it also occurs in straight channels, parallel shear flows, porous media, shell models without physical geometry, and—at high global Reynolds number—as a small-scale regime coexisting with inertial turbulence (Buel et al., 2018, Ray et al., 2016, Garg et al., 22 Jul 2025).

1. Constitutive basis and control parameters

The standard continuum setting for elastic turbulence is an incompressible viscoelastic fluid with solvent viscosity and polymeric stress contributions. In Oldroyd-B form, the momentum balance couples the velocity field to the polymeric stress tensor, while the constitutive dynamics are governed by an upper-convected derivative that advects, stretches, and rotates the stress. A representative formulation is

$$\rho\left(\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}\right) = -\nabla p + \eta_s \nabla^2\mathbf{u} + \nabla\cdot\boldsymbol{\tau}, \qquad \nabla\cdot\mathbf{u}=0,$$

together with

$$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$

In this setting, the upper-convected nonlinearity is the source of purely elastic instability: stretched polymers feed back on the flow through stress growth and transport (Buel et al., 2019).

The basic dimensionless groups are the Reynolds number,

$Re = \frac{\rho U L}{\eta},$

and the Weissenberg number,

$Wi = \lambda \dot{\gamma},$

with geometry-specific versions such as $Wi=\lambda\Omega$ in Taylor–Couette flow or $Wi=\lambda U_b/d$ in channel and cross-slot configurations. When the forcing itself is time-dependent, a Deborah number comparing polymer relaxation to forcing period also becomes central; in the shear-modulated Taylor–Couette problem this is

$De = \frac{\lambda}{\delta},$

with $\delta$ the modulation period (Buel et al., 2019).

A recurring empirical feature is that elastic turbulence appears in regimes with $Re \ll 1$ and large $Wi$. In the two-dimensional Taylor–Couette simulations of van Buel and Stark, for example, the Reynolds number is estimated as $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$0, so the transition is purely elastic rather than inertial (Buel et al., 2019). The shell-model study strengthens the same point from the opposite direction: it shows chaotic elastic-turbulent dynamics in a geometry-free spectral model, indicating that the core mechanism does not rely on specific boundaries or mean-flow curvature, but on generic nonlinear coupling between velocity and polymer degrees of freedom (Ray et al., 2016).

2. Transition scenarios and onset routes

There is no single universal route to elastic turbulence. Depending on geometry, constitutive model, dimensionality, and forcing protocol, the onset can be supercritical, subcritical, finite-amplitude, or linearly unstable.

Configuration	Route to onset	Representative threshold/result
2D Taylor–Couette, Oldroyd-B	Supercritical	$$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$1, with $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$2, $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$3 (Buel et al., 2018)
2D Taylor–Couette, refined DNS	Supercritical	$$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$4 with $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$5 (Hou et al., 15 Oct 2025)
3D von Kármán swirling flow	Subcritical, bistable, hysteretic	Transition to bistable flow at $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$6 (Buel et al., 2022)
Straight pressure-driven channel	Subcritical finite-amplitude	sustained chaotic state for $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$7 (Lellep et al., 2023)
Plane Couette with stress diffusion	Linear instability	polymeric diffusive instability leading to self-sustaining 3D chaos (Beneitez et al., 2022)

In two-dimensional Taylor–Couette flow, the onset has been quantified by a secondary-flow order parameter measuring deviation from the laminar azimuthal base state. One study reported a continuous transition at $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$8 with exponent $$\boldsymbol{\tau} + \lambda\,\overset{\nabla}{\boldsymbol{\tau}} = \eta_p\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathrm T}\right].$$9 (Buel et al., 2018). A later high-resolution study of the same geometry reported $Re = \frac{\rho U L}{\eta},$0 and a square-root scaling of the radial secondary flow, again identifying a supercritical purely elastic instability (Hou et al., 15 Oct 2025). This suggests that the qualitative bifurcation type can be robust even when the numerical threshold is sensitive to resolution and formulation.

Other setups behave differently. In pressure-driven plane channel flow of an sPTT fluid at $Re = \frac{\rho U L}{\eta},$1, the laminar state is linearly stable for all $Re = \frac{\rho U L}{\eta},$2 studied, yet finite-amplitude perturbations above a threshold trigger a chaotic state for $Re = \frac{\rho U L}{\eta},$3; near onset, localized spot-like structures appear, and sudden relaminarization events occur at $Re = \frac{\rho U L}{\eta},$4 and $Re = \frac{\rho U L}{\eta},$5, all characteristic of a subcritical transition (Lellep et al., 2023). In three-dimensional von Kármán swirling flow with Oldroyd-B rheology, the order parameter and flow resistance both display bistability and hysteresis across $Re = \frac{\rho U L}{\eta},$6, again indicating a subcritical transition (Buel et al., 2022).

Parallel shear flows add a further route. In inertialess plane Couette flow with small but finite conformation diffusion, a linear polymeric diffusive instability appears and evolves into a self-sustained three-dimensional chaotic state (Beneitez et al., 2022). In concentrated entangled-polymer and wormlike-micelle models, two-dimensional linear instabilities of homogeneous planar Couette flow arise in both Johnson–Segalman and Rolie–Poly dynamics, including cases with monotonic constitutive curves; nonlinear evolution yields elastic turbulence with narrow shear bands that coalesce, split, and interact (Lewy et al., 8 Jun 2025).

3. Spatial organization, localization, and coherent structures

Elastic turbulence is often strongly structured in space rather than statistically homogeneous. In curvilinear channel experiments, Jun and Steinberg identified a boundary layer associated with nonuniform elastic stresses across the channel. Its characteristic width is independent of $Re = \frac{\rho U L}{\eta},$7 and proportional to the channel width, and correlation functions in the bulk reveal spatial scales of the same order, suggesting ejection of rare parcels of excessive elastic stress from the boundary layer into the interior (Jun et al., 2011). A closely related picture emerges in the two-dimensional Taylor–Couette problem: the fully nonlinear state is weakly anisotropic but strongly nonhomogeneous, with most activity confined to a dynamically active region adjacent to the inner wall, akin to an elastic boundary layer (Hou et al., 15 Oct 2025).

Localized structures also organize straight-channel elastic turbulence. In pressure-driven channel flow, the transition first produces spot-like structures localized near the midplane; farther from onset these proliferate and spread through the domain. The turbulent dynamics appear organized around unstable coherent states localized close to the channel midplane, with state-space trajectories clustering around points and loops suggestive of nearby invariant solutions (Lellep et al., 2023). In a quasi-two-dimensional straight channel with dilute polymer solution, experiments revealed a self-organized cycling process of co-existing streaks and stream-wise vortices, followed by a Kelvin–Helmholtz-like instability of the streaks and a temporary chaotic breakdown before regeneration. The cycle repeats stochastically and is synchronized by elastic waves (Jha et al., 2020).

Cross-slot flow supplies a different spatial archetype. In two-dimensional Oldroyd-B simulations, the extensional zone near the stagnation point produces first a steady symmetry-breaking bifurcation for sufficiently concentrated solutions, then a supercritical Hopf bifurcation, and at larger $Re = \frac{\rho U L}{\eta},$8 an irregular state with intermittent switching of global asymmetry and localized regions of strong polymer stretching (Canossi et al., 2021). In porous media, elastic turbulence is heterogeneous at the pore scale: different pores become unstable at different interstitial Weissenberg numbers, so the global state is a patchwork of unstable and laminar regions over an extended range of forcing (Browne et al., 2020).

Taken together, these results indicate that elastic turbulence frequently occupies stress-rich subregions—boundary layers, streak interfaces, stagnation-point neighborhoods, or selected pores—rather than filling space uniformly. This suggests that wall effects and local constitutive loading are often central to the developed state, not only to its onset.

4. Spectral and statistical signatures

A hallmark of elastic turbulence is a steep, broad-band spectrum. In the shell model of polymer solution, once elasticity and concentration are sufficiently large, the kinetic energy spectrum follows

$Re = \frac{\rho U L}{\eta},$9

while the polymer field is concentrated at large scales and does not display a clear power law (Ray et al., 2016). This steep spectrum is consistent with the classical picture of smooth velocity fields with intense stress dynamics rather than an inertial $Wi = \lambda \dot{\gamma},$0 cascade.

Wall-bounded continuum simulations show comparable behavior but with geometry-dependent exponents. In two-dimensional Taylor–Couette flow, the temporal power spectrum of velocity fluctuations decays as a power law with exponent larger than two, and the exponent depends strongly on radial position (Buel et al., 2018). In three-dimensional von Kármán swirling flow, the maximum measured exponents are $Wi = \lambda \dot{\gamma},$1 in time and $Wi = \lambda \dot{\gamma},$2 in space, both close to experimental values (Buel et al., 2022). In a straight pressure-driven channel, the centerline streamwise velocity spectrum scales as

$Wi = \lambda \dot{\gamma},$3

while the temporal spectrum of centerline velocity fluctuations scales as

$Wi = \lambda \dot{\gamma},$4

again indicating steep but not universal decay (Lellep et al., 2023).

Cross-slot simulations emphasize temporal statistics. At high elasticity the outlet-velocity spectrum follows

$Wi = \lambda \dot{\gamma},$5

with $Wi = \lambda \dot{\gamma},$6 between about $Wi = \lambda \dot{\gamma},$7 and $Wi = \lambda \dot{\gamma},$8; the velocity-fluctuation PDFs remain close to Gaussian, whereas local accelerations exhibit pronounced non-Gaussian tails (Canossi et al., 2021). This combination—weakly non-Gaussian velocity increments but strongly intermittent acceleration statistics—appears repeatedly in experimental and numerical work.

Recent homogeneous-isotropic DNS extend the same diagnostics to high global Reynolds number. At scales larger than the Kolmogorov length, polymeric turbulence remains Newtonian-like, with $Wi = \lambda \dot{\gamma},$9; at scales smaller than the Kolmogorov length, the flow enters an elastic–dissipation range with

$Wi=\lambda\Omega$0

and with three-point structure functions, kurtosis, dissipation correlations, and velocity-gradient PDFs matching those of inertialess elastic turbulence (Garg et al., 22 Jul 2025). This demonstrates that elastic turbulence is not confined to globally low-$Wi=\lambda\Omega$1 systems: it can persist as a small-scale regime whenever local inertia becomes weak while local elasticity remains strong.

5. Flow resistance, control, and transport consequences

Elastic turbulence usually increases drag or flow resistance. In the shear-modulated Taylor–Couette problem, van Buel and Stark defined a dimensionless flow resistance from the polymeric shear stress at the outer cylinder and showed that steady driving above the instability threshold yields irregular stress and enhanced resistance, whereas sufficiently fast shear-rate modulation suppresses the turbulence and recovers a laminar response described by a linear Maxwell model (Buel et al., 2019). In the $Wi=\lambda\Omega$2 plane, the laminar–turbulent boundary is approximately

$Wi=\lambda\Omega$3

for square-wave modulation and

$Wi=\lambda\Omega$4

for sinusoidal modulation (Buel et al., 2019). This provides an explicit control law: fast modulation at large $Wi=\lambda\Omega$5 can suppress elastic turbulence even when the mean $Wi=\lambda\Omega$6 is well above the steady-flow threshold.

In porous media, direct visualization linked macroscopic resistance growth to pore-scale elastic turbulence. The apparent viscosity rises sharply above an interstitial threshold and reaches about $Wi=\lambda\Omega$7; porewise instability thresholds lie in the range $Wi=\lambda\Omega$8 to $Wi=\lambda\Omega$9, and the corresponding Pakdel–McKinley thresholds are $Wi=\lambda U_b/d$0 to $Wi=\lambda U_b/d$1 (Browne et al., 2020). The fraction of time that a pore remains unstable scales near onset as

$Wi=\lambda U_b/d$2

consistent with a directed-percolation interpretation of the transition (Browne et al., 2020). The same study showed that the extra dissipation generated by the fluctuating strain field quantitatively accounts for the anomalous pressure drop through the whole medium.

Energetics are also explicit in the three-dimensional von Kármán problem. There, flow resistance was defined as the total work performed on the fluid, and it jumps sharply at the transition; the main increase arises from elastic work at the lateral boundary rather than from viscous work at the rotating plate (Buel et al., 2022). Across geometries, the transport consequence is consistent: elastic turbulence enhances mixing and can enhance heat transfer, but it often does so by paying an elastic dissipation cost that appears macroscopically as additional resistance (Buel et al., 2019, Browne et al., 2020).

6. Universality, modeling limits, and unresolved questions

Several common misconceptions are now untenable. Elastic turbulence does not require inertia; it also does not require streamline curvature in any strict sense. Geometry-free shell models display elastic-turbulent spectra and positive Lyapunov exponents once elasticity and concentration are large enough (Ray et al., 2016). Straight channels exhibit subcritical elastic turbulence organized around localized coherent structures (Lellep et al., 2023), and inertialess plane Couette flow can become linearly unstable once stress diffusion is included (Beneitez et al., 2022). Highly entangled polymer and wormlike-micelle models show elastic turbulence in planar Couette and planar Poiseuille flows, even for monotonic constitutive curves in which the homogeneous base state is stable in one dimension (Lewy et al., 8 Jun 2025). Conversely, the phenomenon is not always homogeneous, and it is not confined to globally low Reynolds numbers, as shown by the elastic–dissipation range in high-$Wi=\lambda U_b/d$3 polymeric turbulence (Garg et al., 22 Jul 2025).

Theoretical quantification of complexity remains difficult. For the two-dimensional Oldroyd-B model, an attractor-based estimate of the Lyapunov dimension was derived under a numerically supported boundedness assumption, giving

$Wi=\lambda U_b/d$4

up to logarithmic factors, and implying a smallest dynamically relevant length scale that shrinks as $Wi=\lambda U_b/d$5 (Plan et al., 2017). This connects the growth of elastic degrees of freedom to polymer-stress gradients rather than to inertial cascades. At the same time, experiments in bounded geometries show boundary-layer effects and nonuniform stress distributions that are not captured by homogeneous theories; the curvilinear-channel work explicitly found that the predicted saturation of normalized rms velocity gradients is contradicted by experiment (Jun et al., 2011).

Model dependence is another unresolved issue. Oldroyd-B captures many signatures of elastic turbulence but assumes infinite polymer extensibility; more realistic FENE-P, Giesekus, Johnson–Segalman, Rolie–Poly, and sPTT descriptions can change thresholds, stress localization, and bifurcation type (Beneitez et al., 2022, Lewy et al., 8 Jun 2025, Lellep et al., 2023). The polymeric diffusive instability in plane Couette flow is especially sensitive to how diffusion is modeled: a simple diffusive conformation law yields a strong linear instability, whereas a more detailed two-fluid model can suppress it for realistic mobility parameters (Beneitez et al., 2022). This suggests that onset mechanisms are not only geometry-dependent but also constitutively non-universal.

A plausible synthesis is that elastic turbulence is best understood as a family of low-inertia chaotic states generated by nonlinear stress transport and relaxation, with geometry selecting where the active region forms and constitutive physics selecting how it destabilizes. The currently open problems—boundary-layer theory in confined flows, fully three-dimensional continuation from coherent states, constitutive robustness, and the coupling between local elastic and global inertial regimes—are therefore not peripheral details but central to a general theory of elastic turbulence (Hou et al., 15 Oct 2025, Garg et al., 22 Jul 2025).