[Paper Review] Multiple states of turbulence at vanishing inertia

Ziyin Lu, Björn Hof|arXiv (Cornell University)|Feb 26, 2026

Rheology and Fluid Dynamics Studies0 citations

TL;DR

The paper demonstrates two distinct elastic turbulence states in viscoelastic pipe flows at vanishing inertia: a center-mode driven state and a hoop-stress wall-mode state, whose interplay and curvature dependence reveal two separate turbulent states that persist to zero inertia.

ABSTRACT

Based on everyday experience fluid flows tend to be ordered and quiescent if inertial forces are low and held in check by viscosity. This intuition spectacularly fails in the case of complex macromolecular fluids like polymer melts, paints and biofluids. In such cases elastic fluid properties can drive turbulent motions at moderate and even vanishing Reynolds numbers. By studying viscoelastic flows in curved pipes we demonstrate that this low inertia phenomenology results from the competition of two hydrodynamic instabilities and respectively from the co-existence and interdependence of two distinct turbulent states. Unexpectedly the established categories of elastic and elasto-inertial turbulence (ET and EIT) fail to demarcate the actual turbulent states, fundamentally changing the perception of this phenomenon a century after its discovery.

Motivation & Objective

• Motivate understanding of turbulence in viscoelastic fluids where elasticity dominates over inertia.
• Identify and characterize distinct turbulent states arising at low Re and finite Wi in curved pipes.
• Investigate how streamline curvature and elasticity control onset thresholds and state selection.
• Clarify how ET and EIT classifications relate to actual dynamical states in curved versus straight geometries.

Proposed method

• Conduct viscoelastic flow experiments in straight and curved pipes using PAAM in glycerol solutions.
• Vary Reynolds number, Weissenberg number, and curvature to map instabilities.
• Use pressure fluctuation measurements and PIV to diagnose onset and flow structures.
• Apply the Pakdel–McKinley criterion to collapse onset data for hoop-stress instabilities.
• Compare flow-field structures and power spectra to distinguish center-mode and hoop-stress turbulence.

Figure 1: Onset of multiple turbulent states and transformation from a center to a wall mode. (a) Normalized pressure fluctuation levels as a function of Reynolds number used to determine the onset. The fluctuation levels are calculated by $(\sigma_{p}-\sigma_{p}^{lam})/\sigma_{p}^{lam}$ , where $\s

Experimental results

Research questions

• RQ1 Do multiple turbulent states exist at vanishing inertia in viscoelastic pipe flows?
• RQ2 What instabilities (center mode vs hoop-stress) drive these states and how do they depend on curvature?
• RQ3 How does curvature influence the threshold and nature of each instability?
• RQ4 How are elastic turbulence (ET) and elasto-inertial turbulence (EIT) related when inertia is very small?
• RQ5 What are the structural and spectral signatures that differentiate the two turbulent states?

Key findings

• Two distinct turbulent states are observed: a center-mode instability near the pipe center and a hoop-stress-driven wall-mode with strong near-wall fluctuations.
• The center mode threshold is curvature-insensitive, while the hoop-stress instability requires curvature and becomes primary at higher curvature, approaching zero inertia.
• At high elasticity numbers, the center mode threshold can drop to very small Re, enabling hoop-stress turbulence even in straight pipes as a secondary instability.
• Hoop-stress turbulence exhibits wall-localized, elongated streaks and power spectra with slope ≲ -3, consistent with purely elastic turbulence, while center-mode turbulence shows shallower spectra (~ -2).
• The center mode can curve streamlines, enabling the hoop-stress mode to arise in straight pipes, resolving prior discrepancies between EIT and ET in rectilinear geometries.
• The study argues that ET and EIT do not partition turbulence cleanly; two distinct states can exist at vanishing inertia under the same control parameter (Wi).

Figure 2: Multiple turbulent states and the approach to the inertialess regime. (a) Onset of the center and hoop stress modes as a function of curvature ratio for 85% glycerol-200 ppm PAAM in 4 mm pipe ( $\mathit{E}\approx 50$ ) and 80% glycerol-50 ppm PAAM in 1.6 mm pipe ( $\mathit{E}\approx 80$ ).