[Paper Review] Universality at the onset of turbulence in shear flows

Alberto de Lózar, Bjoern Hof|arXiv (Cornell University)|Jan 14, 2010

Fluid Dynamics and Turbulent Flows22 references3 citations

TL;DR

This study demonstrates universal critical behavior at the onset of turbulence in pipe, duct, and channel flows, revealing a phase transition from localized to globally expanding turbulence at a critical Reynolds number. Using lifetime and front velocity measurements, the authors identify a universal critical exponent α = 0.5, confirming a 20-year-old conjecture linking shear flow turbulence to directed percolation in statistical physics.

ABSTRACT

The volatile transition from quiescent laminar to strongly fluctuating turbulent dynamics in shear flows remains only poorly understood despite its practical importance and more than a century of intense research. The theoretical understanding of the transition process has been complicated by the lack of a linear instability mechanism and additionally by a catastrophic collapse of turbulence which can occur after extremely long lifetimes. Turbulence close to onset is investigated experimentally in three different geometries: pipe, duct and channel flow. The reverse transition from turbulent to laminar flow is observed to be a general feature of shear flows. A critical point is uncovered at slightly higher flow rates, where the nature of the flow changes abruptly to the generic case of expanding turbulence. The critical exponent associated with this phase transition is found to be universal. This confirms a conjecture made over 20 years ago based on an analogy between fluid turbulence and discrete models in statistical physics.

Motivation & Objective

• To investigate the nature of turbulence onset in shear flows across different geometries.
• To determine whether localized turbulent spots exhibit universal scaling behavior near a critical point.
• To test the conjecture that turbulence onset in shear flows resembles phase transitions in discrete statistical physics models.
• To measure the transition from transient to globally expanding turbulence and identify its critical parameters.

Proposed method

• Lifetime distributions of turbulent spots were measured via controlled perturbations in pipe, duct, and channel flows, with survival rates tracked downstream.
• Exponential decay models were fitted to lifetime data to extract decay rates τ⁻¹, revealing super-exponential scaling with Reynolds number.
• Front and rear velocities of turbulent spots were measured using pressure drop traces over a 250 L/D section to determine expansion dynamics.
• Critical Reynolds numbers (Rec) were determined by fitting interface speeds to ε^α scaling, where ε = (Re - Rec)/Rec.
• Data collapse across geometries confirmed universality of the critical exponent.
• High-precision Reynolds number control (<0.2%) ensured reliable lifetime and velocity measurements.

Experimental results

Research questions

• RQ1 Is the transition from localized to globally expanding turbulence in shear flows universal across different flow geometries?
• RQ2 What is the critical Reynolds number at which the nature of turbulence changes abruptly in pipe, duct, and channel flows?
• RQ3 Does the scaling of turbulent front velocities with Reynolds number near the critical point follow a universal power law, and if so, what is the exponent?
• RQ4 Can the observed critical behavior be linked to known universality classes in statistical physics, such as directed percolation?

Key findings

• The critical Reynolds number for the transition to globally expanding turbulence is Rec = 2550 in pipes, 1480 in channels, and 2250 in ducts, indicating a universal phase transition.
• The critical exponent for interface speed scaling is universally α = 0.5, consistent with predictions from directed percolation universality class.
• Turbulent spot lifetimes scale super-exponentially with Reynolds number, approaching infinity only asymptotically, confirming transient turbulence.
• The lower cutoff for self-sustained turbulence is Re₀ = 1526 in pipes, 1085 in channels, and 1221 in ducts, below which lifetimes are negligible.
• The transition from localized to expanding turbulence is marked by a linear increase in front velocity with ε = (Re - Rec)/Rec, confirming a second-order phase transition.
• The critical behavior is robust across three distinct shear flow geometries, confirming universality in the onset of turbulence.