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The rise of fully turbulent flow
Dwight Barkley, Baofang Song, Vasudevan Mukund, Grégoire Lemoult, Marc Avila, Björn Hof
TL;DR
The paper addresses the unexplained route from localized transitional states to fully turbulent flow and combines experiments, theory, and simulations to resolve it. An extended model incorporates nonlinear advection and interprets the flow as bistable, reproducing the sequence of turbulent front states and their dynamics. The resulting scenario connects turbulence onset with fully turbulent pipe flow.
Problem
The origin, front dynamics, and transformation of transitional turbulent states into fully turbulent flow remained unexplained.
Method
The study combines experiments, theory, and computer simulations with an extended model incorporating nonlinear advection.
Results
The model captures the route to fully turbulent flow and explains the dynamics of localized, weak-front, and strong-front states.
Takeaways & Limitations
The findings bridge understanding of turbulence onset and fully turbulent flows through a bistable system with nonlinear turbulent-front propagation.
Takeaways & Limitations
The earlier model failed to reflect the observed continuous onset because it omitted nonlinear advection.
Abstract
from arXiv · showhide
Over a century of research into the origin of turbulence in wallbounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow. At slightly higher speeds the situation changes distinctly and the entire flow is turbulent. Neither the origin of the different states encountered during transition, nor their front dynamics, let alone the transformation to full turbulence could be explained to date. Combining experiments, theory and computer simulations here we uncover the bifurcation scenario organising the route to fully turbulent pipe flow and explain the front dynamics of the different states encountered in the process. Key to resolving this problem is the interpretation of the flow as a bistable system with nonlinear propagation (advection) of turbulent fronts. These findings bridge the gap between our understanding of the onset of turbulence and fully turbulent flows.
2 Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
The supplied passage identifies the Institute of Science and Technology Austria in Klosterneuburg as an institutional affiliation.
- The affiliation is the Institute of Science and Technology Austria.The institution is listed with its location and postal code.
- The institute is located in Klosterneuburg, Austria.
4 Institute of Fluid Mechanics, Friedrich-Alexander-Universit¨at, Erlangen, Germany
A bistable model with nonlinear advection organizes the transition from localized turbulence to fully turbulent pipe flow. Its predicted front states and speed changes agree with experiments and simulations across pipe and duct flows.
- Model and bifurcation scenario: The extended model incorporates advective nonlinearity and captures the sequence of states encountered on the route to fully turbulent flow.It predicts localized, asymmetric expanding, and symmetric expanding turbulent structures.
- Model and bifurcation scenario: At low control parameter values, turbulent excitations remain localized because downstream and upstream fronts maintain a fixed separation.This produces a localized puff-like excitation.
- Model and bifurcation scenario: Bistability appears before sustained expansion: a weak downstream front initially lags the upstream front, then becomes unstable and gives way to a rapidly expanding strong front.The onset of bistability and expansion therefore do not coincide.
- Experimental and numerical validation: As expansion develops, front speeds adjust continuously from weak to strong states, with two curvature changes and eventual strong-front symmetry about the neutral speed.The strong-front asymptotics produce a parabolic relation between downstream and upstream speeds.
- Experimental and numerical validation: R ≈2250 in pipe flow and R ≈2030 in duct flow mark the onset of expansion through formation of weak downstream fronts.The corresponding front-speed data collapse and agree well with the model's parabolic asymptotic scaling.
- Experimental and numerical validation: At R = 4500 turbulence expands through a strong downstream front and the long-time flow becomes fully turbulent, whereas R = 2000 remains localized.At R = 2800 the downstream front has intermediate characteristics and intermittent laminar patches occur within turbulent flow.
1 Materials and methods
Experiments in pipes and square ducts measured laminar–turbulent front speeds, while numerical simulations resolved turbulent-front evolution across Reynolds numbers. The study combined controlled perturbations, long observation distances, averaging, and a spectral–finite-difference pipe-flow solver.
- Experiments: 0.5% was the accuracy achieved for Reynolds-number control in the pipe experiments, while flow rates stayed within 0.5% during measurements.Increasing turbulent drag caused less than 0.5% of the overall pressure drop, limiting changes in flow rate during a measurement.
- Numerical simulations: The simulations solved incompressible pipe flow with no-slip walls and axial periodicity using a toroidal–poloidal spectral-finite-difference formulation.The velocity field was represented through toroidal and poloidal potentials, with Fourier expansions in axial and azimuthal directions and finite differences radially.
- Numerical simulations: Simulations used Reynolds-number-dependent pipe lengths and resolutions, with initial conditions prepared near R = 2000 and shorter domains for low-Reynolds-number cases.At low R ≳ 2000, puffs remain approximately 20D long because puff-splitting is extremely unlikely.
- Numerical simulations: Front positions were identified with a local-intensity cutoff of 5 × 10^-4, and the resulting front speed was insensitive to the cutoff value tested.Long turbulent regions were required before speed statistics became length-independent: L0 > 60D for R < 4000 and L0 > 100D for R ≥4000.
2 Theory
The theory models pipe-flow transition as a bistable, two-component advection–reaction–diffusion system coupling turbulent fluctuations with axial velocity. Its asymptotic front analysis identifies a critical bifurcation that distinguishes localized puffs from expanding turbulence and reproduces collapsed pipe and duct front-speed data.
- Model formulation: The model couples turbulent-fluctuation level q and centreline axial velocity u through local nonlinear reactions, advection, and diffusion.q represents turbulent fluctuations, while u represents axial velocity; r is a scaled Reynolds number.
- Model formulation: The laminar state is the fixed point (u = 2, q = 0), while the laminar and upper turbulent branches are stable.The local ODEs describe the interaction between q and u before spatial derivatives are included.
- Front speeds: Nonlinear advection sets the neutral turbulence speed to 2 − ζ rather than the maximum centreline velocity.This speed is the symmetry point between upstream and strong downstream fronts and represents turbulence advection without front dynamics.
- Bifurcation scenario: Without nonlinear advection, the transition to expanding turbulence is first-order; including it produces the observed bifurcation scenario.Removing first-derivative advection terms gives the weak-front branch a distinct critical point and a discontinuous transition.
- Bifurcation scenario: At the critical point, the upper fixed point reaches q+ = q−, changing downstream-front solutions from infinitely many possible speeds to a finite set.For r below critical, the infinite range permits downstream and upstream speeds to match, keeping puffs localized; above critical, the upper point becomes hyperbolic and the connection is unique.
- Comparison with data: With D = 0.13, the model fits upstream and strong downstream front speeds for both pipe and duct flow, while the weak-front branch does not collapse.The collapsed strong branches match experimental data extremely well using one fitted parameter.
3 Control
The model predicts that fully turbulent flow can be destabilized by removing its turbulent fixed point, and simulations confirm that blunting the velocity profile leaves localized turbulent patches.
- Model control: Smaller ϵ values produce more abrupt transitions between weak and strong branches in the modeled front structure.
- Model control: Removing the turbulent fixed point eliminates the fully turbulent state in the model.An additive forcing term blunts the shear profile and can remove the upper turbulent fixed point.
- Direct numerical simulation: A global body force blunts the velocity profile toward a plug-like form and destabilizes fully turbulent pipe flow at R = 5000.The simulation initially begins in a fully turbulent state before the forcing is applied.
- Direct numerical simulation: The destabilized flow degenerates into localized turbulent patches resembling natural puffs found below ∼2300 without additional forcing.Turbulent intensity decreases before the fully turbulent state breaks down.