Source-linked AI summary
The rise of fully turbulent flow
Dwight Barkley, Baofang Song, Vasudevan Mukund, Grégoire Lemoult, Marc Avila, Björn Hof
TL;DR
A prior model did not reflect the observed continuous behaviour at the onset of fully turbulent flow. The paper introduces an extended model with nonlinear advection that captures the sequence leading to fully turbulent flow.
Problem
A prior model did not reflect the observed continuous behaviour at the onset of fully turbulent flow.
Source: §4 · p. 4Method
The authors formulate a two-component advection-reaction-diffusion model of turbulent fluctuations and centreline axial velocity.
Source: §2.1 · pp. 15–16Results
The extended model incorporating advective nonlinearity captures the sequence encountered on the route to fully turbulent flow.
Source: §4 · p. 4Takeaways & Limitations
The model provides an account of the sequence leading to fully turbulent flow.
Source: §4 · p. 4The earlier model omitted nonlinear advection, an intrinsic feature of fluid dynamics, and did not reflect the observed continuous behaviour at fully turbulent onset.
Source: §4 · p. 4
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.
6 June, 2015 · 1 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom · 4 Institute of Fluid Mechanics, Friedrich-Alexander-Universit¨at, Erlangen, Germany
These parts examine the route from localized to fully turbulent pipe flow using experiments and an advective nonlinear model, focusing on front dynamics, stochastic fluctuations, and intermittent laminar patches. They also describe how profile deformation can eliminate the upper turbulent fixed point, relaminarizing the flow and reducing frictional drag.
4 Institute of Fluid Mechanics, Friedrich-Alexander-Universit¨at, Erlangen, Germany
- Pipe and duct turbulence begins expanding at R ≳2250 and R ≳2030, respectively, with downstream fronts accelerating; the spreading shows neither square-root nor percolation-type scaling.Initially, upstream and downstream fronts travel at indistinguishable speeds, indicating localized turbulence.Source: Figure 1 · p. 3, §4 · p. 4
- An extended model incorporating nonlinear advection captures the route to fully turbulent flow, unlike an earlier model that predicted a discontinuous transition.The earlier model omitted nonlinear advection, intrinsic to fluid dynamics.Source: §4 · p. 4
- The model predicts localized, asymmetric expanding, and symmetric expanding turbulent states, with the localized state arising when the downstream front follows the upstream front at a fixed distance.At higher r, a second stable equilibrium makes the system bistable; the downstream front then lags, growing the turbulent region.Source: §4 · pp. 5, 7
- Bistability begins before expansion: the downstream front initially remains slower than the upstream front, and expansion starts only when the localized-state and weak-front speed curves intersect.At larger r, the weak front becomes unstable and gives way to a rapidly expanding strong front.Source: §4 · p. 7
- Pipe and duct front-speed data agree with the model’s localized-flow parabolic scaling, while the model captures two curvature changes as fronts adjust from weak to strong.The strong-front limit predicts a parabolic relation between downstream and upstream front speeds.Source: §4 · pp. 7–8
- Stochastic fluctuations make downstream fronts alternate between weak and strong, so the predicted bifurcation appears only in averages and turbulent flow can contain intermittent laminar patches.Source: §4 · p. 8
- The model's simplicity enables investigation of fully turbulent flow, and explaining its origin is a decisive step toward connecting transitional and high-Re turbulence.The paper presents this as a bottom-up approach to turbulence.Source: §4 · pp. 8, 10
- Deforming the pipe-flow profile can eliminate the upper turbulent fixed point, relaminarize turbulence, and provide a simple, robust way to reduce frictional drag.Source: §4 · p. 8
- R = 2000 turbulence remains localized, R = 4500 expands with a strong downstream front into fully turbulent flow, and R = 2800 fluctuates between weak and strong front branches.At R = 2000 the upstream and downstream front speeds are equal; at R = 4500 spreading is symmetric in the neutral reference frame.Source: Figure 3 · p. 9, §4 · p. 9
1 Materials and methods
The study measured laminar–turbulent front speeds with detailed experiments and highly resolved simulations, using long observation periods to average out stochastic fluctuations.
1 Materials and methods
- Front speeds were measured in detailed experiments and highly resolved simulations, with long observation times used to average stochastic turbulence fluctuations.The fluctuations could disguise the underlying transition.Source: §1 · p. 11
1.2 Duct experiments
- Duct perturbation evolution was monitored via pressure recorded at five sensors positioned 100h, 400h, 600h, 800h and 1000h downstream of the perturbation point, with measurements over at least 50 realisations.Perturbations were injected 120h downstream from the inlet and lasted 5h/U in dimensionless units.Source: §1.2 · p. 13
2 Theory
The theory models turbulent fluctuations and centreline velocity with coupled advection–reaction–diffusion dynamics and uses asymptotic analysis to characterize front propagation between laminar and turbulent branches. It examines how downstream fronts govern puff localization and speed changes, and applies a shared model with flow-specific parameters to collapse and fit pipe and duct data.
2.1 Model details
- The model couples turbulent fluctuations q and centreline axial velocity u through advection–reaction–diffusion equations, with scaled Reynolds parameter r and diffusion representing turbulence-triggered excitation of adjacent laminar flow.Source: §2.1 · pp. 15–17
2.2 Asymptotic analysis
- The laminar q0 and upper q+ branches are stable, and for small ϵ the dynamics separate into slow branch-following regions and fast transitions between them.Source: §2.2 · p. 17
- The model’s neutral speed is 2 − ζ: upstream and strong downstream fronts are symmetric around it, the turbulence advection speed absent laminar–turbulent front dynamics.Source: §2.2 · p. 19
- Without nonlinear advection, the weak-front branch has a distinct critical point and the transition to expanding turbulence is first-order (discontinuous).The analysis identifies nonlinear advection as important to the bifurcation scenario.Source: §2.2 · p. 19
- For generic uf, a heteroclinic connection selects a unique front speed, but at q+ = q− infinitely many connections and speeds occur; criticality is where the upper steady state reaches the q-nullcline limit point.Source: §2.2 · p. 21
- At uf=uc, infinitely many downstream-front connections allow downstream speed to match upstream speed, so puffs remain localized while traveling along the pipe.This infinite range occurs when q−=q+ and the upper fixed point is not hyperbolic.Source: §2.2 · pp. 21–22
- Above the critical r, allowed downstream-front speeds become finite; nonlinear advection masks the resulting discontinuity because downstream speeds are below upstream speeds.Without nonlinear advection, the change would appear as a discontinuous change in downstream-front speed.Source: §2.2 · pp. 21–22
2.3 Collapsing pipe and duct data
- The collapse aligns upstream and strong downstream fronts across pipe and duct data, but the weak-front branch does not collapse.The aligned values are pipe (R0=1920, C0=1.06; R1=2250, C1=0.92) and duct (R0=1490, C0=1.12; R1=2030, C1=0.90).Source: §2.3 · pp. 23–24
2.4 Determining model parameters
- With D=0.13 alone, the model fits upstream and strong downstream front speeds for both flows and fixes a simple relationship between model and experimental data.The model uses flow-specific R0, R1, C0, and C1 values to map model Reynolds number and speed to measured quantities.Source: §2.4 · pp. 25–26
- ϵ controls transition sharpness between weak and strong branches, while ζ controls delay; larger ζ delays pipe transition, and the parameters were chosen by eye.The fitting-parameter values differ between pipe and duct flows.Source: §2.4 · p. 26
3 Control
The model predicts that blunting the shear profile can eliminate the turbulent fixed point, and direct numerical simulation shows that forcing destabilizes fully turbulent pipe flow into localized turbulent patches.
3 Control
- The model’s D=0.13 choice agrees well with pipe and duct data, while smaller epsilon values make the transition between weak and strong branches more abrupt.Source: §3 · p. 27
- The global body force was gradually switched on at time 175 and fully applied by time 200; turbulent intensity subsequently decreases, and fully turbulent flow destabilizes into localized patches like natural puffs.Source: §3 · p. 28
- Additive forcing of the shear-profile variable blunts its shape and can remove the upper turbulent fixed point, eliminating fully turbulent flow.In the model, this control corresponds to blunting the shear profile toward a more plug-like form.Source: §3 · p. 28