Ecological collapse and the emergence of travelling waves at the onset of shear turbulence

Abstract

The mechanisms and universality class underlying the remarkable phenomena at the transition to turbulence remain a puzzle 130 years after their discovery . The statistical behaviour is thought to be related to directed percolation (DP; refs 6,11-13). Attempts to understand transitional turbulence dynamically invoke periodic orbits and streamwise vortices [14][15][16][17][18][19] , the dynamics of long-lived chaotic transients 20, and model equations based on analogies to excitable media 21. Here we report direct numerical simulations of transitional pipe flow, showing that a zonal flow emerges at large scales, activated by anisotropic turbulent fluctuations; in turn, the zonal flow suppresses the small-scale turbulence leading to stochastic predator-prey dynamics. We show that this ecological model of transitional turbulence, which is asymptotically equivalent to DP at the transition 22, reproduces the lifetime statistics and phenomenology of pipe flow experiments. Our work demonstrates that a fluid on the edge of turbulence exhibits the same transitional scaling behaviour as a predator-prey ecosystem on the edge of extinction, and establishes a precise connection with the DP universality class.Turbulent fluids are ubiquitous in nature, arising for sufficiently large characteristic speeds U , depending on the kinematic viscosity ν and a characteristic system scale, such as the diameter of a pipe D. Turbulent flows are complex, stochastic, and unpredictable in detail, but transition at lower velocities to a laminar flow, which is simple, deterministic and predictable. This transition is controlled by the dimensionless parameter known as the Reynolds number, which in the pipe geometry of interest here is given by Re ≡ UD/ν, and occurs in the range 1,700 Re 2,300. The laminar-turbulence transition has presented a challenge to experiment and theory since Osborne Reynolds' original observation of intermittent 'flashes' of turbulence 1 . To explore this transitional regime, we have performed direct numerical simulations of the Navier-Stokes equations in a pipe of length L = 10D, using the open-source code 'Open Pipe Flow' 23 , as described in Supplementary Methods. The Reynolds number at which transitional turbulence occurs is higher for short pipes 23 , and the simulations reported here for L = 10D were performed at a nominal value Re = 2,600, which we estimate to be equivalent to Re 2,200 in long pipe data 7 based on estimates of when puff decay transitions to puff splitting. We confirmed that our results did not qualitatively change for a longer pipe with L = 20D. We denote the time-dependent velocity deviation from the Hagen-Poiseuille flow by u = (u z , u θ , u r ). Because we were interested in transitional behaviour, we looked for a decomposition 2,6,24,25 of large-scale modes that would indicate some form of collective behaviour, as well as small-scale modes that would be representative of turbulent dynamics. In particular, we report here the behaviour of the velocity field (u z , u θ , u r ), where the bar de...

Theoretical perspective on the route to turbulence in a pipe

“…Goldenfeld et al (2010) have shown that a double-exponential scaling of lifetimes naturally results from the assumption that extreme events drive the collapse of puffs. Similar reasoning presumably applies to puff splitting as well (Barkley 2011a;Shih et al 2016).…”

Transition to turbulence in wall-bounded flows: Where do we stand?

“…In the same vein, Goldenfeld and collaborators (Shih et al, 2015) have proposed an interpretation of 'activator-inhibitor' reactions as species interactions of 'prey-predator' type in ecology and constructed a corresponding (quasi-)1D probabilistic automaton with rules adapted from beforehand numerical simulations of pipe flow in a very short tube. Appropriate choice of probabilities led to general agreement with experimental findings and equivalence with 1D DP was demonstrated without surprise in view of the way the model was constructed, still somewhat far from a derivation from first principles.…”

“…The nonlinear terms in (Barkley, 2011) or (Barkley et al, 2015), as well as the rules in (Shih et al, 2015), were chosen according to phenomenological considerations. In (Manneville, 2012b), nonlinearities were borrowed from (Waleffe, 1997), where they were derived from the NS equation within a MFU assumption via appropriate truncation of a Galerkin expansion.…”

Temporal acceleration of a turbulent channel flow

“…Mathur and others the transient nature of turbulence (Hof et al 2006) and criteria for the onset of sustained turbulence (Avila et al 2011;Shi, Avila & Hof 2013;Barkley et al 2015). The nature of transition is better understood by linking it to a more general non-equilibrium stochastic process, the so-called directed percolation phase transition (Pomeau 1986;Lemoult et al 2016;Sano & Tamai 2016;Shih, Hsieh & Goldenfeld 2016).…”