Finite lifetime of turbulence in shear flows

Abstract

Generally, the motion of fluids is smooth and laminar at low speeds but becomes highly disordered and turbulent as the velocity increases. The transition from laminar to turbulent flow can involve a sequence of instabilities in which the system realizes progressively more complicated states, or it can occur suddenly. Once the transition has taken place, it is generally assumed that, under steady conditions, the turbulent state will persist indefinitely. The flow of a fluid down a straight pipe provides a ubiquitous example of a shear flow undergoing a sudden transition from laminar to turbulent motion. Extensive calculations and experimental studies have shown that, at relatively low flow rates, turbulence in pipes is transient, and is characterized by an exponential distribution of lifetimes. They also suggest that for Reynolds numbers exceeding a critical value the lifetime diverges (that is, becomes infinitely large), marking a change from transient to persistent turbulence. Here we present experimental data and numerical calculations covering more than two decades of lifetimes, showing that the lifetime does not in fact diverge but rather increases exponentially with the Reynolds number. This implies that turbulence in pipes is only a transient event (contrary to the commonly accepted view), and that the turbulent and laminar states remain dynamically connected, suggesting avenues for turbulence control.

Dynamical systems and the transition to turbulence in linearly stable shear flows

“…Plane Couette flow is the historically first example, where lifetimes have been studied (Bottin & Chaté 1998;Bottin et al 1998b) and coherent structures have been identified (Nagata 1990(Nagata , 1996Clever & Busse 1992, 1997Eckhardt et al 2002). Similar investigations of pipe flow came later: lifetimes are studied in the papers by , Hof et al (2005Hof et al ( , 2006, Mullin & Peixinho (2006a,b) and Peixinho & Mullin (2006), and coherent states in the papers by Faisst & Eckhardt (2003), Hof et al (2004), and Schneider et al (2007b). The status of pipe flow has been reviewed by Kerswell (2005) and Eckhardt et al (2007) and the related studies of other flows may be traced from Mullin & Kerswell (2004).…”

“…In particular, it can well happen that some of the initial conditions in this regime decay anytime after the maximal integration time, which here was typically limited to t max Z 3000 dimensionless units. While earlier studies suggested that there indeed was a Reynolds number beyond which no trajectories would ever decay (Bottin & Chaté 1998), more recent studies give evidence that a decay will always be possible (see Hof et al (2006) and below).…”

“…An important feature of the probabilities in figure 2 is their curvature and S-shape: as was emphasized by Hof et al (2006), a divergence of the lifetimes at a finite Re would imply that these curves reach a maximal value at a Reynolds number that is independent of the observation time. However, all curves bend towards increasing Re, suggesting that there is always a small but non-negligible fraction of initial conditions which decays.…”

A doubly localized equilibrium solution of plane Couette flow

“…Schneider et al (2010b) found the first known localized solutions, a pair of spanwise-localized, streamwise-periodic equilibrium and traveling-wave solutions of plane Couette flow, further investigated in Schneider et al (2010a). Avila et al (2013) found a streamwise-localized relative periodic orbit of pipe flow that closely resembles the transient turbulent puffs of Hof et al (2006). Deguchi et al (2013) and Gibson & Brand (2014) independently found spanwise-localized forms of the periodic EQ7/HVS solution of Itano & Generalis (2009);Gibson et al (2009).…”

Extraction of 3D vortex structures from a turbulent puff in a pipe using two-color illumination and flakes

“…More recent statistical studies on the creation and advection of a localized turbulence called turbulent puff (Wignanski and Champagne 1973) in controlled pipe flow facilities have provided a physical insight into this phenomenon: Peixinho and Mullin 2006 mentioned that the lifetimes of puffs, and the persistence times of the advections, increase exponentially with increasing Re. Meanwhile, Hof et al 2006 indicated that the variation of lifetime is super exponential: It means that a turbulent puff is fundamentally transient and has a finite lifetime, even at high values of Re. An increasing Re also accompanies the splitting of a puff toward formation of a turbulent slug (Wignanski and Champagne 1973), where the localized turbulence then grows into persistent turbulence.…”