The collapse of turbulence, observable in shear flows at low Reynolds numbers, raises the question if turbulence is generically of a transient nature or becomes sustained at some critical point. Recent data have led to conflicting views with the majority of studies supporting the model of turbulence turning into an attracting state. Here we present lifetime measurements of turbulence in pipe flow spanning 8 orders of magnitude in time, drastically extending all previous investigations. We show that no critical point exists in this regime and that in contrast to the prevailing view the turbulent state remains transient. To our knowledge this is the first observation of superexponential transients in turbulence, confirming a conjecture derived from low-dimensional systems. [4][5][6]. Surprisingly, at relatively low Reynolds numbers (Re & 2000) the turbulent state is not stable and after long times suddenly collapses [7][8][9][10][11][12]. This behavior is reminiscent of memoryless processes in nonlinear systems. In phase space the dynamics can be described by a complex structure giving rise to the disordered dynamics, a socalled chaotic repeller [13]. Underlying such a structure are unstable states and for pipe flow unstable solutions to the governing equations have been identified in the form of traveling waves [14,15]. Surprisingly clear transients of such traveling waves were observed in experiments [16,17] confirming their relevance to the turbulent dynamics. More recently traveling wave transients were also reported in numerical studies [18,19].A way to probe the validity of this model is to measure the lifetime of turbulence in the transient regime. Previous experimental and numerical lifetime measurements have shown approximately exponential probability distributions [8,10,11,20,21] which suggests that the probability for a turbulent structure to decay is independent of its age and hence that this process is memoryless as would be expected for the escape from a chaotic saddle. Here the probability for a flow to still be turbulent after a time t at a fixed Reynolds number (Re) is then given bywhere is the characteristic lifetime ( À1 can be also interpreted as the escape rate) and t 0 is the initial time period required for turbulence to form after the disturbance has been applied to the laminar flow at t ¼ 0. an infinite lifetime is only reached in the asymptotic limit Re ! 1. Subsequently a number of studies have questioned this finding and again entertained the occurrence of a boundary crisis [11,22,23]. A clear constraint of all previous investigations is the limited range in lifetimes measured. Typically scaling laws were postulated from data covering 2 orders of magnitude. Numerical simulations are particularly problematic because in order to capture the quantitatively correct behavior computations have to be carried out in large domains, which severely restricts the number of realizations N that are manageable (N < 50) [11]. Consequently the statistics are often insufficiently resolved resulting in a...

The transitional regime of a sinusoidal pulsatile flow in a straight, rigid pipe is investigated using particle image velocimetry. The main aim is to investigate how the critical Reynolds number is affected by different pulsatile conditions, expressed as the Womersley number and the oscillatory Reynolds number. The transition occurs in the region of Re ¼ 2250-3000 and is characterized by an increasing number of isolated turbulence structures. Based on velocity fields and flow visualizations, these structures can be identified as puffs, similar to those observed in steady flow transition. Measurements at different Womersley numbers yield similar transition behavior, indicating that pulsatile effects do not play a role in the regime that is investigated. Variations of the oscillatory Reynolds number also appear to have little effect, so that the transition here seems to be determined only by the mean Reynolds number. For larger mean Reynolds numbers, a second regime is observed: here, the flow remains turbulent throughout the cycle. The turbulence intensity varies during the cycle, but has a phase shift with respect to the mean flow component. This is caused by a growth of kinetic energy during the decelerating part and a decay during the accelerating part of the cycle. Flow visualization experiments reveal that the flow develops localized turbulence at several random axial positions. The structures quickly grow to fill the entire pipe in the decelerating phase and (partially) decay during the accelerating phase.

Transition to turbulence in a pipe is characterized by the increase of the characteristic lifetimes of localized turbulent spots ('puffs') with increasing Reynolds number (Re). Previous experiments are based on visualization or indirect measurements of the lifetime probability. Here we report quantitative direct measurements of the lifetimes based on accurate pressure measurements combined with laser Doppler anemometry (LDA). The characteristic lifetime is determined directly from the lifetime probability. It is shown that the characteristic lifetime does not diverge at finite Re, and follows an exponential scaling for the observed range 1725 6 Re 6 1955. Over this small Re range the lifetime increases over four orders of magnitude. The results show that the puff velocity is not constant, and the rapid disintegration of puffs occurs within 20-70 pipe diameters.

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