Repeller or Attractor? Selecting the Dynamical Model for the Onset of Turbulence in Pipe Flow

Abstract

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...

Statistics of Intermittent Switching between Potential Flow and Turbulence around an Oscillating Sphere in Superfluid 4He below 0.5 K

“…When the finite time of experimental observation is shorter than the lifetime a turbulent phase, turbulence only appears to be stable. An analogous situation has recently been observed in classical pipe flow where turbulence is shown to have lifetimes that grow very rapidly with the Reynolds number without diverging at some critical value [10].…”

Amplitude-Phase Synchronization at the Onset of Permanent Spatiotemporal Chaos

“…Experimental evidence of chaotic saddles has been obtained in lasers [2] and pipe flows [3]. Numerical simulations of turbulent shear flows have observed a turbulent flow behaving as a chaotic saddle coexisting with a laminar flow separated by an edge of chaos [4].…”

Patterned Turbulence in Liquid Metal Flow: Computational Reconstruction of the Hartmann Experiment

“…There is a clear analogy between our high-Ha turbulent spots and turbulent puffs in the HD pipe flow [22][23][24][25][26], except for some difference in the typical length (about 40 in the HD pipe). The term 'puff' is used in the following.…”

“…We see the main feature of the transitional regimes -localized turbulent spots with laminar flow between them. Localized turbulent spots are well known from HD wall-bounded shear flows [22][23][24][25][26][27] and can be considered in the broader context of patterned turbulence. The best known example is pipe flow, where the spots are known since [9].…”

“…Two types of spots have been identified [22][23][24][25][26]: turbulent puffs with nearly constant length and speed existing in low-Re flows with strong perturbations and turbulent slugs at high Re characterized by aggressive length growth leading to a completely turbulent state. A discussion of their fascinating features, including vorticity dynamics, their nature as a chaotic saddle and the random character and sensitivity to initial conditions, can be found in [22][23][24][25][26].…”