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

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.

Figure 1

(a) Sketch of the general pipe setup. Four different pipes were used, two of them with a 10 mm diameter ($L/D=690$ and 600) and two with a 4 mm diameter ($L/D=2000$ and 3600). Pipes were gravity driven and turbulence could be induced by injection and withdrawal of fluid through small holes in the pipe wall at various downstream positions. Turbulence was detected by monitoring the outflow angle and by LDA velocity measurements. (b) LDA velocity trace obtained at the center line of the $D=10\mathrm{mm}$ ($L/D=690$) pipe during the passage of a turbulent event. The trace shows the well-known [25] signature of a turbulent puff for the axial velocity (top) and the radial velocity (bottom).

Figure 2

Probabilities for the flow to still be turbulent after traveling a fixed distance $x$. Viewed from left to right the five data sets shown correspond to the following distances: $x=140$ (down triangles), 270 (diamonds), 930 (up triangles), 1900 (circles), 3500 (squares). The fitted curves follow directly from the superexponetial scaling shown in Fig. 3. The dotted curves show the scaling that would be expected for the critical behavior suggested by Willis and Kerswell [11].

Figure 3

(a) Decay rates plotted on a log linear scale. Circles represent data obtained in the $D=10\mathrm{mm}$ pipes while the full symbols were measured in the $D=4\mathrm{mm}$ pipe. The dashed line and the open squares are a reproduction of the data points and the linear fit given by Willis and Kerswell [11]. The black line has the same slope as the exponential scaling observed by Hof et al. [12]. The light solid curve assumes a superexponential dependence of the decay rate on Re. The dash-dotted line shows an alternative superexponential fit (see text for details). (b) Data on a log-log linear scale. The data could be fitted by a straight line over the entire regime. This two parameter fit was then used to plot the five curves in Fig. 2 as well as the light solid curve in Fig. 3a.