Folded Edge of Turbulence in a Pipe

The results of an experimental investigation into the threshold boundary between laminar and disordered pipe flow are presented. Complex features have been uncovered using a highly refined experimental approach where an intermediate periodic state forms an integral part of the transition sequence. In accord with the suggestions produced by a numerical investigation, the boundary is found to be folded with a complicated structure. This raises important questions about accepted definitions of threshold amplitudes in this long-standing problem.

Figure 1

Images of waves formed at $\mathrm{Re}=1900$ when Poiseuille flow is disturbed by a small jet. (a) Dye visualization with ${\Phi }_{\mathrm{inj}}/{\Phi }_{\mathrm{pipe}}=0.17\%$. The scale is indicated in units of distance $D$ from the injection point. Image taken at 1.3 sec after the injection. (b) Closeup of breakdown at $\sim 1.6\sec$. (c) A “timeline” evolution of a set of waves towards a puff. The location of the vertical video line was fixed at $6D$. (Arrow indicates approximate location of breakdown.)

Figure 2

Probability of observation of a puff as a function of distance $x/D$ from the disturbance injection point at $\mathrm{Re}=1900$. Measurements were made every $15D$ from $17D$ after the injection point up to $137D$ and further observations were made at $250D$ and $565D$, the end of the pipe. Each data point was obtained from 24 runs of the experiment.

Figure 3

Probability of observation of waves or puffs or slugs as a function of ${\Phi }_{\mathrm{inj}}/{\Phi }_{\mathrm{pipe}}$ at (a) $\mathrm{Re}=1900$ and (b) $\mathrm{Re}=2200$ (c) $\mathrm{Re}=3000$. The vertical strip indicates the threshold for the appearance of waves (Fig. 1). Observations made at $250D$ and $565D$.

Figure 4

Nonmonotonic probability of inducing turbulence in the simulations plotted as a function of perturbation amplitude $A_0$. Probabilities are estimated from relative frequencies in overlapping “sliding” bins (blue curve) and in nonoverlapping statistically independent bins (boxes). (a) Data for a periodically continued domain of length $L=5D$ at $\mathrm{Re}=3878\pm 8$ (raw data from [11]). (b) Data for a longer domain with $L=15D$ at $\mathrm{Re}=1,900$ and perturbations of different spatial structure (raw data from [19]).