Subcritical versus supercritical transition to turbulence in curved pipes

Abstract

Transition to turbulence in straight pipes occurs in spite of the linear stability of the laminar Hagen-Poiseuille flow if the amplitude of flow perturbations as well as the Reynolds number exceed a minimum threshold (subcritical transition). As the pipe curvature increases centrifugal effects become important, modifying the basic flow as well as the most unstable linear modes. If the curvature (tube-to-coiling diameter d/D) is sufficiently large a Hopf bifurcation (supercritical instability) is encountered before turbulence can be excited (subcritical instability). We trace the instability thresholds in the Re − d/D parameter space in the range 0.01 d/D 0.1 by means of laser-Doppler velocimetry and determine the point where the subcritical and supercritical instabilities meet. Two different experimental setups were used: a closed system where the pipe forms an axisymmetric torus and an open system employing a helical pipe. Implications for the measurement of friction factors in curved pipes are discussed. †

Modal instability of the flow in a toroidal pipe

“…Moreover, they observed an hysteresis in the value of the Reynolds number separating the steady and periodic regimes. This observation is inconsistent with both the experimental results of Kühnen et al (2015), and the numerical data presented here.…”

“…All curvatures in the grid revealed spectra with unstable eigenvalues for Re < 6 × 10 3 . This result confirms the numerical observations by Di Piazza & Ciofalo (2011) as well as the experiments by Kühnen et al (2014Kühnen et al ( , 2015 and extends these studies, which are limited to the low curvature range. Most importantly, it proves that this flow is linearly unstable, in obvious contrast to the flow inside a straight pipe (see Schmid & Henningson (1994) and citations therein).…”

“…The eigenvalues belonging to this family can be observed in figure 3(a), where they form the branches just below the critical ones. This family was the one found by Kühnen et al (2015), in the range 0.026 δ 0.1. It appears, from this experimental investigation, that F4S constitutes the instability boundary not only for toroidal pipes, but for low-pitch helical pipes as well.…”

“…Two values of curvature were investigated (0.1 and 0.3) and both showed a transition from stationary to periodic, quasi-periodic and then chaotic flow. However, as indicated by Kühnen et al (2015) and as will be shown herein, these results should be considered inaccurate, with the exception of the symmetry characteristics observed in the flow. The only experiments employing toroidal pipes are those by Kühnen et al (2014Kühnen et al ( , 2015.…”

“…However, as indicated by Kühnen et al (2015) and as will be shown herein, these results should be considered inaccurate, with the exception of the symmetry characteristics observed in the flow. The only experiments employing toroidal pipes are those by Kühnen et al (2014Kühnen et al ( , 2015. These authors sacrificed the 2π (streamwise) periodicity and the mirror symmetry of the system by introducing a steel sphere in the tube to drive the fluid.…”