Subcritical versus supercritical transition to turbulence in curved pipes

Transition to turbulence in straight pipes occurs in spite of the linear stability of the laminar Hagen–Poiseuille flow if both the amplitude of flow perturbations and the Reynolds number Re 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 set-ups are 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.