Subcritical versus supercritical transition to turbulence in curved pipes

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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.