Subcritical versus supercritical transition to turbulence in curved pipes

TL;DR

This study investigates how pipe curvature influences the transition to turbulence, revealing a shift from subcritical to supercritical instability modes as curvature increases, with experimental validation.

Contribution

It identifies the conditions under which subcritical and supercritical transitions occur in curved pipes and maps their thresholds in the Reynolds number and curvature parameter space.

Findings

Transition thresholds depend on pipe curvature and flow conditions.

Supercritical bifurcation occurs before turbulence at high curvature.

Experimental setups confirm the coexistence and meeting point of instabilities.

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 $0.01\leq\ d/D \leq0.1$ 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.