Fig. 1: Approach towards a discontinuous transition.

a, Reynolds number dependence of the TF in a curved (helical) pipe in experiments. Curves show the TFs measured at different downstream locations. The error bars indicate the standard deviation. The TF continues to adjust with downstream location and has not reached the statistical steady state at the end of the pipe (24,000R). The adjustment from laminar (TF = 0) to fully turbulent (TF = 1) flow occurs across a decreasing Re interval. b, An initially fully turbulent flow gradually laminarizes. The flow structure is visualized by adding small amounts of mica flakes (50 μm in size), turbulent regions show up in dark grey, whereas laminar ones appear bright. Turbulent transients persist over the first 8,000 pipe radii, until eventually the flow fully relaminarizes. c, TF as a function of Re, in a vertically heated pipe in direct numerical simulations. The error bars indicate the standard deviation. Here the TF initially adjusts and reaches a statistically steady state after approximately 10,000 advective time units. TFs below 0.8 cannot be sustained and the transition is, hence, discontinuous.