Fig. 3: Discontinuity, hysteresis and metastability.

a, TF as a function of the reduced Reynolds number ϵ. The critical Reynolds number \({{\rm{Re}}}_{{\rm{c}}}\) is 2,040, 3,716.5, 7,090, 3,746, 6,750 and 2,778.75 for the unforced, curved, MHD, heated, plug and parabolic cases, respectively. The error bars indicate the standard deviation. The TFs for ordinary (unforced) pipe flow (black) are taken from experiments³⁹ and simulations⁴⁰ (via conversion of the reported friction factor data assuming the Blasius friction factor relation). Compared with this unforced case (black), the transition in the presence of the various body forces is sharp. The inset highlights the width of the LTC regimes (that is, the ϵ interval in which 0 < TF < 1). b,c, For parabolic forcing, the transition is hysteretic (b). Full symbols show the TF at which the flow settles down, if initialized with TF = 0.5. Additionally, runs were carried out starting from fully turbulent flow (open symbols), which, in this case, persists even below the critical point. Here turbulence is metastable and after the nucleation of a laminar gap (c), the stable laminar state invades turbulence and the flow fully laminarizes.