Scaling of the turbulence transition in pipe flow

Despite significant experimental and theoretical advances, a complete physical picture for the turbulence transition in pipe flow remains elusive. Here we present a theoretical model for a flow disturbance’s critical amplitude to trigger sustained turbulent puffs downstream as a function of the pipe’s Reynolds number Re. Current theory (Trefethen 1993) suggests that flow disturbances excite a spectrum of pseudo modes, which grow transiently in time before exponential decay. These modes may in turn excite a sustained travelling wave packet in the pipe, marking the transition to turbulence. We find this yields a simple scaling for the critical disturbance amplitude u ~ g/Re, where g plays the role of an amplitude response function for the travelling wave. We computed g for several pipe disturbance methods and successfully collapsed the data from corresponding experimental papers onto our scaling; to date, flow obstacles (Nishi 2008) and pulsed fluid injections (Hof 2003 & 2005, Peixinho 2007). We find the travelling wave manifests with the same frequency in the direction of flow regardless of the disturbance method, evidencing the same pathway to turbulence.

Grant funding provided by Shell and NSF (CMMI-1346638 to A.J.H., CMMI- 1727565 to J.W.M.B.)