Completing Moody's friction diagram in the turbulent transitional regime

The Moody diagram, a plot of friction factor versus flow rate, is a well-known engineering tool for estimating head loss in pipe flows. It comprises well-defined relationships between friction factor and flow rate over the majority of parameter space, but there is a gap in the transitional regime between laminar and turbulent flows. It is often left hatched because in this parameter range the friction is deemed indefinite, which Moody remarked could at least partially be due to the different initial conditions used to establish the flow. Here we investigate this issue and seek a systematic dependence for friction in the transitional regime. The novel method we use is to approach the transitional regime from above by reducing the flow speed from a turbulent flow state. We find that in different pipe flow setups, both driven by gravity, a single curve corresponding to a maximum density of the transitional flow structures is found. We test the generality of this result using an alternative method to drive the flow through the pipe, using a mass displacement device. Our investigation of the flow driven by a syringe produces yet a different curve, indicating that the method of driving the flow has a significant impact on both the final states and the paths leading to them in the transitional regime.