Completing Moody's friction diagram in the turbulent transitional regime

TL;DR

This paper investigates the transitional flow regime in pipe flows to systematically characterize friction factors, revealing that flow driving methods significantly influence the transitional behavior and filling the gap in Moody's diagram.

Contribution

It introduces a novel approach to analyze the transitional regime by gradually reducing flow speed from turbulence and compares different flow driving methods to understand their effects.

Findings

A maximum density of transitional flow structures is identified.

Flow driving method significantly affects transitional flow states.

Flow reduction from turbulence reveals a continuous curve in the transitional regime.

Abstract

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 to them in the transitional regime.