Revisiting a Reynolds experiment: head loss in non-fully developed, unsteady flow

Head loss plays a central role in fluid flows. The head loss in fully developed, steady flows is relatively clear, and its calculation has matured and been applied widely in practical problems with success. However, there is a lack of understanding and effective computational methods to estimate head loss in non-fully developed, unsteady flows. This paper revisits a Reynolds experiment, which, unlike the original experiment by Reynolds in 1883, adopts unsteady flows (non-constant water heads) exhibiting negative inertia heads. It discusses their head loss in a tube section where the flows have not fully developed. It shows that the head loss is higher in unsteady flows than in steady flows, and the difference is about two magnitudes of the inertia head. Also, it illustrates that the Darcy-Weisbach equation cannot replicate the measurement data for the head loss in both steady and unsteady, non-fully developed flows. Moreover, the paper presents expressions for head loss in unsteady flows as functions of steady-flow head loss and inertia head.