On an analytical explanation of the phenomena observed in accelerated turbulent pipe flow

Abstract

This research presents a new theory that explains analytically the behaviour of any fully developed incompressible turbulent pipe flow, steady or unsteady. We propose the name of theory of underlying laminar flow (TULF), because its main consequence is the description of any turbulent pipe flow as the sum of two components: the underlying laminar flow (ULF) and the purely turbulent component (PTC). We use the framework of the TULF to explain analytically most of the important and interesting phenomena reported in He & Jackson (J. Fluid Mech., vol. 408, 2000, pp. 1–38). To do so, we develop a simple model for the pressure gradient and Reynolds shear stress that could be applied to the linearly accelerated pipe flow described by He & Jackson (2000). The following features of the unsteady flow are explained: the deformation undergone by the mean velocity profiles during the transient, the velocity overshoot observed in the more rapid excursions, the dual deformation of mean velocity profiles when overshoots are present, the laminarisation effects described during acceleration, the rapid decrease of the Reynolds shear stress upon approaching the wall that brings forth the laminar sublayer, and some other minor effects. A new field is defined to characterise the degree of turbulence within the flow, directly calculable from the theory itself. Arguably, this new field would describe the degree of turbulence in a pipe flow more accurately than the familiar turbulence intensity parameter. Finally, a paradox is found in the deformation of the unsteady mean velocity profiles with respect to equal-Reynolds-number steady profiles, which is duly explained. The research also predicts the occurrence of mean velocity undershoots if the flow is decreased rapidly enough.