On a complete analytical solution of transient friction in pipe flow

TL;DR

This theoretical study analyzes transient friction in pipe flow, revealing how laminar and turbulent components interact asynchronously, leading to phenomena like overshoots and variable friction levels, using the TULF framework.

Contribution

It provides an analytical explanation for transient friction phenomena in pipe flow using the TULF, including effects of asynchrony and turbulence evolution.

Findings

Transient friction can exceed or be lower than steady-state values.

The TULF accurately predicts transient skin-friction and velocity overshoots.

Turbulence development destroys the logarithmic layer during transient.

Abstract

The present research is a theoretical study about the transient friction created in circular pipe mean flow, whenever an incompressible Newtonian fluid is accelerated through a monotonously-increased mean-pressure gradient. The resulting friction stress is the sum of two components, one laminar and the other purely turbulent, not synchronised between them. Each component is analysed separately, in a series of theoretical experiments that explore various possibilities, depending on the degree of asynchrony between them. It is found that in some cases the transient friction is higher than in equal-Re steady-sate flow, but in some others it is noticeably lower. This work provides an analytical explanation for most of the important and interesting phenomena reported in the literature. To do so, it takes advantage of the Theory of Underlying Laminar Flow (TULF), already introduced in previous works of same authors. The TULF predicts quite approximately what is observed in experiments, including the transient skin-friction coefficient and the presence of mean-velocity overshoots. Additionally, the role of the time constant in turbulent mean flow is examined and related to the turbulence's frozen time. Finally, a study of the logarithmic layer evolution in the transient flow is accomplished, which results destroyed during the increase of turbulence occurring along the transient. In summary, the present work unveils new knowledge about transient friction in unsteady flows.