On a complete analytical solution of transient friction in pipe flow

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 synchronized between them. Each component is analyzed separately, in a series of theoretical exercises 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-Reynolds-number steady-state 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 the 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 in turbulence occurring along the transient. In summary, the present work unveils new knowledge about transient friction in unsteady flows.