On a complete analytical solution of transient friction in pipe flow

TL;DR

This work addresses transient friction in circular-pipe flow under a monotonically increasing mean-pressure gradient by treating friction as the sum of an underlying laminar flow (ULF) and a purely turbulent component (PTC) that evolve non-synchronously. It develops and applies the frozen-turbulence analytical model (FTAM) within the Theory of Underlying Laminar Flow (TULF) to derive an exact, modal, analytic description of the mean velocity field and skin-friction evolution, including the canonical bathtub phenomenon and its dependence on laminar and turbulent time scales. The study reveals that the transient friction behavior hinges on the timing of turbulence development (parameters such as $ au_0$, $ au_2$, and $ au_c$), the formation and persistence of a hyperlaminar sublayer (HSL), and the deformation and recovery of the log-layer, providing causal explanations for observed DNS-like friction evolutions and offering actionable predictions for experiments and flow-control strategies. The results have practical implications for designing transient pumping, cleaning, and flow-control schemes to achieve desired friction characteristics, and they underscore the importance of accounting for turbulence timing rather than just its intensity in unsteady pipe flows.

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.

Figures (8)

• Figure 1: Idealised schematic view of the 'canonical' bathtub.
• Figure 2: $Re(\tau)$, $|\sigma_w(\tau)|$, $C_f(Re)$ and $C_f(\tau)$ evolution for laminar U-flows TE1-TE6.
• Figure 3: Evolution of $C_f$, $C_{f_L}$ and $|\sigma_w|$ for the PFT U-flow ($\Delta \tau \mathop{=}0.0015$, $\mathring{\tau}_b\mathop{=}0.005458$).
• Figure 4: $C_f$ and $|\sigma_w|$ evolution for the FTAM. $Re$ is also shown in the 'canonical' bathtubs TEg & TEh.
• Figure 5: Time evolution of mean velocity and turbulence index fields for the FTAM. The hyperlaminar sublayer (HSL) is the space-time domain where $\mathfrak{I}(\tau,\alpha)>1$.