An extension of the momentum transfer model to time-dependent pipe turbulence

TL;DR

This paper extends the momentum transfer model for steady turbulent pipe flow to account for time-dependent and spatially varying turbulence, incorporating flow acceleration history and spatial derivatives into the wall stress expression.

Contribution

It introduces a novel extension of the momentum transfer model that includes dynamic and spatial effects in turbulent pipe flow analysis.

Findings

Derived an expression for wall stress with steady and dynamic components.

Incorporated flow acceleration history into the stress model.

Included spatial derivatives of mean flow in the model.

Abstract

We analyze a possible extension of Gioia and Chakraborty's momentum transfer model of friction in steady turbulent pipe flows (Phys. Rev. Lett. 96, 044502 (2006)) to the case of time and/or space dependent turbulent flows. The end result is an expression for the stress at the wall as the sum of an steady and a dynamic component. The steady part is obtained by using the instantaneous velocity in the expression for the stress at the wall of a stationary flow. The unsteady part is a weighted average over the history of the flow acceleration, with a weighting function similar to that proposed by Vardy and Brown (Journal of Sound and Vibration 259, 1011 (2003); ibid. 270, 233 (2004)), but naturally including the effect of spatial derivatives of the mean flow, as in the Brunone model (B. Brunone et al., J. of Water Resources Planning and Management 236 (2000)).