Global energy budgets in turbulent Couette and Poiseuille flows

TL;DR

This paper compares turbulent Couette and Poiseuille flows using energy budgets, revealing that Couette flows are less efficient but more effective at producing flow rate due to their laminar component and large-scale structures.

Contribution

It extends the constant power input framework to compare different turbulent flows and introduces analytical energy-flux identities for flow efficiency and effectiveness.

Findings

Couette flows are less efficient but more effective than Poiseuille flows.

Couette flows develop stronger large-scale structures that influence mean flow and dissipation.

When fed with the same power, Couette flows dissipate less turbulent energy.

Abstract

Turbulent plane Poiseuille and Couette flows share the same geometry, but produce their flow rate owing to different external drivers, pressure gradient and shear respectively. By looking at integral energy fluxes, we pose and answer the question of which flow performs better at creating flow rate. We define a flow {\em efficiency}, that quantifies the fraction of power used to produce flow rate instead of being wasted as a turbulent overhead; {\em effectiveness}, instead, describes the amount of flow rate produced by a given power. The work by Gatti \emph{et al.} (\emph{J. Fluid Mech.} vol.857, 2018, pp. 345--373), where the constant power input (CPI) concept was developed to compare turbulent Poiseuille flows with drag reduction, is here extended to compare different flows. By decomposing the mean velocity field into a laminar contribution and a deviation, analytical expressions are derived which are the energy-flux equivalents of the FIK identity. These concepts are applied to literature data supplemented by a new set of direct numerical simulations, to find that Couette flows are less efficient but more effective than Poiseuille ones. The reason is traced to the more effective laminar component of Couette flows, which compensates for their higher turbulent activity. It is also observed that, when the fluctuating fields of the two flows are fed with the same total power fraction, Couette flows dissipate a smaller percentage of it via turbulent dissipation. A decomposition of the fluctuating field into large and small scales explains this feature: Couette flows develop stronger large-scale structures, which alter the mean flow while contributing less significantly to dissipation.