Turbulence and secondary motions in square duct flow

TL;DR

This study uses direct numerical simulations to analyze turbulence and secondary motions in square duct flow, revealing their characteristics, minimal impact on mean velocity, and implications for flow prediction models.

Contribution

It provides detailed insights into secondary motions in turbulent square duct flow and demonstrates their limited influence on the overall mean velocity field.

Findings

Secondary motions have an intensity of 1-2% of bulk velocity.

Secondary motions are well approximated by Laplace eigenfunctions at high Reynolds numbers.

Hydraulic diameter-based models effectively predict duct friction coefficient.

Abstract

We study turbulent flows in pressure-driven ducts with square cross-section through direct numerical simulation in a wide enough range of Reynolds number to reach flow conditions which are representative of fully developed turbulence. Numerical simulations are carried out over extremely long integration times to get adequate convergence of the flow statistics, and specifically high-fidelity representation of the secondary motions which arise. The intensity of the latter is found to be in the order of 1-2% of the bulk velocity, and unaffected by Reynolds number variations. The smallness of the mean convection terms in the streamwise vorticity equation points to a simple characterization of the secondary flows, which in the asymptotic high-Re regime are found to be approximated with good accuracy by eigenfunctions of the Laplace operator. Despite their effect of redistributing the wall shear stress along the duct perimeter, we find that secondary motions do not have large influence on the mean velocity field, which can be characterized with good accuracy as that resulting from the concurrent effect of four independent flat walls, each controlling a quarter of the flow domain. As a consequence, we find that parametrizations based on the hydraulic diameter concept, and modifications thereof, are successful in predicting the duct friction coefficient.