Exact coherent states with hairpin-like vortex structure in channel flow

TL;DR

This paper introduces the first nonlinear traveling wave solutions to the Navier-Stokes equations in channel flow that exhibit hairpin-like vortex structures, providing insight into wall turbulence dynamics.

Contribution

It presents the discovery of exact coherent states with hairpin vortices in channel flow, a novel finding in the study of turbulence structures.

Findings

Solutions emerge at Re=666 via saddle-node bifurcation.

Hairpin vortex structures develop with increasing Reynolds number.

Upper branch solutions at Re=1800 resemble turbulent mean profiles.

Abstract

Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear traveling wave solutions to the Navier--Stokes equations in the channel flow geometry -- exact coherent states (ECS) -- that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number Re=666. At the bifurcation, the solution has a highly symmetric quasistreamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented "head" near the channel centerplane where the mean shear vanishes connected to counter-rotating quasistreamwise "legs" that extend toward the channel wall. At Re=1800, the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.