Coherent structures in the linearized impulse response of turbulent channel flow

TL;DR

This paper investigates how isolated impulses in turbulent channel flow generate coherent vortex-streak structures that grow self-similarly and are consistent with the attached eddy hypothesis, using linearized Navier-Stokes equations with eddy viscosity.

Contribution

It introduces a linearized model with eddy viscosity to analyze the formation and evolution of coherent structures in turbulent flow due to impulsive forces.

Findings

Impulses produce long streaks flanked by vortices, some forming hairpin structures.

Structures grow self-similarly with an aspect ratio of about 10.

Reynolds stresses scale with wall distance, supporting the attached eddy hypothesis.

Abstract

We study the evolution of velocity fluctuations due to an isolated spatio-temporal impulse using the linearized Navier-Stokes equations. The impulse is introduced as an external body force in incompressible channel flow at $Re_\tau=10000$. Velocity fluctuations are defined about the turbulent mean velocity profile. A turbulent eddy viscosity is added to the equations to fix the mean velocity as an exact solution, which also serves to model the dissipative effects of the background turbulence on large-scale fluctuations. An impulsive body force produces flowfields that evolve into coherent structures containing long streamwise velocity streaks that are flanked by quasi-streamwise vortices; some of these impulses produce hairpin vortices. As these vortex-streak structures evolve, they grow in size to be nominally self-similar geometrically with an aspect ratio (streamwise to wall-normal) of approximately $10$, while their kinetic energy density decays monotonically. The topology of the vortex-streak structures is not sensitive to the location of the impulse, but is dependent on the direction of the impulsive body force. All of these vortex-streak structures are attached to the wall, and their Reynolds stresses collapse when scaled by distance from the wall, consistent with Townsend's attached eddy hypothesis.