# Characteristic scales of Townsend's wall attached eddies

## TL;DR

This paper introduces new characteristic scales for wall-attached eddies in turbulence, supported by DNS simulations, challenging traditional assumptions about the necessity of wall boundary conditions for turbulence structure.

## Contribution

The study proposes a new scaling framework for wall-attached eddies based on mean momentum flux and shear, supported by numerical simulations that alter boundary conditions.

## Key findings

- New velocity, length, and time scales for eddies are consistent with DNS results.
- Outer-layer turbulence statistics are similar with or without wall boundary conditions.
- Wall boundary conditions are not essential for reproducing turbulence structures far from the wall.

## Abstract

Townsend (1976) proposed a structural model for the logarithmic layer (log-layer) of wall turbulence at high Reynolds numbers, where the dominant momentum-carrying motions are organised into a multi-scale population of eddies attached to the wall. In the attached eddy framework, the relevant length and velocity scales of the wall-attached eddies are the friction velocity and the distance to the wall. In the present work, we hypothesise that the momentum-carrying eddies are controlled by the mean momentum flux and mean shear with no explicit reference to the distance to the wall and propose new characteristic velocity, length, and time scales consistent with this argument. Our hypothesis is supported by direct numerical simulation (DNS) of turbulent channel flows driven by non-uniform body forces and modified mean velocity profiles, where the resulting outer-layer flow structures are substantially altered to accommodate the new mean momentum transfer. The proposed scaling is further corroborated by simulations where the no-slip wall is replaced by a Robin boundary condition for the three velocity components, allowing for substantial wall-normal transpiration at all lengths scales. We show that the outer-layer one-point statistics and spectra of this channel with transpiration agree quantitatively with those of its wall-bounded counterpart. The results reveal that the wall-parallel no-slip condition is not required to recover classic wall-bounded turbulence far from the wall and, more importantly, neither is the impermeability condition at the wall.