Asymptotic limits of the attached eddy model derived from an adiabatic atmosphere

TL;DR

This study examines the asymptotic behavior of the attached-eddy model in high Reynolds number atmospheric turbulence, confirming some theoretical predictions and identifying Reynolds number dependencies in key coefficients through extensive field measurements.

Contribution

It provides empirical evidence for the asymptotic limits of the attached-eddy model coefficients in atmospheric surface layers at very high Reynolds numbers.

Findings

The Townsend-Perry coefficient converges between 1 and 1.25 at high Reynolds numbers.

Vertical velocity variance remains constant relative to wall-normal distance in the inertial sublayer.

A closer relationship between energy spectrum scaling and velocity variance logarithmic behavior is observed at high Reynolds numbers.

Abstract

The attached-eddy model (AEM) predicts mean velocity and streamwise velocity variance profiles that follow a logarithmic shape in the overlap region of high Reynolds number wall-bounded turbulent flows. Moreover, the AEM coefficients are presumed to attain asymptotically constant values at very high Reynolds numbers. Here, the logarithmic behaviour of the AEM predictions in the near-neutral atmospheric surface layer is examined using sonic anemometer measurements from a 62-m meteorological tower located in the Eastern Snake River Plain, Idaho, US. Utilizing an extensive 210-day dataset, the inertial sublayer (ISL) is first identified by analyzing the measured momentum flux and mean velocity profile. The logarithmic behaviour of the streamwise velocity variance and the associated `-1' scaling of the streamwise velocity energy spectra are then investigated. The findings indicate that the Townsend-Perry coefficient ($A_1$) is influenced by mild non-stationarity that manifests itself as a Reynolds number dependence. After excluding non-stationary runs and requiring a Reynolds number higher than $4 \times 10^7$, the inferred $A_1$ converges to values ranging between 1 and 1.25, consistent with laboratory experiments. Moreover, the independence of the normalized vertical velocity variance from the wall-normal distance in the ISL is further checked and the constant coefficient value agrees with reported laboratory experiments at very high Reynolds numbers as well as many surface layer experiments. Furthermore, nine benchmark cases selected through a restrictive quality control reveal a closer relationship between the `-1' scaling in the streamwise velocity energy spectrum and the logarithmic behaviour of streamwise velocity variance at higher Reynolds numbers, though no direct equivalence between them is observed.