# Two-dimensional energy spectra in high Reynolds number turbulent boundary layers

## TL;DR

This study measures two-dimensional energy spectra of streamwise velocity in high Reynolds number turbulent boundary layers, revealing a transition from square-root to linear relationship between spanwise and streamwise scales, supporting the attached eddy hypothesis.

## Contribution

It provides novel high Reynolds number measurements of 2-D spectra and demonstrates the scale relationship transition predicted by turbulence theories.

## Key findings

- At low Reynolds numbers, spectra follow a square-root scale relationship.
- At high Reynolds numbers, the relationship becomes linear, indicating self-similarity.
- Results support the attached eddy hypothesis at high Reynolds numbers.

## Abstract

Here we report the measurements of two-dimensional (2-D) spectra of the streamwise velocity ($u$) in a high Reynolds number turbulent boundary layer. A novel experiment employing multiple hot-wire probes was carried out at friction Reynolds numbers ranging from 2400 to 26000. Taylor's frozen turbulence hypothesis is used to convert temporal-spanwise information into a 2-D spatial spectrum which shows the contribution of streamwise ($\lambda_x$) and spanwise ($\lambda_y$) length scales to the streamwise variance at a given wall height ($z$). At low Reynolds numbers, the shape of the 2-D spectra at a constant energy level shows $\lambda_y/z \sim (\lambda_x/z)^{1/2}$ behaviour at larger scales, which is in agreement with the existing literature at a matched Reynolds number obtained from direct numerical simulations. However, at high Reynolds numbers, it is observed that the square-root relationship tends towards a linear relationship ($\lambda_y \sim \lambda_x$) as required for self-similarity and predicted by the attached eddy hypothesis.