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

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 26 000. Taylor’s frozen turbulence hypothesis is used to convert temporal-spanwise information into a 2-D spatial spectrum which shows the contribution of streamwise ( [STIX]x1D706ₓ ) and spanwise ( [STIX]x1D706y ) 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 [STIX]x1D706y/z~ ([STIX]x1D706ₓ/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 ( [STIX]x1D706y~ [STIX]x1D706ₓ ), as required for self-similarity and predicted by the attached eddy hypothesis.