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 ( $\unicode[STIX]{x1D706}_{x}$ ) and spanwise ( $\unicode[STIX]{x1D706}_{y}$ ) 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 $\unicode[STIX]{x1D706}_{y}/z\sim (\unicode[STIX]{x1D706}_{x}/z)^{1/2}$ 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 ( $\unicode[STIX]{x1D706}_{y}\sim \unicode[STIX]{x1D706}_{x}$ ), as required for self-similarity and predicted by the attached eddy hypothesis.