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 temporalspanwise information into a 2-D spatial spectrum which shows the contribution of streamwise (λ x ) and spanwise (λ 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 λ y /z ∼ (λ 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 (λ y ∼ λ x ) as required for self-similarity and predicted by the attached eddy hypothesis.

An assessment of self-similarity in the inertial sublayer is presented by considering the wall-normal velocity, in addition to the streamwise velocity component. The novelty of the current work lies in the inclusion of the second velocity component, made possible by carefully conducted subminiature ×-probe experiments to minimise the errors in measuring the wall-normal velocity. We show that not all turbulent stress quantities approach the self-similar asymptotic state at an equal rate as the Reynolds number is increased, with the Reynolds shear stress approaching faster than the streamwise normal stress. These trends are explained by the contributions from attached eddies. Furthermore, the Reynolds shear stress cospectra, through its scaling with the distance from the wall, are used to assess the wall-normal limits where self-similarity applies within the wall-bounded flow. The results are found to be consistent with the recent prediction from the work of Wei et al. ["Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows," J. Fluid Mech. 522, 303-327 (2005)], Klewicki ["Reynolds number dependence, scaling, and dynamics of turbulent boundary layers," J. Fluids Eng. 132, 094001 (2010)], and others that the self-similar region starts and ends at z + ∼ O(√ δ +) and O(δ +), respectively. Below the self-similar region, empirical evidence suggests that eddies responsible for turbulent stresses begin to exhibit distance-from-the-wall scaling at a fixed z + location; however, they are distorted by viscous forces, which remain a leading order contribution in the mean momentum balance in the region z + O(√ δ +), and thus result in a departure from self-similarity.

A combination of cross-wire probes with an array of flush-mounted skin-friction sensors are used to study the three-dimensional conditional organisation of large-scale structures in a high-Reynolds-number turbulent boundary layer. Previous studies have documented the amplitude modulation of small-scale motions in response to conditionally averaged large-scale events, but the data are largely restricted to the streamwise component of velocity alone. Here, we report results based on all three components of velocity and find that the small-scale spanwise and wall-normal fluctuations (v and w) and the instantaneous Reynolds shear stress (−uw) are modulated in a very similar manner to that previously noted for the streamwise fluctuations (u). The envelope of the small scale fluctuations for all velocity components is well described by the large-scale component of the u fluctuation. These results also confirm the conditional existence of roll modes associated with the very large-scale or 'superstructure' motions.

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